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Projection schemes for high-dimensional problems instochastic structural dynamics
by
Lin Gao
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Institute for Aerospace StudiesUniversity of Toronto
c© Copyright 2018 by Lin Gao
Abstract
Projection schemes for high-dimensional problems in stochastic structural dynamics
Lin Gao
Doctor of Philosophy
Graduate Department of Institute for Aerospace Studies
University of Toronto
2018
The focus of the present thesis is to formulate efficient schemes to solve high-dimensional
stochastic ordinary differential equations (SODEs) encountered in stochastic structural dy-
namics. Most of the methods for the parametric uncertainty analysis of SODEs suffer from
the curse of dimensionality. To alleviate it, we investigate a few different methods.
Firstly, we formulate a Generalized Spectral Decomposition (GSD) method for linear
SODEs. It is a stochastic Galerkin method proposed to alleviate the curse of dimensionality
faced by the classical gPC-based stochastic Galerkin projection scheme. Numerical studies
suggest that it is not straightforward to scale the GSD method to large-scale problems since
it is a sequential iterative scheme and there is no clear guidance on how to optimize the
number of expansion terms.
Secondly, we propose an anchored ANOVA Petrov-Galerkin (AAPG) scheme for linear
SODEs. The main idea of the AAPG scheme is to approximate the dynamic response using
a Hoeffding functional ANOVA decomposition along with appropriate constraints to ensure
the uniqueness of the decomposition. We show that when the test functions in the weighted
residual form are chosen appropriately, the original high-dimensional stochastic problem can
be decoupled into a sequence of low-dimensional stochastic subproblems that can be solved
independently of each other. We also extend the AAPG scheme to nonlinear SODEs and
reach the conclusion that the AAPG scheme holds significant potential to alleviate the curse
of dimensionality for linear and nonlinear SODEs, as confirmed by numerical studies.
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Finally, we carry out a theoretical analysis of the AAPG scheme focusing on a priori
error estimation and computational cost. This analysis motivates the formulation of an
adaptive version of the AAPG scheme that enables additional improvements in computa-
tional efficiency by reducing the number of subproblems to be solved. Numerical studies on
test problems with up to 80 stochastic degrees-of-freedom suggest that the adaptive AAPG
can provide a similar level of accuracy as the original scheme while providing a substantial
reduction in computational cost.
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To my wife and parents.
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Acknowledgements
First and foremost, I would like to express my sincere gratitude to my advisor Prof. Prasanth
B. Nair. He has supported me since I started seven years ago with his patience, motivation,
enthusiasm, and immense knowledge. His guidance helped me in all the time of research and
writing of this thesis. I could not have imagined having a better advisor and mentor for my
Ph.D study.
Besides my advisor, I would like to thank the rest of my thesis committee: Prof. Zingg
and Prof. Steeves for their encouragement, insightful comments, and hard questions.
My sincere thanks also goes to Dr. Audouze, who has been very helpful by proofreading
draft material and providing comments and suggestions.
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Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
1.1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research objectives and methodology . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Literature review 6
2.1 Linear stochastic structural dynamics . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Time domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Free vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Nonlinear stochastic structural dynamics . . . . . . . . . . . . . . . . . . . . 10
2.3 Uncertainty modeling and discretization of random fields . . . . . . . . . . . 11
2.4 Simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Variance reduction methods . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Quasi-Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3 Multilevel Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . 19
2.4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Sparse quadrature method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Response surface method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Polynomial chaos expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
v
3 Stochastic Galerkin methods 28
3.1 Generalized polynomial chaos expansion . . . . . . . . . . . . . . . . . . . . 28
3.1.1 gPC-based stochastic Galerkin projection method . . . . . . . . . . . 31
3.1.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Generalized spectral decomposition scheme . . . . . . . . . . . . . . . . . . . 34
3.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Spring-mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Two-dimensional linear beam problem . . . . . . . . . . . . . . . . . 41
3.3.3 Three-dimensional hexahedron problem . . . . . . . . . . . . . . . . . 45
3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Anchored ANOVA Petrov-Galerkin scheme for linear stochastic structural
dynamics 49
4.1 Anchored ANOVA decomposition . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Anchored ANOVA Petrov-Galerkin (AAPG) projection scheme . . . . . . . . 52
4.2.1 Approximating the weighted residual solution of SODE . . . . . . . . 52
4.2.2 Computational and implementation aspects . . . . . . . . . . . . . . 58
4.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Spring-mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Two-dimensional beam problem . . . . . . . . . . . . . . . . . . . . . 62
4.3.3 Three-dimensional hex problem . . . . . . . . . . . . . . . . . . . . . 63
4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Anchored ANOVA Petrov-Galerkin scheme for nonlinear stochastic struc-
tural dynamics 67
5.1 Mathematical derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Application of gPC Galerkin scheme . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Single-dof Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 Application of gPC Galerkin scheme . . . . . . . . . . . . . . . . . . 77
5.3.2 Pseudo-spectral approach . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.3 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Multi-dof Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
vi
6 Theoretical analysis and adaptive AAPG scheme for structural dynamics101
6.1 A priori error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1.1 Background and strategy . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.1.2 Mathematical background and notations . . . . . . . . . . . . . . . . 102
6.1.3 Spectral decomposition error . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.4 Error estimation for gPC approximations of SODEs . . . . . . . . . . 106
6.1.5 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1.6 Discussion on error constants E1 and E2 in Theorem 6.1.1. . . . . . 112
6.1.7 A priori error estimate for the nonlinear SODE system . . . . . . . . 114
6.2 Analysis of the computational cost . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Adaptive AAPG Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3.1 Numerical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7 Concluding remarks and future work 130
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8 Appendix 134
8.1 Non-zero terms in Dijk = 〈φi(ξ)φj(ξ)φk(ξ)〉 . . . . . . . . . . . . . . . . . . . 134
8.2 Examples applying the AAPG scheme to a linear SODE system . . . . . . . 136
8.2.1 Single-dof linear system with stochastic initial conditions . . . . . . . 136
8.2.2 Two-dof linear undamped system with stochastic stiffness coefficients 137
Bibliography 141
vii
List of acronyms
AAPG anchored ANOVA Petrov-Galerkin
AAPG1/AAPG2/AAPG3 first-/second-/third-order AAPG scheme
ANOVA analysis of variance
dof degree of freedom
FEA Finite element analysis
FEM Finite element method
gPC generalized polynomial chaos
gPC1/gPC2 first-/second-order gPC scheme
GSD generalized spectral decomposition
i.i.d. independent and identically distributed
ISD importance sampling density
LHS Latin hypercube sampling
MCS Monte Carlo simulation
MLMC multilevel Monte Carlo
ODE ordinary differential equation
PC polynomial chaos
PDE partial differential equation
pdf probability density function
QMC Quasi-Monte Carlo
RK4 Runge-Kutta fourth-order method
RSM response surface method
SODE stochastic ordinary differential equation
SPDE stochastic partial differential equation
SQ sparse quadrature
SSFEM spectral stochastic finite element method
viii
KL Karhunen-Loeve
UQ uncertainty quantification
ix
Nomenclature
M number of stochastic dof
n number of spatial dof of the system
ξ set of independent random variables, ξ = (ξ1, ξ2, . . . , ξM)T
u(t; ξ) n-dimensional displacement response vector in the stochastic structural dynamics
system
T integration time, T < +∞t temporal coordinate, t ∈ [0, T ]
M(ξ) stochastic mass matrix
C(ξ) stochastic damping matrix
K(ξ) stochastic stiffness matrix
U vector of gPC expansion coefficients
M(ξ) coefficient matrix assembled from stochastic mass matrix
C(ξ) coefficient matrix assembled from stochastic damping matrix
K(ξ) coefficient matrix assembled from stochastic stiffness matrix
Θ(u(t; ξ)) coefficient matrix assembled from nonlinear restoring force
f(t; ξ) time-dependent external force
N number of examples in sampling methods
σ standard deviation of the corresponding function
H number of strata in stratified sampling method
Hp gPC basis functions of order p from the Askey family
NMCS sample size of MCS
Nl MCS sample size at the l-th MLMC discretization level
ρ(ξ) joint pdf of ξ
ΓM joint image of ξ
Ω sample space
F σ−algebra associated with Ω
x
P probability measure associated with FD domain of interest for the multi-dimensional integral
d dimensionality of multi-dimensional integral
q(ξ) probability density function on DZ0(ξ) initial value of u(t; ξ), Z0(ξ) = u(0; ξ)
Z1(ξ) initial value of u(t; ξ), Z1(ξ) = u(0; ξ)
U(t; ξ) solution to the first order ODE system, U(t; ξ) = (u(t; ξ); u(t; ξ))T
Z(ξ) initial value of U(t; ξ), Z(ξ) = U(0; ξ)
γ(u(t; ξ); ξ) nonlinear stochastic restoring force
p gPC expansion order
Nξ number of terms in the gPC expansion of a random variable or random field
ϕi(ξ) set of orthonormal gPC basis functions
λm,Ψm eigenvalue and eigenfunctions of the Fredholm integral equation of second kind
K(u(t; ξ)) stochastic nonlinear stiffness matrix
ζ(ξ) damping ratio of the single-dof stochastic structural dynamics system
η(ξ) stochastic nonlinearity coefficient
w0 undamped characteristic frequency of single-dof dynamics system, w0 =√
km
τ unitless temporal coordinate, τ ∈ [0, w0T ]
ξ vector of M independent random variables, ξ : Ω→ RM
K number of terms in GSD approximation
Ik indices of the set of active random variables, Ik = j1, . . . , jkξa anchor point, ξa = (ξa1 , ξ
a2 , . . . , ξ
aM)T ∈ ΓM
ξaj1...jk M -dimensional vector with random entries ξi(i ∈ Ik) and ξi = ξai (i 6∈ Ik)〈·〉 expectation operator with respect to ρ
ULANOV A L-th order truncated ANOVA trial space, L ≤M
V0, . . . , Vj1...jL subspace of test functions
V L admissible tensor product of subspace test functions
uL(t; ·) L-th order truncated ANOVA solution, uL(t; ·) ∈ ULANOV A
u0, . . . ,uj1...jL ANOVA component functions of uL. uj1...jk |ξi=ξai = 0,∀i ∈ Ik.uj1...jk solution of the low-dimensional auxiliary SODE systems with random variable ξaj1...jk∆t time marching step in the solution of ODEs
pξ gPC expansion order
e error in solution of multi-dof SODE defined in (5.53)
α1,2 pre-defined constants in adaptivity conditions for AAPG
xi
Φi effective dimensions in the i-th order ANOVA components, i = 1, . . . , L
θ1,2 pre-defined thresholds associated with different criteria for selecting active terms in
the adaptive AAPG scheme
xii
Chapter 1
Introduction
1.1 Motivation and background
In traditional structural dynamics, the assumption of determinism is implicitly invoked when
deriving partial differential equations (PDE) governing the system response. However, some
level of uncertainty is inevitable in modeling real world engineering systems. The source
of uncertainties typically encountered in engineering practice include spatial variability in
material constitutive laws due to statistically inhomogeneous microstructure, variations in
nominal geometry due to manufacturing tolerances, in-service degradation or thermal effects,
uncertainty in loading due to fluctuations in the operating environment, and unmodeled
terms in the governing PDEs.
In the face of uncertainty, deterministic or nominal response predictions may be mislead-
ing and designs based on deterministic models can potentially involve significant risk and are
likely to violate design requirements. To consider uncertainties in engineering design, it is
common practice to introduce the factor of safety, which depends on prior experience and in-
troduces unfavourable consequences such as more conservative design and weight penalty [1].
Such disadvantages lead to the development of various approaches in robust design subject
to uncertainties. Tools such as stochastic calculus [2] have been developed extensively to
model stochastic systems where random inputs can be treated as idealized processes such as
Wiener processes, Poisson processes, etc. However, a more general approach is needed in the
analysis of engineering systems due to their complex nature. Non-parametric approaches are
typically used to deal with uncertainty in unmodeled behaviour [3, 4, 5, 6, 7]. Fuzzy analysis
is useful in the context of subjective knowledge or insufficiently known parameters [8, 9].
1
Bayesian analysis suits well when the system is monitored and data are collected [10, 11].
There are also situations where a non-probabilistic model of uncertainty is more appropri-
ate [12, 13]. A comparison between these methods can be found in [14].
The focus of the present thesis is on parametric uncertainty quantification (UQ) wherein
the main goal is to estimate the joint probability density function (pdf) of the response, given
the joint pdf of the stochastic coefficients, initial conditions and boundary conditions in the
governing stochastic differential equations. There are two main types of problems in UQ.
The first concerns the forward propagation of uncertainty from model parameters to model
outputs, which is the focus of the present thesis. The second type, which involves estimation
of parametric uncertainties using experimental data or stochastic micro-mechanical analysis,
although important, is outside the scope of the present thesis.
Computational methods to compute the response statistics of complex stochastic systems
broadly fall into two classes: sampling and non-sampling methods. Sampling methods such
as the Monte Carlo simulation (MCS) are very versatile and general-purpose. MCS can be
used to approximate the response statistics to an arbitrary degree of accuracy by simulat-
ing the stochastic system repeatedly with different realizations of the stochastic parameters.
Nevertheless, its slow convergence rate has motivated research into better sampling strategies
that can provide computational cost savings; see, for example, [15, 16, 17, 18, 19, 20] and the
references therein. Non-sampling methods often offer higher computational efficiency at the
cost of generality. For example, with the assumption that the coefficients of variation of the
random inputs and outputs are relatively small, perturbation methods [21, 22] and the Neu-
mann expansion method [23] can be used to compute the first two statistical moments of the
response with significantly lower cost than sampling methods. However, the assumption of
small randomness limits their application, especially for nonlinear systems where small input
randomness can result in large randomness in the output [24]. Response surface methods [25]
based on linear and quadratic models have been applied to many problems in UQ. However,
this approach does not scale well to problems with large number of variables [26, 27, 28].
Stochastic projection schemes based on polynomial chaos (PC) decomposition were pro-
posed as an alternative to sampling and approximation methods by Ghanem & Spanos [29],
who applied PC expansions in conjunction with stochastic Galerkin projection to solve a
range of stochastic operator equations. In their original work Hermite polynomials were
2
used as basis functions. As shown later by Xiu and Karniadakis [30], orthogonal polyno-
mials from the Askey family can be used to construct a generalized PC (gPC) expansion.
This approach has been shown to provide good approximations with computational effort
significantly lower than sampling methods for a large class of problems [31, 32, 33, 34].
However, the computational cost associated with solving the deterministic equations arising
from stochastic Galerkin projection can become prohibitive when employing high-order gPC
expansions for large-scale systems with even a modest number of random variables. This
has motivated research into alternative decomposition strategies that can be more efficient
than gPC-based stochastic Galerkin methods; see, for example, sparse gPC expansions [35],
the generalized spectral decomposition (GSD) technique and its variants [36, 37, 38, 39],
and dynamically orthogonal field equation methods [40, 41]. These approaches are yet to be
applied in the context of stochastic structural dynamics, the only exception being the work
of Chevreuil and Nouy [42] which proposes a proper generalized decomposition scheme for
frequency domain analysis of structural systems.
Another function decomposition scheme that has been widely studied in various fields is
the Hoeffding functional analysis of variance (ANOVA) representation [43, 44, 45, 46]. It can
be combined with the sparse tensor product collocation method and adaptivity procedures
to solve UQ problems nonintrusively [47, 48, 49, 50]. Alternatively, ANOVA decomposi-
tion can be used together with stochastic Galerkin projection scheme to solve UQ problems
intrusively [51]. More recently, Audouze and Nair [52] proposed the anchored ANOVA
Petrov-Galerkin (AAPG) projection scheme for solving high-dimensional parabolic stochas-
tic partial differential equations (SPDEs). It was shown that by using an anchored ANOVA
decomposition in conjunction with an appropriate test space, the original high-dimensional
weak form can be decoupled into low-dimensional subproblems that can be solved indepen-
dently of each other. Theoretical analysis and numerical studies conducted so far suggest
that AAPG schemes provide accuracy comparable to the gPC-based stochastic Galerkin
projection scheme while offering significant computational cost savings for high-dimensional
SPDEs. This approach will be the main focus of the present thesis.
3
1.2 Research objectives and methodology
Most of the methods available for the parametric uncertainty analysis of stochastic ordinary
differential equations (SODEs) arising in dynamic analysis of stochastic structural problems
suffer from the curse of dimensionality, that is, their computational cost grows quickly and
becomes infeasible for problems with a large number of stochastic degrees of freedom (dof).
The focus of this thesis is to formulate efficient algorithms for the solution of high-dimensional
linear and nonlinear SODEs in the most general form, where the coefficient matrices, exci-
tation and initial conditions are modeled as random variables or fields.
We first carefully examine the gPC-based stochastic Galerkin method and the general-
ized spectral decomposition (GSD) scheme. These two methods are applied to a few linear
stochastic structural dynamic problems to showcase their advantages and disadvantages. Fol-
lowing this, we propose the AAPG method that decomposes the original high-dimensional
linear and nonlinear SODEs into smaller subproblems that can be solved independently of
each other. It is worth mentioning that although specific forms of linear and nonlinear SODE
systems were used in the theoretical development, the key properties of the AAPG scheme
applies to general high-dimensional linear and nonlinear SODE systems. Extensive numeri-
cal studies are conducted for problems with different sizes and different kinds of randomness
in excitation, coefficient matrices and/or initial conditions.
Finally, the sources of error for the AAPG method are identified and a priori error
estimates are derived for linear and nonlinear SODEs. Combined with theoretical analysis
of the computational cost, we identified difficult situations for the AAPG scheme when it
is applied to high-dimensional systems with high ANOVA expansion order. The adaptive
ANOVA scheme is proposed to reduce the number of subproblems and numerical studies are
included to demonstrate its performance.
1.3 Thesis outline
This thesis is organized as follows:
Chapter 2 introduces the SODEs governing the dynamics of linear and nonlinear stochas-
tic structural systems and reviews available methods to solve them. To facilitate paramet-
ric uncertainty analysis, the random fields in the system need to be discretized and the
4
Karhunen-Loeve expansion is introduced as an example. Both sampling and non-sampling
methods can be used to approximate the response statistics. In the first category we include
MCS and its variance reduction variations, as well as Quasi-MCS and Multilevel MCS. Non-
sampling methods such as sparse quadrature, response surface and PC-based methods are
also discussed.
Chapter 3 focuses on stochastic Galerkin methods for SODEs. The gPC-based stochastic
Galerkin projection scheme uses a gPC expansion of the response together with the stochas-
tic Galerkin projection scheme to approximate the response statistics. Despite its success
in a wide range of problems, this scheme is inefficient when applied to high-dimensional
problems. The GSD method is proposed as an alternative with reduced cost. A few test
cases are included to demonstrate the performance of gPC-based stochastic Galerkin and
GSD schemes.
Chapter 4 formulates the AAPG scheme for the solution of second-order linear SODEs. It
is proved that by using an appropriate test space, the original high-dimensional linear weak
form can be decoupled into low-dimensional subproblems that can be solved independently
of each other. Detailed numerical studies are presented to show that the AAPG scheme can
efficiently alleviate the curse of dimensionality.
Chapter 5 extends the AAPG scheme to nonlinear SODEs. A single-dof system is first
discussed to pave the way for more complicated multi-dof systems. Both the gPC-based
stochastic Galerkin projection scheme and the AAPG scheme are derived for these nonlinear
systems and we discuss numerical issues that require evaluating the stochastic terms using
the pseudo-spectral approach. Extensive numerical tests are included to demonstrate the
performance of the AAPG scheme when applied to single-dof and multi-dof nonlinear SODEs.
Chapter 6 presents a theoretical analysis of the AAPG scheme focusing on a priori error
estimation and computational cost. This analysis and the numerical tests in Chapter 4
and 5 point out a difficult situation for the AAPG scheme when there might be too many
subproblems to solve. To address this issue, the adaptive AAPG scheme is proposed and
numerical tests show it can effectively reduce the number of subproblems while retaining a
similar level of accuracy.
Chapter 7 summarizes the main contributions of this thesis and discusses some directions
for future research.
5
Chapter 2
Literature review
This chapter introduces governing equations for stochastic structural dynamic systems in the
form of SODEs. We also present an outline of general methods available for the parametric
uncertainty analysis of such SODEs, i.e. to calculate the response statistics given stochas-
tic coefficients, excitation and initial conditions in the form of random variables or random
fields. We begin with the governing equations arising in linear and nonlinear stochastic
structural dynamics. This is followed by a brief discussion of the Karhunen-Loeve expan-
sion as an example of a discretization method for random fields. We subsequently review
the Monte Carlo simulation method and its variants. Alternative methods such as sparse
quadrature, response surface and polynomial chaos expansion methods are also discussed,
while a few other methods are excluded because of their limitations. For instance, perturba-
tion and sensitivity-based methods can be highly inaccurate even for moderate coefficients of
variation of the random variables, particularly for dynamic response calculations [21, 1] and
nonlinear systems. Moment methods [53, 54] attempt to compute the moments of the random
solution directly and experience difficulties when the derivation of a moment requires infor-
mation of higher moments, which is almost always the case. The Fokker-Planck-Kolmogorov
(FPK) [55, 56] method assumes white noise excitation and is so far not applicable to large-
scale systems. In this chapter, a multidimensional integration problem is used to demonstrate
different methods except for the case of multilevel Monte Carlo (MLMC) simulation, which
is designed to optimize resources in order to minimize discretization and sampling error
simultaneously in the solution of differential equations.
6
2.1 Linear stochastic structural dynamics
In this section we will introduce the governing equations arising in linear stochastic structural
dynamics. The response of a stochastic structural dynamic problem can be described in either
the time domain, in the form of u(t; ξ) as a function of time t and random vector ξ, or in the
frequency domain, where the process is specified by its random amplitude U(ω; ξ) (generally
a complex number representing the phase as well) as a function of the excitation frequency
ω and ξ. Both are different representations of the same function and can be converted to
each other by means of the Fourier transformation. Time domain analysis is usually used
to study the dynamic response of systems subject to complex non-stationary excitation, or
when the transient solution is of interest. Since a harmonic time function is preserved under
linear transformations, a frequency response function can be used to represent the one-to-
one mapping from excitation to response in the steady state. Frequency domain analysis is
often used in structural reliability analysis when the excitation can be characterized in the
frequency domain.
It is worth mentioning that there is extensive literature dedicated to the study of deter-
ministic systems subject to stochastic external forcing, where white noise models are most
commonly considered [57, 58, 59, 60, 61]. In contrast, the algorithms in this thesis are capa-
ble of handling the most general case, where the material and geometric properties, external
force and/or initial conditions are considered uncertain.
2.1.1 Time domain analysis
The dynamics of a linear stochastic structural system in the time domain are governed by
the following system of second-order SODEs:
M(ξ)u(t; ξ) + C(ξ)u(t; ξ) + K(ξ)u(t; ξ) = f(t; ξ) a.s. in [0, T ]× ΓM , (2.1)
where u ∈ Rn is the displacement vector, t ∈ [0, T ] denotes time (T < +∞) and n is the total
number of dof. M(ξ) ∈ Rn×n, K(ξ) ∈ Rn×n and C(ξ) ∈ Rn×n denote the stochastic mass,
stiffness and damping matrices, respectively. The external force f(t; ξ) ∈ Rn is assumed to be
a time-dependent stochastic process. The components of the vector ξ = (ξ1, ξ2, . . . , ξM)T ∈RM are assumed to be a set of independent random variables whose joint pdf can be written
as the product of its marginal densities, i.e., ρ(ξ) =∏M
i=1 ρ(ξi). We denote the joint image
of ξ by ΓM = Γ1 × · · · × ΓM and the probability space by the triplet (Ω,F ,P), where Ω is
7
the sample space, F is the σ−algebra associated with Ω and P : F → [0, 1] is a probability
measure. The governing equation (2.1) is supplemented with the following stochastic initial
conditions
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (2.2)
The above SODE features discretized random fields and arises after spatially discretizing the
underlying SPDE using the finite element method (FEM). The model structure considered
here is fairly general and allows for the consideration of uncertainties in constitutive model
parameters, geometry, initial conditions, boundary conditions, forcing functions, etc. It is
worth noting that the solution of (2.1) is parameterized in terms of the same set of random
variables used in the parameterization of the matrices and the source term. Indeed, from
an SPDE point of view it follows from the Doob-Dynkin lemma [57] that when the SPDE
coefficients and/or the source term depend on ξ ∈ RM , then the SPDE solution can be
described using the same set of random variables.
It is sometimes convenient to convert (2.1) into the following first-order ODE system
with 2n states:
U = F(t,U; ξ), (2.3)
where
U(t; ξ) =
(u(t; ξ)
u(t; ξ)
)∈ R2n, F(t,U; ξ) = −A(ξ)U +
(M−1(ξ)f(t; ξ)
0
)∈ R2n,
and A(ξ) =
(M−1(ξ)C(ξ) M−1(ξ)K(ξ)
−I 0
),
with the initial conditions on the state vector given by
U(0; ξ) = Z(ξ), where Z(ξ) ≡
(Z1(ξ)
Z0(ξ)
). (2.4)
This first-order ODE system is particularly useful in numerical studies and a priori error
estimation, as we can utilize a broader selection of well-studied time-marching methods in-
cluding the explicit Euler method, θ-weighted methods, Runge-Kutta method, etc.
8
2.1.2 Frequency domain analysis
If we assume the excitation has random amplitude and deterministic frequency, it can be
written in the form f(t; ξ) = F(ω; ξ)eiωt, where i =√−1 and the steady state response
is given by u(t; ξ) = U(ω; ξ)eiωt. Substituting these expressions into the linear governing
equation (2.1) we obtain the following governing equation in the frequency domain:
[−ω2M(ξ) + iωC(ξ) + K(ξ)
]U(ω; ξ) = F(ω; ξ), (2.5)
which can be written compactly in the form
H(ω; ξ)U(ω; ξ) = F(ω; ξ), (2.6)
where the stochastic dynamic matrix H(ω; ξ) = [−ω2M(ξ) + iωC(ξ) + K(ξ)]. Numerical
studies have shown that small randomness in the stiffness and mass matrices could change
the response characteristics significantly in systems with low damping, or when the driving
force excites the medium and high frequency modes of the system [31].
Classical approaches to solve (2.6) involve the inversion of stochastic matrix H(ω; ξ) that
can be done using perturbation, Neumann expansion, optimal series expansions, optimal lin-
earization and digital simulation methods [62]. In particular, various perturbation methods
exist [21, 63, 64]. These methods solve for the structural normal modes individually thus
are feasible only when the total response can be represented by a limited number of modes.
More recently, stochastic reduced basis methods [31, 65, 66] were proposed to address the
performance issues faced by existing approaches.
2.1.3 Free vibration
For systems free of external excitation, the system response is characterized by the undamped
natural frequencies and mode shapes that can computed by solving the following generalized
random eigenvalue problem:
K(ξ)φ(ξ) = λ(ξ)M(ξ)φ(ξ), (2.7)
where λi and φi, i = 1, . . . , n denote the eigenvalue and eigenvector (mode shape) of a par-
ticular mode. The undamped natural frequency ωi =√λi for i = 1, . . . , n. In most practical
problems, the structure is subject to conservative loadings and the above eigenvalue prob-
9
lem is self-adjoint. For systems involving follower forces, aerodynamic damping or gyroscopic
couples the above problem can be non-self adjoint [67, 68].
Classical approaches for the probabilistic analysis of the above random eigenvalue prob-
lem include sampling methods [69] and mean centered first/second order perturbational
approaches [70]. Recent developments focus on general methods that can be applied to com-
plex systems with high randomness. When the random matrix K can be decomposed into
mean and random perturbations the statistical moments in closed-form is available in a few
exceptional cases [71, 5]. In a more general setting the polynomial chaos expansion scheme
can be used to approximate the solution of (2.7) [32].
2.2 Nonlinear stochastic structural dynamics
The linear SODE in the previous section may not suffice to describe more complicated
structural systems, for example, when the system is subject to large deformations or when
the material constitutive properties are nonlinear. This kind of geometrical nonlinearity will
be our focus.
Nonlinear structural dynamics is fundamentally different from linear systems since the
principle of linear superposition does not apply, making various well-established linear tech-
niques inapplicable to nonlinear systems. While in free vibration, nonlinear systems are
frequency-energy dependent, which lead to Liapunov instability of the free periodic responses
of undamped nonlinear oscillators [72], no analytical solutions for free damped nonlinear re-
sponses [73], and other complex nonlinear phenomena. It is typical for a nonlinear system
to have multiple, co-existing stable equilibrium positions, each with its own separate domain
of attraction. Another key character of nonlinear systems is the bifurcations of equilib-
rium positions or periodic orbits of nonlinear systems. Bifurcations lead to various nonlinear
phenomena including sudden nonlinear transitions between stable attractors (jumps), transi-
tions of regular motions to chaotic, and chaotic explosions. Because of bifurcations, nonlinear
systems are very sensitive to initial conditions.
In nonlinear structural dynamics, frequency domain analysis is generally not practical
because there is typically multi-frequency response to a single-frequency excitation and the
one-to-one mapping between the excitation and frequency response function no longer exists.
The algorithms we study in this thesis apply to both linear and nonlinear problems, thus in
the theoretical development and numerical studies it is sought to approximate the statistics
of the dynamic response, u(t; ξ), as a function of time. The following SODE describes the
10
dynamics of a structural system with geometrical nonlinearity:
M(ξ)u(t; ξ) + C(ξ)u(t; ξ) + K(ξ)u(t; ξ) + γ(u(t; ξ); ξ) = f(t; ξ) a.s. in [0, T ]× ΓM , (2.8)
with the stochastic initial conditions in the form of
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (2.9)
Compared to the linear SODE (2.1), the additional term γ(u; ξ) ∈ Rn in (2.8) denotes the
nonlinear restoring force that is a deviation from the linear restoring force vector K(ξ)u(t; ξ).
Statistical linearization [74, 75, 76] can be used to approximate this nonlinear term and it
works well if the goal is to predict the mean-squared response. However, this method is
restricted to systems with weak nonlinearity since the response is assumed to be quasi-
Gaussian [77]. Similarly, equivalent quadratization [78] and equivalent cubicization methods
[79, 80] use second- and third-order polynomials to approximate the nonlinear behaviour.
The equivalent quadratization method works well when the nonlinearity is statistically
symmetric [79], while equivalent cubicization is able to capture both symmetric and non-
symmetric nonlinear characteristics [80]. Fatica et al. [81] conducted detailed numerical
studies to investigate higher order polynomial approximations. It was shown that third or-
der polynomials provide good agreement with numerical simulations over a wide range of
loads [82]. As a result we choose to represent the nonlinear restoring force γ(u; ξ) in the
following form of third order polynomial
γi(u; ξ) ≈n∑
j1≤j2
aij1j2uj1uj2 +n∑
j1≤j2≤j3
bij1j2j3uj1uj2uj3 , i = 1, 2, . . . , n, (2.10)
where γi(u; ξ) and ui are the components of γ(u; ξ) and u at the ith dof, respectively.
aij1j2 , bij1j2j3
are pre-defined scalars. A procedure to approximate an arbitrary nonlinearity by
such a polynomial expansion can be found in [80].
2.3 Uncertainty modeling and discretization of ran-
dom fields
Uncertainty can be classified into two categories: aleatoric and epistemic. Aleatoric uncer-
tainty results from variability and is usually represented statistically. Epistemic uncertainty
11
is due to insufficient knowledge of the underlying system and can be reduced or eliminated.
The present thesis focuses on aleatoric uncertainty where the pdf of the underlying properties
is assumed to be given, from which the corresponding model parameters can be represented as
random variables, or random fields to take into account spatial or temporal inhomogeneities.
Such random fields need to be represented in terms of a finite set of independent random
variables for parametric UQ and a few popular techniques can be used, including spectral
series [83], Karhunen-Loeve (KL) expansion [84, 85, 86, 87, 88] and orthogonal series [89].
Detailed discussions on these random field discretization methods can be found in [90, 15].
In the dynamical analysis of structural systems, it is typical to find very small correlation
length for the stochastic field modelling of realistic random heterogeneous media. Numerical
tests have shown that when the ratio of the length of the random process over correlation
parameter is of small value (highly correlated), KL expansion is much more efficient com-
pared to the spectral series method [91]. As a result, we choose to use the KL expansion
scheme in this thesis and briefly review this method in this section.
The KL expansion scheme identifies a set of basis functions to characterize a random
process and can be viewed as a special case of orthogonal series expansion. It is based on the
spectral decomposition of the bounded covariance function C(x,y) of a random field κ(x, θ)
in the following form
κ(x, θ) = κ0(x) +∞∑m=1
ξm(θ)κm(x), (2.11)
where x ∈ D denotes the spatial coordinates and θ ∈ Ω is an element of the sample space.
In practice, the following truncated expansion is used
κ(x, θ) ≈ κ0(x) +M∑m=1
ξm(θ)κm(x), (2.12)
where κ0(x) is the mean value of κ(x, θ). κm(x) =√λmΨm(x),m = 1, 2, . . . ,M are a set of
basis functions, where λm and Ψm(x) are the eigenvalues and eigenfunctions, respectively, of
the following Fredholm integral equation of the second kind∫D
C(x,y)Ψm(y)dy = λmΨm(x). (2.13)
The eigenvalues λm are ordered such that λ1 ≥ λ2 ≥ · · · ≥ λM and satisfy∫D
∫D|C(x,y)|2dxdy =∑M
m=1 λm. The eigenfunctions Ψm(x) are mutually L2(D) orthogonal and are basis of L2(D)
12
if the covariance function is positive definite. ξm in (2.12) are mutually uncorrelated random
variables centred with unit variance.
The KL expansion provides the optimal representation of κ(x, θ) in the mean square
sense, i.e. for any other linear combination of M functions, the error ||κ − κM ||L2(D×Ω)
is not smaller than for the KL expansion [29]. A closed form solution of (2.13) is only
available for certain types of C(x,y) on simple geometries [92, 85]. Numerical approaches
can be found in [93, 94] to approximate the dominant eigenpairs in (2.13) and efficient
implementations can be found in [95, 96]. Note that because of the truncation in (2.12),
the KL expansion always under-represents the true variance of the random field. The KL
expansion is mathematically well founded and guaranteed to converge. In general, using more
eigenfunctions in the KL expansion results in finer spatial resolutions. A detailed theoretical
analysis of the convergence properties of the KL expansion can be found in [97, 91, 98].
2.4 Simulation methods
We shall now briefly review methods that can be used to solve the SODEs arising in lin-
ear and nonlinear stochastic structural dynamics. Since parametric uncertainty analysis of
SODEs involves computing the statistics of some quantity of interest, the following form of
d-dimensional integral is used to demonstrate different methods in this section:
Id =
∫Df(ξ)q(ξ)dξ, ξ ∈ Rd. (2.14)
Here D ⊂ Rd is the domain of interest and q(ξ) is a probability density function on D.
q(ξ) ≥ 0 and∫D q(ξ)dξ = 1.
A commonly used approach to numerically approximate the above integral is the MCS
technique. In this method, we draw samples from q(ξ) repetitively to generate realizations
of f(ξ), using which the integral (2.14) can be approximated. The law of large numbers
guarantees the convergence of MCS. MCS is versatile, straightforward to implement, easily
parallelizable and provides access to any order of response statistics and thus is usually used
to get benchmark solutions. The expected value of the integral (2.14) estimated by MCS is
Id ≈ In =1
N
N∑i=1
f(ξi), (2.15)
13
where the samples ξi are generated from the density q(ξ) and N is the sample size. The
following strong law of large numbers guarantees the convergence of MCS:
Pr( limn→∞
In = I) = 1. (2.16)
The error (In − I) satisfies
ε(n) = O(σN−1/2), (2.17)
where σ is the standard deviation of f . The biggest drawback of MCS is its slow convergence
rate as a function of the sample size. To improve its convergence rate, a few alternative sam-
pling methods were developed [99, 100]. Variance reduction methods accelerate convergence
by lowering the value of variance σ in (2.17). One can also modify the statistics by replac-
ing the realization of ξ with an alternative sequence in order to improve the convergence
rate, examples include Quasi-Monte Carlo [101, 102, 103] and multilevel Monte Carlo [20]
methods.
2.4.1 Variance reduction methods
Variance reduction methods are a group of methods proposed to improve the convergence
speed of MCS by reducing the variance of the integrand. We discuss here stratified sam-
pling [104], Latin Hypercube sampling [105], importance sampling [106], line sampling [107]
and subset sampling [108] methods.
Stratified Sampling
Stratified sampling involves dividing the domain of ξ into a union of strata D =⋃Hh=1Dh
where Di⋂Dj = ∅ if i 6= j. The integral is estimated in each stratum and then combined,
that is
Id ≈ ISTRAT =H∑h=1
|Dh|Nh
Nh∑i=1
f(ξhi), (2.18)
where |Dh| is the volume of stratum Dh. Samples ξhi, i = 1, . . . , Nh are taken from within
stratum Dh. To illustrate, we use a simple setting H = 2, D1 = D2 and Nh1 = Nh2. Then
the mean and variance estimates of f(ξ) are given by
14
〈f〉 =1
2[〈f〉1 + 〈f〉2] , (2.19)
σ2STRAT (f) =
1
2
[σ2
1(f) + σ22(f)
]. (2.20)
As a comparison, the variance estimated using MCS is
σ2 (f) =1
2
[σ2
1(f) + σ22(f)
]+
1
4[〈f〉1 − 〈f〉2]2 . (2.21)
σ2STRAT (f) is always smaller than σ2 (f) whenever the means of the stratified samples 〈f〉1
and 〈f〉2 are different. In general, let f(ξ) = µh(ξ) be the mean in strata and define residual
fRES(ξ) = f(ξ) − f(ξ), stratified sampling reduces the MCS variance from σ2(f)/N to
σ2(fRES)/N .
The performance of stratified sampling largely depends on the choice of strata and al-
location of samples. The simplest choice of strata are of same size and roughly speaking,
the efficiency of stratified sampling increases as H2 [109]. In practice, increasing H leads
to increasing cost, and recursive stratified sampling is used to refine the division of strata
where most needed [110]. A better choice would be to choose strata so that variance of
f is the same in each stratum [111]. The jittered sampling approach [112] is particularly
well suited for the case when D is a unit cube. Optimal allocation of sampling points take
Nh ∝ |Dh|σh. In practice, poor estimation of σh can result in larger variance than MCS. As
a result, Nh ∝ |Dh| is commonly used.
Latin Hypercube Sampling
The Latin hypercube sampling (LHS) method is a multivariate stratified sampling technique
first introduced by McKay, Conniver and Beckman [105]. Considering a d-dimensional ran-
dom variable ξ ∈ V ⊂ Rd and sample size N , the range of each variable ξ1, . . . , ξd is
divided into N non-overlapping intervals on the basis of equal probability. One value xjifrom the j-th interval (j = 1, . . . , N) in the i-th dimension (i = 1, . . . , d) is randomly se-
lected following the probability density in the interval. These xji from each dimension are
then combined in a shuffling operation to create a set of d-tuplets ξj (j = 1, . . . , N) with a
specified correlation structure, which ensures every row and column in the hypercube of par-
titions has exactly one sample. Figure 2.1 shows a two-dimensional example with N = 10.
Assuming two random variables are both evenly distributed, LHS partitions the parameter
15
space into N = 10 bins in both dimensions and assigns one sample to each bin according to
the criteria described above. Note that there is no restriction on N but the number of bins
must be the same in all dimensions.
(a) One sample in each of 10 horizon strata. (b) One sample in each of 10 vertical strata.
Figure 2.1: Same set of LHS sampling (d = 2, N = 10) with (a) horizon strata and (b)vertical strata.
Owen [113] shows that for all N ≥ 2, d ≥ 1 and square integrable f , variance in LHS
satisfies
σ2LHS ≤
σ2
N − 1, (2.22)
where σ2 is the variance of f . Compared to (2.17), it is clear that in the worst case LHS is like
MCS but with one less sample. It was proved in [114] that variance of LHS is approximately
σ2RES/N where σ2
RES is the smallest variance of fRES for any decomposition of the following
form
f(ξ) =d∑i
f(ξi) + fRES, (2.23)
where f(ξi) depends on ξi only. LHS is computationally cheaper than direct MCS and well
suited for various input distributions, but can still be prohibitively expensive. It covers the
volume well without replicated values, but behaves poorly at untried input locations.
16
Importance Sampling
Importance sampling [106, 115, 116] is particularly useful when the sample distribution is not
standard. For example, in the reliability analysis of dynamical systems, we are interested in
failure events with small probability. Direct MCS is not efficient when the estimated failure
probabilities is smaller than 10−3 since it usually requires many samples before one such
failure sample occurs. In importance sampling, the samples are placed in important regions
of the space (e.g. in or near the failure regions) to reduce the necessary sample size N
for a specified accuracy, then the samples are weighted appropriately to obtain an unbiased
estimate of the failure probability. In the example of (2.14), importance sampling introduces
a probability density p(ξ) on D such that p(ξ) > 0 whenever q(ξ)|f(ξ)| > 0, then
Id =
∫Df(ξ)q(ξ)dξ =
∫D
f(ξ)q(ξ)
p(ξ)p(ξ)dξ, (2.24)
and we can approximate Id using
Id ≈ IpN =1
N
N∑i=1
f(ξi)q(ξi)
p(ξi), (2.25)
with ξi sampled from probability distribution p(ξ). The efficiency of importance sampling
depends on the choice of importance sampling density p(ξ) and it is popular to make use of
design points that are local most probable points within the failure region in the stochastic
space. The search for the design point involves the solution of a constrained optimization
problem, for which the computation of gradient is needed [117]. Numerical studies show that
the cost to construct importance sampling density (ISD) is much less than the gain in effi-
ciency from importance sampling in static or time-invariant problems. However, it becomes
more expensive to construct ISD for time-varying and/or nonlinear dynamics systems and
importance sampling becomes less favourable [118].
Line Sampling and Subset Sampling
Line sampling [107], an alternative to importance sampling, was proposed to treat high-
dimensional reliability analysis problems with an implicitly available performance function
obtained from deterministic finite element analysis. Its efficiency depends on the important
direction, which points toward the failure domain nearest to the origin. It was shown that
line sampling will always be faster compared to direct Monte Carlo, even with additional
17
cost associated with finding the important direction.
The important region in importance sampling and important direction in line sampling
may be prohibitively expensive to construct in time-varying and/or nonlinear dynamics
systems. Subset simulation [108] is particular advantageous in these cases. It was proposed
to address the slow convergence of MCS in estimation of small failure probabilities. The basic
idea is to replace a small probability with a product of larger conditional probabilities by
introducing intermediate events. The conditional probabilities are estimated using Markov
chain Monte Carlo (MCMC) simulation based on the Metropolis algorithm. Because the
conditional failure probabilities can be made sufficiently large, the cost of subset MCS is
significantly smaller than direct MCS. The convergence is generally guaranteed by chains
with a Metropolis-Hasting transition probability kernel.
2.4.2 Quasi-Monte Carlo simulation
Quasi-Monte Carlo (QMC) methods [119] utilize quasi-random sequences that are correlated
elements chosen at deterministic locations to provide greater uniformity. An example is
presented in Figure 2.2. It is a logical extension of LHS in the sense that the sampling
points are designed to balance in any hyper-dimensional strata. To do that, QMC methods
initialize segments of the generated quasi-random sequence to fill space uniformly and later
fill the “holes” in the initial segment. This systematic way of refining sample points at
deterministic locations guarantees its convergence. The Koksma-Hlawka inequality [120]
proves such deterministic law of large numbers can be much better than the random one.
For functions with bounded variation, QMC methods have convergence rate of [121, 103]
ε(N) = O((log(N))dN−1), (2.26)
Compared to O(σN−1/2) for the MCS, it is almost half an order better. However, the
dimension d enters through the logarithmic term.
Common methods to generate quasi-random multi-variate sequences include Halton se-
quence [122], Sobol’s sequences [123, 124], Faure sequence [125] and integration lattices [126,
127]. Because of the way QMC generate multi-variate sequences, there may be unwanted
correlations between variables and certain techniques can be used to reduce the correlation
and improve uniformity [128]. QMC can be randomized [129, 130] to get sample based error
estimation. Compared to (2.26), some forms of randomized QMC can achieve the following
18
(a) Direct Monte Carlo simulation (b) Quasi-Monte Carlo simulation
Figure 2.2: Comparison of sample point distribution between direct MCS and QMC.
improved convergence rate for smooth enough f :
ε(N) = O((log(N))(d−1)/2N−3/2). (2.27)
2.4.3 Multilevel Monte Carlo simulation
The idea of multilevel Monte Carlo (MLMC) [20] is borrowed from the multi-grid method [131,
132] for the iterative solution of linear systems arising from the discretization of elliptic par-
tial differential equations. For a general engineering problem, the stochastic function of
interest depends on spatial and/or temporal coordinates and its governing differential equa-
tion must be discretized to be numerically solved, which introduces discretization error. For
a general spatial/temporal tessellation with m dofs, the approximate solution of f is referred
to as fm and fm → f when m → ∞. In the case of SODE, m → ∞ represents the process
of refining time-marching steps. Using MCS we have the following estimate of f
fMCSm,N =
1
N
N∑i=1
f im, (2.28)
19
with the following error in mean
ε(N) =
√σ2N−1 + 〈fm − f〉2. (2.29)
This error includes two independent terms from the MCS estimator and discretization. As
a result, accurate estimation requires large number of MCS samples and sufficiently fine
discretization that makes MCS too expensive when each evaluation of f is computationally
expensive.
MLMC replaces the statistics of fm with a sequence of evaluations at progressively refined
discretization levels. The number of discretization nodes on each level is optimized to reach
balance between the discretization and sampling errors, reflecting the goal to not over-resolve
one or the other. In this way, most of the simulations are performed at coarser levels and
relatively few simulations are performed at finer levels. The same level of accuracy associated
with the smallest step is retained but overall computational complexity is reduced. To
demonstrate, we first introduce a sequence of discretization levels ml : l = 0, . . . , L with
m0 < m1 < · · · < mL = m and write the expected value 〈f〉 as
〈f〉 = 〈fm0〉+L∑l=1
⟨fml − fml−1
⟩=
L∑l=0
〈yl〉 ,
where yl = fml − fml−1for 1 ≤ l ≤ L are the differences between levels and y0 = fm0 . An
unbiased MLMC estimator of f can be written as
fMLMCL =
L∑l=0
yl =L∑l=0
1
Nl
Nl∑i=1
yil , (2.30)
where MCS is applied at each level l to estimate yl. The estimator variance is∑L
l=0 σ2(yl)N
−1l
and total cost is∑L
l=0NlCl, where Cl is the cost for evaluating yil . The following constrained
minimization problem is formulated:
f(Nl, λ) =L∑l=0
NlCl + λ
(L∑l=0
σ2(yl)N−1l − ε
2/2
), (2.31)
20
where λ is a Lagrange multiplier. The equality constraint enforces a stochastic error from
MLMC estimator variance equal to the residual bias error ε2/2 and reflects the goal of
balancing the discretization and sampling errors. The result of the minimization is
Nl =2σ(yl)√Clε2
L∑k=0
σ(yk)C1/2k . (2.32)
This is the optimal sample allocation per discretization level. The advantage of MLMC
lies in the fact that σ(yl) decreases when l→ L (because fm → f and yl → 0). As a result, the
number of simulations Nl decreases with l where the simulations are relatively more expensive
to compute. Giles [133] proved that if there exists positive constants α, β, γ, c1, c2, c3 such
that α ≥ 12
min(β, γ) and
1. |E(fl − f)| ≤ c12−αl,
2. σ2(yl) ≤ c22−βl,
3. Cl ≤ c32γl,
then for any error bound ε < e−1 there exist a positive constant c4 such that there are values
l and Nl for which the computational complexity has the boundc4ε−2, β > γ,
c4ε−2(log ε)2, β = γ,
c4ε−2−(γ−β)/α, β < γ.
(2.33)
In the first case when β > γ, the coarsest level has the dominant computational cost.
The third case when β < γ corresponds to a situation when significant portion of the
computational cost is on the finest levels. Note that in the best situation, regardless which
sampling method is used, the cost approaches O(ε−2) for MCS with i.i.d. samples. This
could happen if the discretization error in (2.29) is small (α and β are large) or when the
function evaluation is very cheap (γ → 0). While the cost for MCS is O(ε−2−γ/α) (different
from the case for i.i.d. samples because of the additional discretization error in (2.29)) under
certain assumptions [134], MLMC has lower computational cost in all three cases and its
advantage is more pronounced when β ≤ γ, that is, when the discretization error is large
or the function evaluation is expensive. In high dimensional (d 1) problems we almost
always have β ≤ γ since the rate of convergence β of the multilevel variance will usually be
21
independent of d, and γ, the rate of increase in the computation cost, will increase at least
linearly with d. As a result, MLMC is particularly useful when MCS experience difficulties.
More recently, there has been developments to further improve the computational cost when
β ≤ γ, for an example, combining MLMC with Richardson-Romberg extrapolation [135] and
extending MLMC to the multi-index Monte Carlo [136].
2.4.4 Remarks
Some of the sampling techniques we discussed in this section can be applied together. For
an example, whenever pseudo-random numbers are used, we can replace them with quasi-
random numbers to provide greater uniformity.
2.5 Sparse quadrature method
In the last section we have outlined the MCS technique and how its alternatives can be used
to evaluate the multi-variate integral (2.14). While the alternative sampling methods have
faster convergence rate than direct MCS, researchers have been looking for more efficient
non-sampling methods. Quadrature methods approximate an univariate integral, i.e. (2.14)
with d = 1, as follows:
I(f) ≈ Il(f) =
nl∑i=1
cif(ξi), (2.34)
where ξi and ci are evaluation nodes and their corresponding weights for quadrature level
l and nl is the total number of quadrature points at quadrature level l. Compared to
sampling methods, the evaluation points in quadrature methods are pre-fixed and by choosing
sufficiently high l it is guaranteed to compute the integral of certain order polynomial exactly.
The error bound for functions f ∈ Cr is O(2−lr), where r is the smoothness of f . The choice
of ξi, ci depends on the quadrature rules and a few commonly used ones are listed in Table
2.1.
Clenshaw-Curtis [137] uses nested Chebyshev points that are roots or extrema of Cheby-
shev polynomials. Such nested quadrature rules have the advantage that all points from lower
quadrature levels are reused in higher levels. Another family of popular choices are the Gauss
rules that use the roots of orthogonal polynomials, such as Legendre polynomial, as evalua-
tion points. Gauss rules are typically non-nested with the exception of Gauss-Patterson [138].
22
Table 2.1: List of quadrature rules. nl expression in table is valid for l ≥ 2. For l = 1, allquadrature rules listed has nl = 1.
Quadrature rule nl Polynomial degree of accuracy NestedClenshaw-Curtis 2l−1 + 1 nl − 1 YesGauss-Patterson 2l − 1 1.5nl + 0.5 YesGauss-Legendre 2l − 1 2nl − 1 No
Regardless of the choice of quadrature rule, the required number of evaluations nl is very
small compared to sampling methods. As an example, the Gauss-Legendre rule with nl = 10
can evaluate 19-th order polynomial exactly. Note that because the Gauss-Legendre rule is
non-nested, evaluating lower order polynomials exactly requires more evaluation points from
lower quadrature levels, which result in O(2l) growth of total number of evaluation points.
The same quadrature rules can be extended to d dimensional integration (2.14) by ten-
sorization, resulting in∏d
i nil number of quadrature points. Such full tensor product quadra-
ture can still be very efficient compared to sampling methods for small d, but it quickly
becomes too expensive with modestly large d. Sparse quadrature (SQ) was proposed by
Smolyak [139] to minimize the number of quadrature nodes in the multi-dimensional space
for a certain level of approximation accuracy. The number of sparse grid points for d di-
mensional integration is ndl = O(2lld−1) for all quadrature rules listed in Table 2.1. This
is significantly lower compared to O(2ld) for the full quadrature rule. Figure 2.3 includes
a comparison between the 2D full tensor product quadrature grid and sparse grid utilizing
Clenshaw-Curtis points of the same order.
While the quadrature rules listed scale with the same order, nested quadrature rules are
advantageous since the constants are considerably lower given the same number of abscissas
for the univariate quadrature formulas are used [140]. The following Figure 2.4 demonstrates
2D sparse grids with Gauss-Patterson and Gauss-Legendre of the same order.
The sparse grid method scales much more gracefully with d compared to the full tensor
product grid. It leverages the regularity of stochastic variables and has error of the order
of n−rl (log nl)(r+1)(d−1), where nl is the total number of quadrature points. For r > 1, its
convergence is faster than QMC and for very smooth functions (r →∞) the convergence is
almost exponential. For high dimensional problems (d 1), the approximation error grows
quickly and to have a certain level of accuracy requires a very high quadrature level and large
number of evaluation points. This quick growth of computational cost with increased di-
23
Figure 2.3: Grid comparison between full tensor product grid (left) and sparse grid (right)utilizing Clenshaw-Curtis points with l = 5.
Figure 2.4: Comparison between Gauss-Patterson (left) and Gauss-Legendre (right) sparsegrid of l = 6. When d = 1, these two rules have the same quadrature nodes for the same l.For d = 2, l = 6, Gauss-Patterson has 321 quadrature nodes, while Gauss-Legendre has 637quadrature nodes.
mensionality is commonly observed in uncertainty quantification methods and often referred
to as the curse of dimensionality [141]. Fortunately, the effective dimension of the function
may be low and a low-level sparse quadrature rule may be sufficiently accurate. For cases
when the effective random dimension is large, cubature rules [142, 143] would scale better.
It is different from full tensor product or SQ in that they are not based on combinations of
one-dimensional quadrature rules. But they are limited to homogeneous random variables
and restricted in integrand order.
Recent developments in SQ have shown it to be an efficient alternative to full tensor
product quadrature while keeping the same level of accuracy in numerical integration [140],
stochastic collocation [144, 145, 146, 147, 148, 149] and non-intrusively solving coefficients
in the polynomial chaos expansion method [150]. Despite of these developements, this class
of method can only delay the onset of the curse of dimensionality.
24
2.6 Response surface method
The Response Surface Method (RSM), also known as surrogate modeling or metamodeling,
uses inexpensive approximations to capture the salient features of an expensive computer
model. RSM can be used to explore the response over the parameter space, or can be used
as a replacement for more expensive optimization or uncertainty quantification. It has a
broad range of applications in uncertainty analysis including fuzzy analysis [151], reliability
problems [152] and robust design optimizations [153]. In uncertainty quantification, directly
applying sampling methods can be prohibitively expensive, particularly when detailed finite
element analysis (FEA) models are used. A solution is to use RSM to create a computa-
tionally inexpensive model from an initial set of FEA runs and applying sampling methods
to it [154]. However, the additional cost of the initial set of FEAs must be justified by the
saving against MCS, which is shown in a few experiments [155]. It is also to be noted that
this approximation process introduces additional error. Nevertheless, case studies show that
more accurate results can be expected from RSM in comparison to MCS, except for very low
sample sizes where the quality of prediction of the two are comparable [156]. Unlike MCS,
RSM suffer from the curse of dimensionality like most other methods.
The performance of RSM relies heavily on the selection of experiment points to run
FEA, and the fit methods used to recover the surrogate model from data (response values,
gradients and Hessians). Depending on the number of points used for generating the data
fit, RSM method can be further divided into local, multipoint and global approximation
techniques. An incomplete list of techniques includes: first or second-order Taylor series
(local), two-point exponential approximation (multipoint) [157, 158], Kriging interpolation
(global) [159] and multivariate adaptive regression splines (global) [160]. Interested readers
are refered to [161] for a more complete list of data fit models and recommendations in
different applications.
2.7 Polynomial chaos expansions
Polynomial chaos (PC) representations of the stochastic processes were proposed by Wiener
as a generalization of Fourier series expansion [162]. This spectral representation was adopted
by Ghanem and Spanos [29] to approximate the solution of stochastic systems. Considerable
work has been accomplished following this work to improve convergence speed for non-
Gaussian problems [30, 163, 164] and alleviate the requirement of globally smooth basis
25
polynomials [165, 166, 167, 168, 164]. In this section we briefly introduce the PC expansion
and will discuss it in depth later in Chapter 3.
The first step in solving for the unknown process ui(t; ξ) in the SODE systems (2.1) and
(2.8) involves approximating it with the PC expansion
u(t; ξ) ≈ u(t; ξ) =
Nξ∑i=1
ui(t)ϕi(ξ), (2.35)
where ui(t) ∈ Rn are undetermined vector functions of time and ϕi(ξ), i = 1, 2, . . . , Nξ
denotes orthonormal PC basis functions up to order p. ϕ0 = 1, 〈ϕiϕj〉 = δij, where 〈·〉 =∫Γ·ρ(ξ)dξ and δij denotes the Kronecker delta [29]. The number of terms in the expansion
is given by Nξ = (M +p)!/(M !p!). The PC expansion is guaranteed by the Cameron-Martin
theorem [169] to converge for any arbitrary random process with finite second-order moment.
The coefficients ui(t) can be computed non-intrusively or intrusively, from which the
statistics of the response can be assembled. There are two different non-intrusive implemen-
tations. The first recast the gPC coefficients as multidimensional integrals [170, 171] and
apply efficient sampling methods to estimate their values. Using (2.35) as an example, the
coefficients can be computed using the following steps:
1. Generate samples of ξj, j = 1, 2, . . . , N , according to the chosen sampling strategy.
2. For each sample ξj, evaluate u(t; ξj) using the original code for the corresponding
deterministic problem.
3. Using all N samples to numerically evaluate the expectations for the Galerkin projec-
tion coefficients ui = 〈uϕi〉 / 〈ϕ2i 〉, ∀i ∈ 1, 2, . . . , Nξ.
This process reuses the original code for the corresponding deterministic problem as a black-
box thus is straightforward to implement. However, the computational cost is dominated by
the computation of u(t; ξj) for every ξj and potentially very large number of system evalu-
ations will be needed for accurate approximation. Efficient simulation methods introduced
in section 2.4 or sparse quadrature methods in section 2.5 can be applied to reduce the re-
quired number of evaluations N . The second implementation to estimate the PC coefficients
non-intrusively takes a similar approach to the response surface method outlined in section
2.6. It reuses the original code for the corresponding deterministic problem to evaluate the
response at selected set of collocation points, from which the response at other points are
26
approximated using regression [172, 173, 174]. This implementation guarantees accurate
representation at collocation points, but has no explicit control over the error elsewhere.
Alternatively, the intrusive approach uses the stochastic Galerkin projection scheme and
the orthogonal properties of the PC basis to convert the original n-dimensional SODE to
nNξ-dimensional deterministic ODE system that can be solved using classical time-marching
methods. This approach is sometimes referred to as the spectral stochastic finite element
method (SSFEM) in the field of stochastic structural mechanics. Although the implementa-
tion of the intrusive approach is more involved, the computational cost can be much lower
than non-intrusive approaches. This is especially true for high dimensional random spaces
(M 1), and/or when the corresponding deterministic system is already time consuming.
As a result, we will focus on the intrusive approach in the next chapter.
27
Chapter 3
Stochastic Galerkin methods
This chapter discusses stochastic Galerkin methods for solving the SODEs encountered in
linear stochastic structural dynamics introduced in the previous chapter. Although sampling
techniques such as MCS are versatile and general-purpose in scope, their convergence rate
is low and they can be computationally very expensive for systems with a large number of
dof. This has motivated research into alternative approximation methods that can provide
computational cost savings; see, for example, [15, 16, 17, 18, 19] and the references therein.
A particularly popular family of methods and our focus in this chapter are the stochastic
Galerkin methods with history dating back to 1991, when Ghanem and Spanos [29] proposed
the application of polynomial chaos (PC) expansions in conjunction with stochastic Galerkin
projection to solve a range of stochastic operator equations. We will start this chapter with
a detailed introduction to the generalized PC (gPC) expansion scheme in section 3.1. Subse-
quently, the use of gPC expansions together with stochastic Galerkin projection to solve the
linear SODEs will be presented. This is followed by discussions on another decomposition
method, namely the generalized spectral decomposition (GSD) scheme, in section 3.2. Nu-
merical studies are presented to demonstrate the performance of the gPC-based stochastic
Galerkin projection scheme and the GSD method in section 3.3.
3.1 Generalized polynomial chaos expansion
Polynomial chaos (PC) representations of stochastic processes were discussed by Wiener in
the integration theory [162]. The idea was to project the process onto a stochastic sub-
space spanned by the Hermite polynomials, which is complete in the Hilbert space. The
use of Hermite polynomials has a sound mathematical foundation, since its convergence is
28
guaranteed for any arbitrary random process with finite second order moments according to
the Cameron-Martin theorem [169]. Ghanem and Spanos [29] adopted this representation
to facilitate parametric uncertainty analysis. It was demonstrated in [175] that Hermite
polynomials are optimal for the expansion of a Gaussian process with an exponential con-
vergence rate. Nevertheless, a slower convergence rate is observed when using Hermite
polynomials in the expansion of non-Gaussian process [176, 177]. To address this issue,
Xiu and Karniadakis [30] proposed a generalized PC (gPC) expansion constructed using
orthogonal polynomials from the Askey family. Table 3.1 includes a list of different types
of continuous and discrete random variables, the corresponding Askey polynomial chaos ba-
sis functions and their support. It was shown that for the listed continuous and discrete
processes, corresponding Askey polynomials provide faster convergence of the error than
Hermite polynomials. Detailed convergence studies of gPC under different assumptions can
be found in [163, 178]. Discussion on chaos representations for random process with arbitrary
probability measure can be found in [163, 164].
Random variables Askey chaos Support
Continuous Guassian Hermite-Chaos (−∞,∞)Gamma Laguerre-Chaos [0,∞]
Beta Jacobi-Chaos [a, b]Uniform Legendre-Chaos [a, b]
Discrete Poisson Charlier-Chaos 0, 1, 2, . . . Binomial Krawtchouk-Chaos 0, 1, . . . , N+
Negative Binomial Meixner-Chaos 0, 1, 2, . . . Hypergeometric Hahn-Chaos 0, 1, . . . , N+
Table 3.1: Correspondence between different types of continuous and discrete random vari-ables, the Askey polynomial chaos basis function and its support. N+ is a non-negative finiteinteger.
Classical gPC expansions use globally smooth basis functions that sometimes lead to
problems, for example, when Hermite polynomials fail to adequately describe complex so-
lutions such as shock formation or an energy cascade [176], or when the dependence of the
solution on the random input data varies rapidly. In fact, any set of complete basis can be a
viable choice in principle. Le Maıtre et al. [165, 166] proposed the wavelet basis expansion
to approximate smooth and well behaved solutions that may change dramatically or even
discontinuously in the stochastic parameter space. Using such wavelet basis expansions, the
29
resulting decomposition of solution is localized and thus more robust at the cost of slower
rate of convergence. Other generalizations of gPC expansion that alleviate the requirement
of globally smooth basis polynomials include piecewise polynomials basis [167] and multi-
element gPC [168, 164]. For demonstration purposes, we will focus on the classical (globally
smooth) gPC expansion in this section.
Let ξ∞ = (ξ1, ξ2, . . . )T ∈ R∞ denote an infinite set of independent random variables
with probability space (Ω,F ,P), where Ω ⊂ R∞ is the sample space, F is the σ−algebra
associated with Ω and P : F → [0, 1] is a probability measure. Then a random variable
X : Ω→ R can be represented in the following form of gPC expansion [162, 29, 30]
X(ξ∞) = a0H0 +∞∑i1=1
ai1H1(ξi1) +∞∑i1=1
i1∑i2=1
ai1i2H2(ξi1 , ξi2) + . . . , (3.1)
where Hp is a set of orthonormal gPC basis functions of order p from the Askey family, i.e.
H0 = 1, 〈HiHj〉 = δij, where 〈·〉 =∫
Γ·ρ(ξ)dξ and δij denotes the Kronecker delta [29]. The
preceding expansion can be written more compactly as
X(ξ∞) =∞∑k=1
Xiϕi(ξ∞), (3.2)
where there is a one-to-one correspondence between the coefficients and functionals in (3.1)
and (3.2) [29]. The same gPC expansion can be applied to the random vector u ∈ Rn in the
following form:
u(t; ξ∞) =∞∑k=1
ui(t)ϕi(ξ∞). (3.3)
In practice, the gPC expansion is truncated in both order p and stochastic dimension M ,
resulting in the following gPC approximation:
u(t; ξ∞) ≈ u(t; ξ) =
Nξ∑i=1
ui(t)ϕi(ξ), (3.4)
where ξ = (ξ1, ξ2, . . . , ξM)T ∈ RM . ui(t) ∈ Rn are undetermined vector functions of time and
ϕi(ξ), i = 1, 2, . . . , Nξ denotes orthonormal gPC basis functions up to order p. The number
30
of terms in the expansion is given by
Nξ = (M + p)!/(M !p!). (3.5)
With the above gPC expansion (3.4), the unknown system response can be approximated
and the propagation of uncertainty can be characterized by the time-evolving coefficients
ui(t). We have discussed the drawbacks of the non-intrusive approach to solve the expansion
coefficients in section 2.7 and will focus on the intrusive gPC-based Galerkin projection
approach in this section.
3.1.1 gPC-based stochastic Galerkin projection method
The gPC expansion coefficients in (3.4) can be computed using the stochastic Galerkin
projection scheme. To demonstrate, we use the matrix system of second order SODEs arising
in dynamic analysis of linear stochastic structural systems introduced earlier in section 2.1.1
and reproduced here for convenience
M(ξ)u(t; ξ) + C(ξ)u(t; ξ) + K(ξ)u(t; ξ) = f(t; ξ), (3.6)
with initial conditions
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (3.7)
Here M,C and K ∈ Rn×n denote the stochastic mass, stiffness and damping matrices re-
spectively. The external force f(t; ξ) ∈ Rn is a time dependent stochastic process that can
be discretized using the KL expansion scheme outlined in section 2.3. Applying the same
discretization to the random fields used to model material and geometrical variation of the
structure and assuming proportional damping, we arrive at the following equations:
M(ξ) ≈NM∑m=0
Mmϕm(ξ), C(ξ) ≈NC∑m=0
Cmϕm(ξ),
K(ξ) ≈NK∑m=0
Kmϕm(ξ), f(t; ξ) ≈Nf∑m=0
fm(t)ϕm(ξ).
(3.8)
Note that the preceding expansions coincide with the form of gPC expansion. If we sub-
stitute these expansions and the gPC approximation of the solution in (3.4) into (3.6), the
31
undetermined coefficients ui(t) can be solved by applying the stochastic Galerkin projection
scheme that involves the enforcement of the following orthogonality conditionsM(ξ)¨u(t; ξ) + C(ξ) ˙u(t; ξ) + K(ξ)u(t; ξ)− f(t; ξ) ⊥ ϕi(ξ),
u(0; ξ)− Z0(ξ) ⊥ ϕi(ξ),˙u(0; ξ)− Z1(ξ) ⊥ ϕi(ξ).
(3.9)
where i = 1, 2, . . . , Nξ. The resulting second order governing equation is of the form
MU(t) + CU(t) + KU(t) = F(t), (3.10)
where U = (u1, . . . ,uNξ)T ∈ RnNξ denotes the vector of undetermined coefficients in the
gPC expansion (3.4). The coefficient matrices M,C,K are all matrices with Nξ ×Nξ blocks
and each block can be defined as:
Mj,i =
NM∑m=0
Mm 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ, (3.11)
Cj,i =
NC∑m=0
Cm 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ. (3.12)
Kj,i =
NK∑m=0
Km 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ, (3.13)
Since 〈ϕiϕj〉 = δij, F(t) can be written as a vector of length nNξ as follows:
F(t) =(F1(t),F2(t), . . .FNξ(t)
)T,Fj(t) =
fj(t)
⟨ϕ2j
⟩when j = 1, 2...Nf
0 when j = Nf + 1, ...Nξ
(3.14)
The equation resulting from the stochastic Galerkin projection scheme in (3.10) is a
second-order deterministic ODE and can be solved using the Newmark integration scheme [179].
The solution U = (u1, . . . ,uNξ)T can then be used to assemble the gPC approximation of the
solution using (3.4). This approach has been shown to provide good approximations with
computational effort significantly lower than simulation methods for a large class of linear
problems [31, 33, 34] and some specific types of nonlinear problems [165, 180, 181, 182].
Nevertheless, the above procedures convert the original stochastic system to a coupled
deterministic system with much larger number of dof (U(t) ∈ RnNξ versus u(t; ξ) ∈ Rn
32
originally). In some special cases, the coupling between gPC Galerkin equations does not
incur much additional computational cost, for example, the Navier-Stokes equations with
random boundary/initial conditions [183]. In some other cases the gPC Galerkin system can
be decoupled, for an example, the stochastic diffusion equations [184, 185, 186]. Efficient
numerical methods such as Krylov-type iterative techniques were proposed to exploit the
properties of the block diagonal-sparse coefficient matrices [187, 188, 189, 190], but it is
difficult to build efficient preconditioners and the memory requirements limit their use in
low stochastic dimensions with large-scale applications. In general, the application of gPC
Galerkin scheme is limited by its quick growth of computational cost for large systems and/or
large number of random variables.
3.1.2 Remarks
The gPC scheme has been successfully applied to a wide range of steady state and tran-
sient problems such as deformation of elasto-plastic bodies [191], fluid flow simulations using
Euler and Navier-Stokes equation [192, 171, 193, 194], nonlinear vibrations [195] and multi-
phase flows in heterogeneous random media [27, 26]. Nevertheless, theoretical and numerical
developments have revealed concerns as follows:
1. When the stochastic solutions exhibit discontinuity in the random space, gPC basis of
piecewise polynomials (versus the global orthogonal polynomials) are required to avoid
accuracy loss [167, 196, 166, 168]. The challenge is that the location of discontinuity in
random space is not known a priori, especially for dynamical problems. The additional
computational cost associated with partitioning the random space into elements and
assembling solutions in each element to the whole multi-dimensional space through
tensor products can also be very high [168].
2. The accuracy of the gPC expansion decreases in long-term integration [197]. Theo-
retical explanation for this phenomenon is provided in [198]. It is not an inherent
deficiency of gPC expansion but rather a result of the classical approximation theory.
As a result, one needs higher order gPC expansions as time evolves to retain a fixed
accuracy [199], which in turn results in higher computational cost.
3. The computational cost becomes prohibitive when employing high-order gPC expan-
sions for large-scale systems with even a modest number of random variables.
33
3.2 Generalized spectral decomposition scheme
The classical gPC decomposition scheme discussed in the previous section provides a use-
ful tool to solve SODEs. However, it is subject to the curse of dimensionality and various
alternative decomposition methods were proposed to provide computational cost savings.
We will present an example in this section: the generalized spectral decomposition (GSD)
scheme. It was first introduced by Nouy for linear elliptic stochastic partial differential equa-
tions [36] and later extended to a wider class of stochastic problems [37] and low dimensional
nonlinear steady problems [200]. Classical definitions of GSD based on Galerkin or minimal
residual formulations and several improvements, including the Minimax Proper Generalized
Decomposition, are proposed in [201] for the solution of time-dependent PDEs. An attempt
to further circumvent the curse of dimensionality by exploiting the tensor product structure
of stochastic function spaces can be found in [202]. For demonstration purposes, we will
focus on the original GSD scheme introduced in [36, 37] in this section.
The GSD scheme is designed to reduce the computational cost of stochastic Galerkin
methods by using an optimal set of K (K Nξ) terms, each being the product of a
stochastic coefficient by a deterministic function, to approximate the unknown solution. In
the linear SODE system (3.6), the GSD approximation to the solution can be written in the
following form:
u(t; ξ) ≈ u(t; ξ) =K∑k=1
λk(ξ)φk(t). (3.15)
Here λk(ξ), ∀k = 1, 2, . . . , K denote stochastic coefficients and φk(t) ∈ Rn are deterministic
time-dependent vectors. Neither λk(ξ) nor φk(t) are fixed a priori. This decomposition
is similar to the KL expansion (section 2.3) in the sense that it can be interpreted as an
“extended” eigenvalue problem. The KL expansion is optimal in the mean square sense, i.e.
it has the least mean square distance to the underlying stochastic field compared to any other
decomposition with M terms.Directly approximating u(t; ξ) using KL is impossible, since we
have no a priori knowledge of the solution. Instead, we can use the classical GSD scheme [36,
37] to find the optimal GSD expansion by applying ad hoc iterative techniques to build
the approximation and transform the problem into the resolution of alternate deterministic
linear equations and ODEs. To be more specific, starting with a random initial value of
φk(t),∀k = 1, 2, . . . , K we can use the following two-step iterative procedure to compute
λk(ξ) and φk(t):
34
• Step 1: Solve for λk(ξ) given φk(t), k = 1, 2, . . . , K. This step involves the application
of the Galerkin condition 1: ε(ξ) ⊥ φj(t), i.e.,∫ T
0φTj (t)ε(ξ)dt = 0. ∀j = 1, 2, . . . , K.
• Step 2: Solve for φk(t) given λk(ξ), k = 1, 2, . . . , K. This step involves the application
of the Galerkin condition 2: ε(ξ) ⊥ λj(ξ), ∀j = 1, 2, . . . , K.
Here ε(ξ) = u(t; ξ)−u(t; ξ) is the residual error of the GSD expansion in (3.15). In order to
ensure that the expansion in (3.15) is unique, we need to apply the Gram-Schmidt process to
orthogonalize φk(t) and λk(ξ) before applying step 1 and 2 respectively. Additional details
on how these two steps can be carried out for the second order ODE system (3.6) is presented
below.
Step 1: Solve for λ(ξ) given φ(t)
A set of random temporal basis functions φk(t),∀k = 1, 2, . . . , K are initially used in this
step. Once the iteration begins, the solution of φk(t), k = 1, 2, . . . , K in step 2 will be used.
For simplicity we assume NM = NK = NC = Nf = M , i.e., the expansions of M,K,C and f
are all truncated at the first (M + 1) terms. Substituting the expansion of u(t; ξ) in (3.15),
gPC decomposition of coefficient matrices in (3.8) and applying Galerkin condition 1 to the
governing equation (3.6) we have:(M∑m=0
Lmϕm(ξ)
)Nξ∑i=1
λiϕi(ξ) =M∑m=0
ϕm(ξ)(
(rm)1, (rm)2, . . . (r
m)K
)T, (3.16)
where
(Lm)j,k =
∫ T
0
φTj Mmφkdt+
∫ T
0
φTj Kmφkdt+
∫ T
0
φTj Cmφ
T
k dt, ∀j, k = 1, 2, . . . , K.
(rm)j =
∫ T
0
φTj fm(t)dt, ∀j = 1, 2, . . . , K.
Both (Lm)j,k and (rm)j in (3.16) are deterministic. Note that∑Nξ
i=1 λiϕi(ξ) is the gPC
expansion of vector λ(ξ) ∈ RK composed of random coefficients λk(ξ),∀k = 1, 2, . . . , K.
Defining Rm = ((rm)1, (rm)2, ...(r
m)K) and applying the Galerkin orthogonality condition
to (3.16) we arrive at the following deterministic algebraic system of equations with KNξ
unknowns:
LH = R, (3.17)
35
where (L)li =∑M
m=0 Lm 〈ϕmϕiϕl〉 , l, i = 1, 2...Nξ. H = (λ1,λ2, ...λNξ)T ∈ RKNξ and
R = (⟨ϕ2
0
⟩R0,
⟨ϕ2
1
⟩R1, . . . ,
⟨ϕ2M
⟩RM , 0, . . . 0)T . R ∈ RKNξ . (3.18)
Step 2: Solve for φ(t) given λ(ξ)
Substituting the GSD decomposition in (3.15) and the gPC expansion of the coefficient
matrices in (3.8) into the governing equation (3.6) and applying the Galerkin condition 2 we
arrive at the following system of second order deterministic ODE,
NM∑m=0
K∑k=1
Mmφk(t) 〈ϕmλkλj〉+
NC∑m=0
K∑k=1
Cmφk(t) 〈ϕmλkλj〉
+
NK∑m=0
K∑k=1
Kmφk(t) 〈ϕmλkλj〉 =
Nf∑m=0
fm(t) 〈ϕmλj〉 ,∀j = 1, 2, . . . , K.
(3.19)
Comparing (3.19) and (3.10) we can see the similarity between Step 2 of GSD scheme
and the gPC scheme. In fact, we can reuse the code used to solve the ODEs resulting
from the classical stochastic Galerkin projection scheme. The difference is to replace the
corresponding coefficients 〈ϕmϕiϕj〉,⟨ϕ2j
⟩in (3.11)-(3.14) with 〈ϕmλkλj〉, 〈ϕmλj〉.
In practice, the accuracy of the GSD method highly depends on K and the optimal value
of this user defined parameter needs to be carefully chosen to trade off between accuracy and
computational efficiency. In addition, the performance of GSD method will depend as well on
the stopping criterion used to terminate the iterations. In the following numerical studies, the
GSD iterations are stopped when the difference between two successive residual norms (the
residual norm being defined as∫ T
0
∫ΓεTερ(ξ)dξdt ) is smaller than 10−8 or when the number
of iterations reaches a maximal value (taken as 15). There can be alternative stopping
criteria, such as the difference in standard deviation between two successive iterations being
less than certain threshold.
In comparison to the gPC projection scheme, GSD dramatically reduces the memory
requirement and computational cost when the number of modes K Nξ, where Nξ is
the number of gPC expansion modes. But its mathematical foundation remains unclear,
especially regarding the convergence of the outcome of the above iterative steps. In addition,
there are no clear guidelines on the setting of K to guarantee convergence. Numerical studies
suggest that different value of K are needed to describe the solution for various definitions
of GSD [203, 201].
36
3.3 Numerical studies
This section presents three model problems in linear stochastic structural dynamics used for
comparing the performance of gPC-based stochastic Galerkin and GSD methods. Results
are also computed using MCS and serve as reference. The average acceleration Newmark
integration scheme [204, 205] will be used as the time-marching scheme in the following
numerical studies. All the numerical tests are conducted using Matlab codes on a machine
with Intel i7-2600 CPU and 16Gb RAM.
The problems are organized with increasing spatial dof to present the performance of
different methods under different circumstances. In the last example, the number of spatial
dof is relatively high and gPC expansion with order p ≥ 3 would result in a deterministic
system too big to solve using our system. Results show that although the GSD method
can effectively reduce the memory requirements, its computational cost is higher than the
classical gPC-based stochastic Galerkin method for comparable level of accuracy in all three
problems. We define the error of gPC Galerkin method in the mean response as
em(t) = |µgPC(t)− µMCS(t)|, (3.20)
where µgPC(t), µMCS(t) are mean solutions obtained using gPC and MCS at the same dof
at time t, respectively. Similarly, the error in the standard deviation of the response is given
by
es(t) = |σgPC(t)− σMCS(t)|. (3.21)
The errors in the mean and standard deviation of the response computed using the GSD
method are defined similarly.
3.3.1 Spring-mass system
This three-dof spring-mass system features stochastic mass, stiffness and damping coefficient
at each dof:
mi = m0(1 + Cξi), (3.22)
ci = c0(1 + Cξ3+i), (3.23)
ki = k0(1 + Cξ6+i), (3.24)
37
here i ∈ 1, 2, 3, m0 = 1 kg, c0 = 0.2 kg/s2, k0 = 1 kg, C=0.5. ξi ∈ [−1, 1], 1 ≤ i ≤ 9 are
independent uniform random variables. A graphical representation of the system is included
in Figure 3.1.
Figure 3.1: Three-dof spring-mass system with a stochastic excitation force applied to m3.
The three spring-mass systems are connected such that the coefficient matrix for the
dynamics of the whole system are:
M(ξ) =
m1 0 0
0 m2 0
0 0 m3
,C(ξ) =
c1 + c2 −c2 0
−c2 c2 + c3 −c3
0 −c3 c3
,K(ξ) =
k1 + k2 −k2 0
−k2 k2 + k3 −k3
0 −k3 k3
.A stochastic excitation force f(t; ξ) is applied to m3 for t ∈ [0, T ], T = 5 s. f(t; ξ) has the
autocorrelation function
Rff (t1, t2) = σ2fe− |t1−t2|
A , A > 0, (3.25)
where A = 0.1 m is the correlation length and σf = 1 N is the standard deviation of the
process. This random process can be decomposed using a truncated KL expansion (see
section 2.3) up to the 6th random dimension as follows
f(t; ξ) = f(t) + σf
6∑j=1
√λjΨj(t)ξj = f(t) +
6∑j=1
fj(t)ξj. (3.26)
Here f(t) = 23− 2
3sin(2πt) exp(−0.1t)(N) is the mean of the process. λi and Ψi(t) are the
eigenvalues and eigenfunctions of a Fredholm integral equation of the second kind given by∫ T0Rff (t1, t2)Ψi(t1)dt1 = λiΨi(t2). λi are ordered such that λ1 ≥ λ2 ≥ · · · ≥ λ6. The
stochastic excitation used for this model problem is presented in Figure 3.2.
With stochastic mass, stiffness and damping coefficient at each dof, plus six stochastic
dof from the excitation force, this three-dof spring-mass system has total stochastic dof
M = 15. Zero initial conditions are specified and the dynamic response of the third dof over
the interval [0, T ], T = 5 s is studied. The CPU time required by MCS of sample size 106 is
5163.96 s for this model problem.
38
Figure 3.2: Stochastic excitation with mean f(t) = 2− 2 sin(2πt) exp(−0.1t) N and stochas-tic correlation length A = 0.1 m, standard deviation σf = 1 N. Grey area indicates themean±standard deviation. The standard deviation is smaller than σf because the KL ex-pansion is truncated at the first 6 terms.
Application of the gPC Galerkin scheme
The absolute error in mean and standard deviation of the solutions obtained using gPC-based
stochastic Galerkin projection with gPC order p = 1, 2, 3 are presented in Figures 3.3 and 3.4.
It can be observed that the higher-order gPC approximations are in better agreement with
those obtained using MCS. The CPU time required by the gPC-based stochastic Galerkin
projection scheme with p = 1, 2, 3 are 0.27, 1.46 and 7.73 s, respectively.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
10-8
10-6
10-4
10-2
Ab
so
lute
err
or
in m
ea
n
gPC,p=1
gPC,p=2
gPC,p=3
Figure 3.3: Spring-mass system: absolute error in mean response as a function of time fordifferent orders of gPC-based stochastic Galerkin projection schemes.
39
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
10-8
10-6
10-4
10-2
100
Ab
so
lute
err
or
in s
tan
da
rd d
evia
tio
n
gPC,p=1
gPC,p=2
gPC,p=3
Figure 3.4: Spring-mass system: absolute error in standard deviation of displacement as afunction of time for different orders of gPC-based stochastic Galerkin projection schemes.
Application of the GSD scheme
The GSD scheme was applied with K = 5, 10 and the absolute error in mean and standard
deviation of solutions are shown in Figures 3.5 and 3.6, where the corresponding errors for
the second-order gPC scheme are also included as a reference. The CPU time required by
GSD with K = 5, 10 are 230.38 and 451.00 s, respectively. This is significantly higher
compared to the CPU time required by the second-order gPC-based stochastic Galerkin
projection scheme (1.46 s). Figures 3.5 and 3.6 show that using GSD with K = 10 to predict
the response results in higher level of error in both mean and standard deviation than the
second-order gPC scheme. Using K > 10 in the GSD approximation could potentially result
in similar level of accuracy to second-order gPC scheme at even higher CPU time.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
10-8
10-6
10-4
10-2
100
Ab
so
lute
err
or
in m
ea
n
gPC,p=2
GSD,Kmodes=5
GSD,Kmodes=10
Figure 3.5: Spring-mass system: absolute error in mean of displacement em(t) = |µGSD(t)−µMCS(t)| as a function of time for the GSD schemes with different values of K.
40
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
100
Ab
so
lute
err
or
in s
tan
da
rd d
evia
tio
n
gPC,p=2
GSD,Kmodes=5
GSD,Kmodes=10
Figure 3.6: Spring-mass system: absolute error in standard deviation of displacement es(t) =|σGSD(t)− σMCS(t)| as a function of time for the GSD schemes with different values of K.
3.3.2 Two-dimensional linear beam problem
In this test case we examine the dynamic response of a two-dimensional linear beam to
periodic excitation. The beam of length L = 10 m, width W = 1 m is cantilevered at one
end. It is made of aluminum with shear modulus G = 26 GPa, Poisson ratio ν = 0.35 and
mass density ρ = 2.70×103 kg/m3. We model the Young’s modulus of the beam as a random
field with mean E0 = 70 GPa. A mass proportional damping model is used in the analysis,
i.e., C = αdMM, where the constant αdM = 10. The finite element (FE) mesh used for this
problem has total number of dof n = 88.
We study the dynamic response over the time interval [0,T], T=2 s with zero ini-
tial conditions for the displacement and velocity. A time-dependent force of the form
f (1− [1− sin(bt)] · e−at) is applied to the upper right tip of the beam in the x2 direction,
where f = 105 N. The constants a and b are used to control the amplitude and the period of
the source term. In our test case we take a = 3T
, b = 20πT
. The time-dependent part of force
is depicted in Figure 3.7(b).
The Young’s modulus is a random field and denoted by E(x, ω), where x = (x1, x2)T
denotes the spatial coordinates and ω ∈ Ω. The covariance function is chosen to be
C(x,y) = σ2 exp
(−|x1 − y1|
c1
− |x2 − y2|c2
), (3.27)
where y = (y1, y2)T , σ is the standard deviation of the field and c1 and c2 are the correlation
lengths in x1 and x2 directions, respectively. In this test case we set σ = 0.1 N and c1 = c2 = 1
m. In section 2.3 we discussed random field discretization methods and the following form
41
0 2 4 6 8 10−1
0
1
2
3
4
5
6
7
x1(m)
x2(m
)Upper−right Tip
(a) Model linear elastic beam at T=2 s.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
τ
force
Mean
Mean+/−std
(b) Time-dependent part in the external forcef(t), t ∈ [0, T ].
Figure 3.7: 2D beam clamped at one end and external force applied to its upper-right tipin the x2 direction. (a) Deformed configuration of the beam at T = 2 s with displacementamplified by a factor of 103. (b) Time-dependent part of the external force with T = 2s, a =3T, b = 20π
T.
of truncated KL expansion is used to discretize E(x, θ):
E(x, θ) ≈ E0 +M∑m=1
ξm(θ)Em(x). (3.28)
Here ξm are independent uniform random variables in the interval [−1, 1]. Em(x),m =
1, 2, . . . ,M are a set of basis functions that can be written as Em(x) = σ√λmΨm(x), where
λm and Ψm(x) are the eigenvalues and eigenfunctions of the Fredholm integral equation of
the second kind. Eigenvalues λm are ordered such that λ1 ≥ λ2 ≥ · · · ≥ λM . The resulting
constitutive matrix can be written in the following form
D(x, ξ) ≈ D0 +M∑m=1
ξmDm(x). (3.29)
The stochastic element stiffness matrix is given by
ke =
∫De
BTD(x, ξ)Bdx, (3.30)
where B is the strain-displacement matrix and De denotes the domain of the element. Sub-
stituting (3.29) in (3.30) leads to the following expression for the stochastic element stiffness
42
matrix ke(ξ) = ke0 +∑M
m=1 kemξm, where ke0 =
∫De
BTD0Bdx and kem =∫De
BTDmBdx. Stan-
dard numerical quadrature schemes can be used to evaluate these integrals (for more details,
see [206, 29]). Assembling the element stiffness matrices and accounting for the specified
boundary conditions result in the following expansion for the global stiffness matrix
K(ξ) = K0 +M∑m=1
Kmξm, (3.31)
where K0 ∈ Rn×n,Km ∈ Rn×n are deterministic matrices. In this case study, the mass
and damping matrices are assumed to be deterministic. We shall compare the accuracy of
different numerical schemes for approximating the statistics of the displacement component
in the x2 direction at the upper right tip of the beam (see Figure 3.7(a)). M is set to be 5
and MCS with sample size of 106 costs 1.96× 105 s.
Application of the gPC Galerkin scheme
The absolute error in mean and standard deviation of solutions obtained using the gPC-
based stochastic Galerkin projection with gPC order p = 1, 2, 3 are presented in Figures
3.8 and 3.9. Similar to the previous problem, it can be observed that the higher-order gPC
approximations result in lower level of error in both mean and standard deviation. The CPU
time required by the gPC-based stochastic Galerkin projection scheme with p = 1, 2, 3 are
0.22, 3.93 and 50.33 s, respectively. It is to be noted that in this problem, the computational
cost associated with p = 3 is significantly higher than that of p = 2. This is often observed
in systems with larger dof and as a result, gPC scheme with p = 2 is typically considered a
balanced choice between accuracy and cost.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210
−10
10−9
10−8
10−7
10−6
10−5
t(s)
Absolu
te e
rror
in m
ean
gPC,p=1gPC,p=2gPC,p=3
Figure 3.8: Two-dimensional beam: absolute error in mean response (3.20) as a function oftime for gPC-based stochastic Galerkin projection schemes.
43
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210
−9
10−8
10−7
10−6
10−5
10−4
t(s)
Absolu
te e
rror
in s
tandard
devia
tion
gPC,p=1gPC,p=2gPC,p=3
Figure 3.9: Two-dimensional beam: absolute error in standard deviation of displacement(3.21) as a function of time for gPC-based stochastic Galerkin projection schemes.
Application of the GSD scheme
The absolute error in mean and standard deviation of solutions obtained using GSD scheme
are shown in Figures 3.10 and 3.11, where the corresponding errors for the second-order gPC
scheme are also included as a reference. It can be observed that approximation errors for
the GSD scheme decrease when increasing the number of modes (K). When K = 20, the
accuracy of GSD is comparable to the second-order gPC method. The CPU time required
by GSD with K = 10, 15, and 20 are 215, 372 and 559 s, respectively. For this particular
problem, GSD turns out to be less efficient than the second-order gPC method which required
3.93 s. In the two problems we tested so far, GSD does not offer any improvement in
performance compared to gPC-based stochastic Galerkin methods. Nevertheless, GSD is
designed to outperform gPC-based Galerkin projection method for much larger systems
with high spatial and/or stochastic dofs, and we will move on to a larger system next.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t(s)
10-10
10-8
10-6
Absolu
te e
rror
in m
ean
gPC,p=2
GSD,Kmodes=10
GSD,Kmodes=15
GSD,Kmodes=20
Figure 3.10: Two-dimensional beam: absolute error in mean of displacement em(t) =|µGSD(t) − µMCS(t)| as a function of time for the GSD schemes with different values ofK.
44
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t(s)
10-10
10-8
10-6
10-4
Absolu
te e
rror
in s
tandard
devia
tion
gPC,p=2
GSD,Kmodes=10
GSD,Kmodes=15
GSD,Kmodes=20
Figure 3.11: Two-dimensional beam: absolute error in standard deviation of displacementes(t) = |σGSD(t)− σMCS(t)| as a function of time for the GSD schemes with different valuesof K.
3.3.3 Three-dimensional hexahedron problem
In this three-dimensional test case, we consider a hexahedron shaped linear structure with
two cylinder shaped holes as shown in Figure 3.12. Spatial discretization is carried out
using second-order (10 node) tetrahedral elements leading to a total of 4, 446 dof. Mass
proportional damping is considered (C = 10M) and the material properties are taken to the
same as in the two-dimensional beam example (see section 3.3.2). One face of the structure
is clamped at x = 0. A time-dependent force f(t) of the same form considered in the beam
example with net magnitude of 105 N is evenly applied to another face (x = 10) in the
y direction, with a = 3T, b = 20π
T. The dynamic response of the node (x, y, z) = (10, 2, 1)
(red point in Figure 3.12) over the interval [0, T ], T=1 s is considered with a null initial
displacement and velocity.
Figure 3.12: Hexahedron shaped structure with two cylinder shaped holes. One cylinder iscentered at x = 1.5, y = 1.5 with radius r = 1, another one at x = 6, y = 1.5 with radiusr = 0.6. Both cylinders have the same height as the hexahedron, which is 1.
45
Similar to the two-dimensional example, the Young’s modulus is treated as a random
field with mean value E0=70 GPa and a covariance function of the form
C(x,y) = σ2 exp
(−|x1 − y1|
c1
− |x2 − y2|c2
− |x3 − y3|c3
), (3.32)
with σ = 0.01 and c1 = c2 = c3 = 1. The random field is discretized using the KL
expansion scheme (see section 2.3) and M = 5 terms are retained in the expansion. The
random variables are considered to be uniformly distributed in the interval [−1, 1]. MCS
with sample size N = 105 cost 1.388× 105 s.
Application of the gPC Galerkin scheme
The absolute error in mean and standard deviation of solutions obtained using the gPC-based
stochastic Galerkin projection with gPC order p = 1, 2 are presented in Figures 3.13 and
3.14. We can see that second-order gPC results in better accuracy in standard deviation. The
CPU time required by the gPC-based stochastic Galerkin projection scheme with p = 1, 2
are 1.408 × 102 and 3.395×103 s, respectively. gPC expansion with p ≥ 3 results in more
expansion terms and a larger deterministic system to solve. In this relatively bigger test case,
p = 3 is not feasible since we have an “out of memory” error from Matlab. It is possible to
supply more memory to test gPC-based Galerkin method with p = 3 or even higher. But
we expect the associated CPU time to increase drastically and we need better methods to
alleviate the curse of dimensionality for even bigger problems.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-12
10-10
10-8
Absolu
te e
rror
in m
ean
gPC,p=1
gPC,p=2
Figure 3.13: Three-dimensional hexahedron: absolute error in mean response (3.20) as afunction of time for gPC-based stochastic Galerkin projection schemes.
46
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-10
10-5
Absolu
te e
rror
in s
tandard
devia
tion
gPC,p=1
gPC,p=2
Figure 3.14: Three-dimensional hexahedron: absolute error in standard deviation of displace-ment (3.21) as a function of time for gPC-based stochastic Galerkin projection schemes.
Application of the GSD scheme
GSD with K = 5, 10, 15 modes are applied to this problem. The absolute errors in the mean
and standard deviation of the response are shown in Figures 3.15 and 3.16, respectively.
Error associated with second-order gPC scheme is also included for comparison. It can be
seen from Figure 3.15 that the error in the mean of the response computed using the different
schemes are comparable. Although higher value of K lead to better accuracy, we can see from
Figure 3.16 that second-order gPC has better accuracy in predicting the standard deviation
of the response than GSD with K = 15. The CPU time required by GSD method with
K = 5, 10, 15 are 2.387×104, 9.582×104 and 2.181×105 s, respectively. Compared with the
second-order gPC scheme that costs 3.395×103 s, it is clear that GSD offers less accuracy at
a much higher cost in this test case.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-12
10-10
10-8
Absolu
te e
rror
in m
ean
gPC,p=2
GSD,K=5
GSD,K=10
GSD,K=15
Figure 3.15: Three-dimensional hexahedron: absolute error in mean of displacement em(t) =|µGSD(t)− µMCS(t)| as a function of time for the GSD schemes with different values of K.
47
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-10
10-8
10-6
Absolu
te e
rror
in s
tandard
devia
tion
gPC,p=2
GSD,K=5
GSD,K=10
GSD,K=15
Figure 3.16: Three-dimensional hexahedron: absolute error in standard deviation of dis-placement es(t) = |σGSD(t) − σMCS(t)| as a function of time for the GSD schemes withdifferent values of K.
3.4 Concluding remarks
In this chapter we introduced the classical gPC-based stochastic Galerkin projection method
and applied it to solve a SODE system arising in structural dynamic analysis. Its application
to systems with large number of stochastic dof is limited by its quickly growing computational
cost, and the GSD method was proposed to reduce the computational cost by using reduced
number of optimal basis in the decomposition.
However, the GSD method relies on ad hoc iterative techniques without clear mathe-
matical description of the convergence properties. In the three test cases it is found to be
slower than the second-order gPC Galerkin projection method for the same or lower level of
accuracy. It is worth mentioning that in the numerical studies we implemented the classical
GSD introduced in [36] and it may be possible to speed up the GSD scheme further using a
different numerical procedure for estimating the component functions [201], or using another
stopping criterion to terminate the GSD iterations. Nevertheless, the computational cost of
the GSD scheme directly hinges on K but there lacks clear guidance on its setting to guar-
antee convergence. Because of these limitations, we will exclude GSD for numerical studies
from this point onwards. The challenges posed by high-dimensional stochastic SODEs call
for methods that can alleviate the curse of dimensionality more effectively and that brings
us to the anchored ANOVA Petrov-Galerkin method presented in the next chapter.
48
Chapter 4
Anchored ANOVA Petrov-Galerkin
scheme for linear stochastic structural
dynamics
In this chapter, we propose anchored ANOVA Petrov-Galerkin (AAPG) projection schemes
to efficiently solve high-dimensional SODEs encountered in linear stochastic structural dy-
namics. The AAPG scheme was originally developed in the context of linear parabolic
SPDEs [52]. In this chapter, we consider the semi-discrete form of the governing equations
in the time-domain that was introduced in section 2.1.1. We propose to approximate the
dynamic response using a Hoeffding functional analysis of variance decomposition. Subse-
quently, we consider the weighted residual form of the governing SODEs and design a set of
test functions for a stochastic Petrov-Galerkin projection scheme that enables the original
high-dimensional problem to be decomposed into a sequence of decoupled low-dimensional
subproblems that can be solved independently of each other. Numerical results are presented
to demonstrate the efficiency and accuracy of AAPG projection schemes and comparisons
are made to approximations obtained using MCS and gPC-based stochastic Galerkin pro-
jection schemes. The results obtained suggest that the proposed approach holds significant
potential for alleviating the curse of dimensionality encountered in tackling high-dimensional
problems in stochastic structural dynamics with a large number of spatial and stochastic dof.
49
4.1 Anchored ANOVA decomposition
We begin with approximating the solution u(t; ξ) of a system of SODEs modeling linear
stochastic structural dynamics using a Hoeffding functional ANOVA decomposition [43, 44,
45, 46] of the form
u(t; ξ)≈u0(t) +M∑j1=1
uj1(t; ξj1) +M∑
j1<j2
uj1j2(t; ξj1 , ξj2) +M∑
j1<j2<j3
uj1j2j3(t; ξj1 , ξj2 , ξj3) + · · ·
(4.1)
The first term u0(t) in the decomposition is the zero-order component function which is a
deterministic function of time. The first-order component function uj1 is the independent
contribution to u(t; ξ) by the random variable ξj1 acting alone. The second-order compo-
nent function uj1j2 denotes the pair correlated contribution to the solution by ξj1 and ξj2 .
Similarly, the higher-order terms in (4.1) denote higher-order correlated contributions by
subsets of random variables. Note that all the component functions in the decomposition
are functions of time.
The main advantage of using a functional ANOVA decomposition arises from the fact
that in various practical applications, higher-order interactions between random variables
can be neglected, meaning that a small truncation order can be used in (4.1). In a wide
range of high-dimensional problems this feature can be exploited to design highly efficient
numerical schemes [46]. Several examples are examined in [46] showing that in practice the
truncation order L is generally small in applications such as molecular dynamics simulations
or statistics (typically 2 ≤ L ≤ 4). However, for data mining applications, the higher-order
interactions may need to be considered (L ≥ 7) [160]. We shall later show via numerical
studies that a second-order functional ANOVA decomposition (L = 2) can provide good
accuracy for a set of case studies in stochastic linear structural dynamics.
The L-th order functional ANOVA decomposition can be compactly rewritten as follows
u(t; ξ) ≈ uL(t; ξ) = u0(t) +L∑k=1
M∑j1<j2<···<jk
uj1j2...jk(t; ξj1 , ξj2 , . . . , ξjk), 1 ≤ L ≤M. (4.2)
To ensure the uniqueness of the ANOVA decomposition, the following orthogonality
50
constraints are imposed on the component functions
⟨uj1···js(t; ξj1 , . . . , ξjs),u
k1···kp(t; ξk1 , . . . , ξkp)⟩µ
= 0, for (j1, . . . , js) 6= (k1, . . . , kp). (4.3)
Here 〈·, ·〉µ denotes the L2 inner product 〈w1,w2〉µ =∫
ΓwT
1 (ξ)w2(ξ)dµ(ξ). The preceding
orthogonality constraint is equivalent to imposing a null integral constraint of the form∫Γk
uj1...jp(t; ξj1 , . . . , ξjp) dµk(ξk) = 0, ∀k ∈ j1, . . . , jp. (4.4)
In an anchored functional ANOVA decomposition, dµ(ξ) is taken to be a Dirac product
measure of the form [46, 45]
dµ(ξ) =M∏j=1
dµj(ξj) =M∏j=1
δ(ξj − ξaj )dξj, (4.5)
where δ(·) denotes the Dirac delta and ξa = (ξa1 , ξa2 , . . . , ξ
aM)T ∈ ΓM is the so-called anchor
point. It was shown in [207, 45] that when using the Dirac product measure (4.5), the ANOVA
component functions in (4.2) can be written in terms of point evaluations as follows:
u0(t) = u(t; ξa),
uj1(t; ξj1) = u(t; ξaj1)− u0(t),
uj1j2(t; ξj1 , ξj2) = u(t; ξaj1j2)− uj1(t; ξj1)− uj2(t; ξj2)− u0(t),
(4.6)
or more generally,
uj1···jk(t; ξj1 , · · · , ξjk) = u(t; ξaj1...jk)
−∑
i1<···<ik−1,il∈Ik
ui1···ik−1(t; ξi1 , · · · , ξik−1)
−∑
i1<···<ik−2,il∈Ik
ui1···ik−2(t; ξi1 , · · · , ξik−2) (4.7)
...
−∑i1∈Ik
ui1(t; ξi1)− u0(t),
with Ik = j1, j2, · · · , jk. We denote by u(t; ξaj1...jk) the evaluation of u at the point ξaj1...jk ,
where ξi, i ∈ Ik are active random variables and ξi = ξai for i 6∈ Ik. In addition, the null
51
integral constraints (4.4) using the Dirac product measure (4.5) becomes
uj1...jk |ξi=ξai = 0,∀i ∈ Ik. (4.8)
The above property will be used later in the proof of Theorem 4.2.1.
We shall next devise a stochastic Petrov-Galerkin projection scheme to set up equations
that govern the evolution in time of the component functions in (4.2).
4.2 Anchored ANOVA Petrov-Galerkin (AAPG) pro-
jection scheme
In this section we introduce the stochastic weighted residual form of (4.9) and show that by
defining an appropriate test space, the original high-dimensional problem can be decoupled
into stochastic low-dimensional subproblems that can be solved independently of each other.
This is an important property of AAPG projection schemes that enables the development of
parallel numerical implementations that scale very well to high-dimensional problems [52].
4.2.1 Approximating the weighted residual solution of SODE
The following systems of second-order stochastic ordinary differential equations (SODEs)
arising in linear stochastic structural dynamics of the form (2.1) will be used throughout
this chapter. The semi-discrete form of the governing equation is reproduced below for the
sake of convenience
M(ξ)u(t; ξ) + C(ξ)u(t; ξ) + K(ξ)u(t; ξ) = f(t; ξ) a.s. in [0, T ]× ΓM , (4.9)
where u(t; ξ) ∈ Rn, M(ξ),C(ξ),K(ξ) ∈ Rn×n. The external force f(t; ξ) ∈ Rn is assumed
to be a time-dependent stochastic process. ξ = (ξ1, ξ2, . . . , ξM)T ∈ ΓM ⊂ RM are a set
of independent and identically distributed (i.i.d.) random variables whose joint pdf can be
written as ρ(ξ) =∏M
i=1 ρi(ξi). Initial conditions are
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (4.10)
52
In order to solve (4.9) we introduce the following stochastic weighted residual form
Find u(t; ·) ∈ U such that 〈v,M(ξ)u + C(ξ)u + K(ξ)u− f(t; ξ)〉 = 0, ∀v ∈ V, (4.11)
where U and V denote the trial and test spaces, respectively. The inner product 〈·, ·〉 is
defined as
〈w1,w2〉 =
∫Γ
w1(ξ)Tw2(ξ)ρ(ξ)dξ. (4.12)
Typically, U can be defined as the space of square integrable vectorial functions on ΓM with
respect to the pdf measure, i.e., U = L2(ΓM)n = w(ξ) ∈ Rn,∫
Γw(ξ)Tw(ξ)ρ(ξ)dξ < +∞.
To approximate the solution using an anchored ANOVA decomposition, the space of trial
functions denoted by UANOVA can be defined as the following direct sum [52]
UANOVA = Vnξ,0 ⊕
(N⊕j1=1
Vnξ,j1
)⊕
(N⊕
j1<j2
Vnξ,j1j2
)⊕ · · · ⊕ Vnξ,j1j2...jN , (4.13)
where the subspaces Vnξ,0, Vnξ,j1 , Vnξ,j1j2
. . . are defined as the following tensor product spaces
Vnξ,0 = 11 ⊗ · · · ⊗ 1N = w ∈ Rn,
Vnξ,j1 = 11 ⊗ · · · ⊗Wξ,j1 ⊗ · · · ⊗ 1N =
w(ξj1) ∈ Rn,∫
Γj1w(ξj1)dµj1(ξj1) = 0
,
Vnξ,j1j2 = 11 ⊗ · · · ⊗Wξ,j1 ⊗ · · · ⊗Wξ,j2 ⊗ · · · ⊗ 1N
=
w(ξj1 , ξj2) ∈ Rn,∫
Γj1
∫Γj2
w(ξj1 , ξj2)dµj1(ξj1)dµj2(ξj2) = 0,
(4.14)
and so on. 1j spans constant functions with respect to the j-th coordinate ξj and Wξ,j =
w : Γj → Rn,∫
Γjw(ξj)dµj(ξj) = w(ξaj ) = 0.
In practical computations, we shall use the following L-th order ANOVA space
ULANOVA = Vnξ,0 ⊕
(N⊕j1=1
Vnξ,j1
)⊕
(N⊕
j1<j2
Vnξ,j1j2
)⊕ · · · ⊕
(N⊕
j1<···<jL
Vnξ,j1...jL
)(4.15)
instead of the full ANOVA expansion in (4.13). The test space V that we shall use was
originally proposed in [52] in the context of parabolic SPDEs. The main difference compared
to [52] is that since we are dealing with SODEs, the trial and test spaces do not depend on
spatial coordinates. The space of test functions corresponding to the L-th order truncated
53
ANOVA decomposition is defined as
V L = V0 ⊕
(N⊕j1=1
Vj1
)⊕
(N⊕
j1<j2
Vj1j2
)⊕ · · · ⊕
(N⊕
j1<···<jL
Vj1...jL
), (4.16)
with
V0 = wδ(ξ − ξa),w ∈ Rn, ξa ∈ Γ ,
Vj1 =
w(ξj1)
N∏i 6=j1
δ(ξi − ξai ),w ∈ L2(Γj1)n, ξai ∈ Γi
,
...
Vj1...jL =
w(ξj1 . . . ξjL)
N∏i 6∈IL
δ(ξi − ξai ),w ∈ L2(Γj1 × · · · × ΓjL)n, ξai ∈ Γi, IL = j1, j2, . . . , jL
,
where δ(ξ − ξa) =∏N
i=1 δ(ξi − ξai ).
In summary, when using the L-th order truncated ANOVA approximation (4.2) for the
solution of the weighted residual form (4.11), the resulting (Petrov-Galerkin) weighted resid-
ual form can be stated as
Find uL(t; ·) ∈ ULANOVA such that⟨
v,M(ξ)uL + K(ξ)uL + C(ξ)uL − f(t; ξ)⟩
= 0, ∀v ∈ V L, (4.17)
where ULANOVA, V L are defined by (4.15) and (4.16), respectively. Initial conditions for uL
and uL can be expressed in terms of Z0 and Z1 evaluated at the anchor point using (4.10),
(4.6), and (4.8) as follows
uL(0; ξ) = Z0(ξa) +M∑j1=1
(Z0(ξaj1)− Z0(ξa)
)+ · · · (4.18)
uL(0; ξ) = Z1(ξa) +M∑j1=1
(Z1(ξaj1)− Z1(ξa)
)+ · · · (4.19)
Next, we shall show that using an L-th order anchored functional ANOVA decomposition
of u(t; ξ) along with the test space defined earlier leads to a system of decoupled low-
dimensional subproblems.
54
Theorem 4.2.1. Consider the L-th order truncated anchored ANOVA approximation uL for
the solution of the weighted residual form (4.17). Let u0,uj1 , . . . ,uj1...jL be the component
functions of the anchored ANOVA decomposition (4.2) that are subject to the null integral
constraints (4.8). If the test functions are chosen from the space V L defined in (4.16) then
the zero-order component function u0 satisfies the deterministic system of ODEs
M(ξa)u0 + C(ξa)u0 + K(ξa)u0 = f(t; ξa), (4.20)
with the initial conditions u0(0) = Z0(ξa) and u0(0) = Z1(ξa), where Z0,Z1 are defined in
(4.10).
The higher order ANOVA component functions uj1...jk , k = 1, 2, . . . , L, are given by
uj1...jk = uj1...jk − u0 −∑l1∈Ik
ul1 −∑
l1<l2,li∈Ik
ul1l2 − · · · −∑
l1<l2···<lk−1,li∈Ik
ul1l2...lk−1 , (4.21)
where Ik = j1, j2, . . . , jk, and the auxiliary variable uj1...jk is the solution of the following
low-dimensional system of SODEs (with k random variables)
M(ξaj1...jk)¨uj1...jk
+ C(ξaj1...jk)˙uj1...jk
+ K(ξaj1...jk)uj1...jk = f(t; ξaj1...jk), (4.22)
with the initial conditions uj1...jk(0; ξj1 , . . . , ξjk) = Z0(ξaj1...jk) and ˙uj1...jk
(0; ξj1 , . . . , ξjk) =
Z1(ξaj1...jk).
Proof. The proof uses ideas from Theorem 1 in [52], which presents a similar result for
parabolic SPDEs. For simplicity of notation, we shall first rewrite (4.17) in the compact
form
Find uL(·, t) ∈ ULANOVA such that
⟨v, a(uL, uL, uL; ξ)− f(ξ, t)
⟩= 0,∀v ∈ V L, (4.23)
where a(uL, uL, uL; ξ) = M(ξ)uL + C(ξ)uL + K(ξ)uL. Expanding uL and by definition of
a, we have
a(uL, uL, uL; ξ) = a(u0, u0, u0; ξ) +L∑k=1
N∑j1<···<jk
a(uj1...jk , uj1...jk , uj1...jk ; ξ). (4.24)
55
Hence, the weighted residual form (4.23) can be written as
⟨v, a(u0, u0, u0; ξ)
⟩+
L∑k=1
N∑j1<···<jk
⟨v, a(uj1...jk , uj1...jk , uj1...jk ; ξ)
⟩= 〈v, f(ξ, t)〉 , ∀v ∈ V L.
(4.25)
To prove (4.20) we consider test-functions v ∈ V L ∩ V0 = V0 of the form v(ξ) = wδ(ξ− ξa),
with w ∈ Rn. Since the ANOVA component functions satisfy the null integral property
(4.8), we have uj1...jk |ξ=ξa = 0,∀k ≥ 1 and similarly uj1...jk |ξ=ξa = 0, uj1...jk |ξ=ξa = 0,∀k ≥ 1.
As a result, (4.25) becomes the deterministic weighted residual equation
wT(a(u0, u0, u0; ξa)− f(ξa, t)
)= 0,∀w ∈ Rn, (4.26)
which implies a(u0, u0, u0; ξa) = f(ξa, t), i.e., equation (4.20). We shall next consider test-
functions v ∈ V L ∩Vj1...jk = Vj1...jk that can be written as v(ξ) = w(ξj1 , . . . , ξjk)∏N
i 6∈Ik δ(ξi−ξai ), with Ik = j1, . . . , jk, w ∈ L2(Γj1 × · · · × Γjk)
n. We expand the first term in (4.25) as
⟨v, a(u0, u0, u0; ξ)
⟩= cj1...jk
∫Γj1×···×Γjk
wTa(u0, u0, u0; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk , (4.27)
with cj1...jk =∏
i 6∈Ik ρi(ξai ) > 0. The first-order terms in (4.25) can be written as
N∑j′1=1
⟨v, a(uj
′1 , uj
′1 , uj
′1 ; ξ)
⟩
= cj1...jk
N∑j′1=1
∫Γj1×···×Γjk
wTa(uj′1 , uj
′1 , uj
′1 ; ξaj1...jk)
∏i∈Ik
ρi(ξi)dξj1 . . . dξjk .
(4.28)
From the null integral constraints (4.8), we have uj′1|ξaj1...jk = 0 for j′1 6∈ Ik, uj
′1 |ξaj1...jk = uj
′1
for j′1 ∈ Ik, and similar conditions hold for the derivatives uj′1 and uj
′1 . Hence (4.28) reduces
to the following summation with k terms
cj1...jk∑l1∈Ik
∫Γj1×···×Γjk
wTa(ul1 , ul1 , ul1 ; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk . (4.29)
56
Similarly, the second-order terms in (4.25) are given by
N∑j′1<j
′2
⟨v, a(uj
′1j′2 , uj
′1j′2 , uj
′1j′2 ; ξ)
⟩
= cj1...jk
N∑j′1<j
′2
∫Γj1×···×Γjk
wTa(uj′1j′2 , uj
′1j′2 , uj
′1j′2 ; ξaj1...jk)
∏i∈Ik
ρi(ξi)dξj1 . . . dξjk ,
(4.30)
which simplify to the following summation with k(k−1)2
terms
cj1...jk∑
l1<l2,li∈Ik
∫Γj1×···×Γjk
wTa(ul1l2 , ul1l2 , ul1l2 ; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk , (4.31)
since the null integral constraints (4.8) gives uj′1j′2 |ξaj1...jk = 0 for j′1 6∈ Ik or j′2 6∈ Ik,
uj′1j′2 |ξaj1...jk = uj
′1j′2 for j′1, j
′2 ∈ Ik (the same conditions hold for the derivatives uj
′1j′2 and
uj′1j′2). Similar arguments can be used when considering higher order terms in (4.25).
Gathering all the terms obtained by expanding (4.25) such as (4.27), (4.29) and (4.31),
using the fact that cj1...jk 6= 0 and introducing the auxiliary variable uj1...jk defined in (4.21),
we recover the following low-dimensional stochastic weighted residual form
Find uj1...jk ∈ L2(Γj1 × · · · × Γjk)n such that⟨
w, a(uj1...jk , ˙uj1...jk
, ¨uj1...jk
; ξaj1...jk)− f(ξaj1...jk , t)⟩
= 0,∀w ∈ L2(Γj1 × · · · × Γjk)n, (4.32)
which implies the strong form (4.17). The initial conditions for u0, u0 and uj1...jk , ˙uj1...jk
, k =
1, . . . , L, follow from the combination of (4.6), (4.8), (4.10) and (4.21). For example, we have
u0(0) = u(ξa, 0) = Z0(ξa), (4.33)
u0(0) = u(ξa, 0) = Z1(ξa), (4.34)
uj1(ξj1 , 0) = uj1(ξj1 , 0) + u0(0) = u(ξaj1 , 0)− u0(0) + u0(0) = Z0(ξaj1), (4.35)
˙uj1
(ξj1 , 0) = uj1(ξj1 , 0) + u0(0) = u(ξaj1 , 0)− u0(0) + u0(0) = Z1(ξaj1). (4.36)
This completes the proof.
It is worth noting that the low-dimensional stochastic subproblems governing the auxil-
iary variables uj1...jk , k = 1, 2, . . . , L, can be solved in parallel, independently of each other.
And the resulting solutions of subproblems are post-processed to get the ANOVA component
57
functions using the following steps:
uj1 = uj1 − u0, j1 = 1, . . . , N,
uj1j2 = uj1j2 − uj1 − uj2 − u0, 1 ≤ j1 < j2 ≤ N,
...
uj1...jk = uj1...jk −∑
l1<···<lk−1,li∈Ik
ul1...lk−1 − · · · −∑l1∈Ik
ul1 − u0.
(4.37)
The above post-processing steps ensure that the component functions uj1 ,uj2 , . . . ,uj1...jL
are orthogonal with respect to the Dirac product measure (4.5) by construction. As an
example, consider the first-order component function uj1(t; ξj1) = uj1(t; ξj1) − u0(t). From
(4.6) we know that uj1(t; ξj1) = u(t; ξaj1) − u0(t), which implies the null integral condition
uj1|ξj1=ξaj1= 0. On the other hand, since uj1 satisfies the weighted residual form
⟨v,M(ξaj1)
¨uj1
+ C(ξaj1)˙uj1
+ K(ξaj1)uj1 − f(t; ξaj1)
⟩= 0,∀v ∈ L2(Γj1)
n, (4.38)
we formally deduce that uj1 coincides with u(t; ξaj1). The component function uj1 recombined
using (4.37) satisfies the null integral constraint. Finally, the approximate solution uL can
be post-processed for its statistical moments or other metrics of interest. This completes the
derivation of the AAPG scheme when considering stochastic ODEs of the form (4.9).
4.2.2 Computational and implementation aspects
In this section, we outline how the subproblems (4.22) arising in the AAPG scheme can
be solved using gPC-based stochastic Galerkin schemes [29] and how these solutions can be
postprocessed to calculate the mean and variance of the response approximation.
Solution of low-dimensional subproblems
To solve the low-dimensional subproblems (4.22) using the classical gPC method, we first
expand the random matrices M(ξ),C(ξ),K(ξ) and the random vector f(t; ξ) using gPC
basis functions as follows
M(ξ) ≈NM∑m=0
Mmϕm(ξ),C(ξ) ≈NC∑m=0
Cmϕm(ξ),K(ξ) ≈NK∑m=0
Kmϕm(ξ), (4.39)
58
with Mm,Cm,Km ∈ Rn×n, and f(ξ, t) ≈∑Nf
m=0 fm(t)ϕm(ξ) with fm(t) ∈ Rn. The orthonor-
mal gPC basis functions are chosen from the Askey family [30].
The gPC expansion of the solution of each subproblem (4.22), uj1...jk , can be written as
follows:
uj1...jk ≈ uj1...jk =
Pk∑i=1
βj1...jki (t)ϕi(ξj1 , . . . , ξjk), (4.40)
where βj1...jki ∈ Rn, i = 1, 2, . . . , Pk, are undetermined expansion coefficients. The number of
terms in the expansion (4.40) is a function of the number of active random variables (k) in
the subproblem and the gPC expansion order (p), i.e., Pk = (k+p)!k!p!
. It can be seen that Pk
increases rapidly with respect to k, especially when high gPC expansion order p is required
to ensure good accuracy. However, this is not an issue in the AAPG formulation since k is
less than or equal to the functional ANOVA expansion order L, which in turn is significantly
smaller than the total number of random variables, M .
Applying the stochastic Galerkin projection scheme [29] to the subproblem (4.22)
M(ξaj1...jk)¨uj1...jk
+ C(ξaj1...jk)˙uj1...jk
+ K(ξaj1...jk)uj1...jk − f(t; ξaj1...jk) ⊥ ϕi, i = 1, 2, . . . , Pk,
(4.41)
leads to a system of Pk coupled deterministic ODEs which govern the coefficients βj1...jki .
Writing the stochastic Galerkin conditions for the initial conditions of (4.22) as
uj1...jk(0; ξj1 . . . ξjk)− Z0(ξaj1...jk) ⊥ ϕi, i = 1, 2, . . . , Pk, (4.42)
˙uj1...jk
(0; ξj1 . . . ξjk)− Z1(ξaj1...jk) ⊥ ϕi, i = 1, 2, . . . , Pk, (4.43)
leads to the following initial conditions for the gPC expansion coefficients in (4.40)
βj1...jki (0) =⟨Z0(ξaj1...jk)ϕi
⟩, i = 1, 2, . . . , Pk, (4.44)
βj1...jki (0) =
⟨Z1(ξaj1...jk)ϕi
⟩, i = 1, 2, . . . , Pk, (4.45)
that can be used to solve the deterministic ODEs arising from (4.41).
Post-processing
In this section we present explicit formulas for the mean and variance of uL when L = 2.
Assuming that the low-dimensional subproblems are solved using the classical gPC Galerkin
59
projection scheme as outlined earlier, the second-order ANOVA approximation can be writ-
ten as
uL(ξ, t) ≈ α0u0(t) +
M∑j1≤j2
P2∑i=1
λj1j2i (t)ϕi(ξj1 , ξj2), (4.46)
with α0 =
(1−M +
M(M − 1)
2
), (4.47)
λj1j2i =
βj1j2i for j1 < j2, i = 1, 2, . . . , P2,
(2−M)βj1I−1(i) for j1 = j2 and i ∈ A,
0 for j1 = j2 and i 6∈ A.
(4.48)
Note that for simplicity of notation, the first and second-order gPC basis and coefficients
in (4.46) are collectively written as ϕi(ξj1 , ξj2) and λj1j2i (t), respectively. The mapping i ∈1, 2, . . . , P1 7→ I(i) ∈ A ⊂ 1, 2, . . . , P2 is introduced to express the first-order gPC terms
in the form of second-order gPC terms. Using (4.46), the mean and the variance of uL are
explicitly given by
µAAPG(t) ≈ α0u0(t) +
N∑j1≤j2
λj1j20 (t), (4.49)
σ2AAPG(t) ≈
P2∑i=2
N∑j1≤j2
diag(λj1j2i (t) ·
(λj1j2i (t)
)T). (4.50)
Similar expressions can be derived for the statistics of the velocity and acceleration vectors.
4.3 Numerical studies
In this section, we will study the three model problems introduced earlier in section 3.3 using
the first- and second-order AAPG schemes, i.e., uL with L = 1, 2 in (4.2). Key specifications
of the problems such as the spatial and stochastic dof will be briefed and details can be found
in section 3.3. The last two test cases involve linear stochastic structures, where the Young’s
modulus is treated as a random field and discretized using the KL expansion scheme. All the
numerical tests are conducted using Matlab codes on a machine with Intel i7-2600 CPU and
16Gb RAM. The low-dimensional subproblems of the form (4.22) are solved sequentially
60
using a second-order gPC-based stochastic Galerkin projection scheme; see Section 4.2.2.
The anchor point (ξa in (4.5)) is set to ξai = 〈ξi〉 = 0, i = 1, . . . ,M in all three test cases.
4.3.1 Spring-mass system
In this test case, we consider a three-dof spring-mass system with stochastic mass, stiffness
and damping coefficient subject to stochastic forcing. The system has total stochastic dof
M = 15. Results obtained using MCS with sample size M = 106 are used as reference
to compute the absolute error in mean and standard deviation of the response. CPU time
for the MCS method is 5.16×103 s. The errors corresponding to AAPG methods and the
second-order gPC scheme are included in Figure 4.1 and 4.2. It can be seen that second-
order AAPG scheme has similar level of accuracy to the second-order gPC method. While
second-order gPC cost 1.46 s, serial implementation of first and second-order AAPG cost
0.53 and 3.66 s, respectively.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
10-6
10-4
10-2
Absolu
te e
rror
in m
ean
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.1: Sping-mass system: Errors |µAAPG(t) − µMCS(t)| as a function of time corre-sponding to the displacement at the third dof.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s)
10-4
10-3
10-2
10-1
Absolu
te e
rror
in s
tandard
devia
tion
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.2: Spring-mass system: Errors |σAAPG(t) − σMCS(t)| as a function of time corre-sponding to the displacement at the third dof.
61
4.3.2 Two-dimensional beam problem
In this test case a two-dimensional linear beam is cantilevered at one end and a time-
dependent force is applied to the other end. FE mesh used for this problem has 88 dof. The
Young’s modulus of the beam is modeled as a random field and KL expansion method is
applied to discretize it, resulting in stochastic dof M = 5. MCS results with sample size
M = 106 are used as reference. The absolute error in mean and standard deviation of the
displacement at the upper right tip of the beam computed using the AAPG schemes are
presented in Figures 4.3 and 4.4. The errors corresponding to the second-order gPC scheme
are also included as a reference.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t(s)
10-12
10-10
10-8
10-6
10-4
Absolu
te e
rror
in m
ean
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.3: Two-dimensional beam: Errors |µAAPG(t) − µMCS(t)| as a function of timecorresponding to the displacement at the upper-right tip in the x2 direction. Error of second-order gPC method is also included for comparison.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t(s)
10-10
10-8
10-6
10-4
Absolu
te e
rror
in s
tandard
devia
tion
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.4: Two-dimensional beam: Errors |σAAPG(t) − σMCS(t)| as a function of timecorresponding to the displacement at the upper-right tip in the x2 direction. Error of second-order gPC method is also included for comparison.
62
It can be seen from Figures 4.3 and 4.4 that the first-order AAPG scheme has a higher
level of error compared to the second-order gPC method while the second-order AAPG
scheme provides accuracy levels comparable to the second-order gPC method. The compu-
tational time required by the second-order AAPG scheme is 2.62 s. In comparison, MCS
with sample size 106 costs 1.96 × 105 s and the second-order gPC method costs 3.93 s.
It is to be noted that in our current implementation the AAPG subproblems are solved
sequentially. Since the AAPG subproblems are decoupled a parallel implementation would
significantly speed up the calculations, particularly for systems with a large number of spatial
and stochastic dof.
4.3.3 Three-dimensional hex problem
This test case features a three-dimensional hex shaped linear structure with two cylinder
shaped holes. One face of the structure is clamped and a time-dependent force is evenly
applied to another face. Spatial discretization result in a total of 4, 446 dof. The Young’s
modulus of the structure is treated as a random field and M = 5 terms are retained in the KL
expansion. Figure 4.5 and 4.6 presents the absolute error compared to results computed using
MCS with sample size 105. Error of the results computed using the second-order gPC method
is also included for comparison. It can be seen that the errors in the mean response computed
using the different schemes are comparable (see Figure 4.5). By contrast, the errors in the
standard deviation computed using first-order AAPG are one order of magnitude higher
compared to second-order gPC and second-order AAPG (see Figure 4.6). The CPU time
required by AAPG schemes are presented in Table 4.1. Compared with MCS and second-
order gPC methods, the sequential second-order AAPG scheme offers the best efficiency
while providing the same level of accuracy compared to the second-order gPC method for
this particular test case.
Table 4.1: CPU time required by different methods. Results are generated using Matlabcodes on a machine with Intel i7-2600 CPU and 16Gb RAM.
CPU time(s)MCS, sample size M = 105 1.388×105
second-order gPC 4.454×103
Sequential first-order AAPG 1.543×102
Sequential second-order AAPG 2.237×103
63
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-12
10-10
10-8
Ab
so
lute
err
or
in m
ea
n
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.5: Three-dimensional hex: Absolute error in mean of displacement at node(x,y,z)=(10,2,1) for different methods.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t(s)
10-10
10-9
10-8
10-7
10-6
Absolu
te e
rror
in s
tandard
devia
tion
1st order AAPG
2nd order AAPG
gPC,p=2
Figure 4.6: Three-dimensional hex: Absolute error in standard deviation of displacement atnode (x,y,z)=(10,2,1) for different methods.
We also conducted some additional numerical studies to examine the performance of the
AAPG schemes when the number of random variables is increased, i.e. for M = 5, 10, 15, 20.
Figure 4.7 shows the time-averaged relative error in the response variance approximation
and the CPU time of the first- and second-order AAPG schemes as a function of M . Similar
trends for the first-order gPC scheme are included for comparison. The computational
cost and memory requirements of the second-order gPC schemes are much higher and not
presented in these figures. Figure 4.7 (a) shows that the level of L2 error for second-order
AAPG scheme is about two orders lower than the first-order AAPG and gPC schemes. The
CPU times shown in Figure 4.7 (b) are for a sequential implementation. For systems with
a larger number of spatial and stochastic degrees of freedom, the decoupled subproblems
arising in the AAPG scheme will need to be solved in parallel to achieve further reductions
in CPU time.
64
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
log10
(N)
log
10(e
rrors
)
gPC,p=1
1st order AAPG
2nd order AAPG
(a) L2 relative error for the standard deviation.
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.51.5
2
2.5
3
3.5
4
4.5
5
5.5
log10
(N)
log
10(C
PU
)
MCS
gPC,p=1
1st order AAPG
2nd order AAPG
(b) CPU time.
Figure 4.7: Performance of first and second order AAPG scheme with regards to the number
of random variables N . Error is computed as Eσ =||σAAPG−σMCS ||L2([0,T ])
||σMCS ||L2([0,T ]), where MCS solution
is used as reference. First-order gPC results are also included for comparison.
4.4 Concluding remarks
In this chapter, we have proposed AAPG projection schemes for solving a class of stochastic
ordinary differential equations encountered in linear stochastic structural dynamics. The
main idea of the proposed formulation is to approximate the dynamic response using a Ho-
effding functional ANOVA decomposition along with appropriate constraints to ensure the
uniqueness of the decomposition. We showed that when the test functions in the weighted
residual form are chosen appropriately, the original high-dimensional stochastic problem can
be decoupled into a sequence of low-dimensional stochastic subproblems that can be solved
independently of each other.
Numerical studies on a set of linear stochastic structural dynamical systems suggest that
the AAPG scheme with second-order truncation provides accuracy that is comparable to the
classical gPC-based stochastic Galerkin approach, while incurring lower computational cost.
For large-scale systems, the AAPG projection scheme will be significantly faster compared
with existing methods, because the low-dimensional subproblems arising in this scheme are
decoupled and can be solved independently of each other. The AAPG projection scheme
is expected to perform very well for systems where the dynamic response has low effective
dimension (i.e. when a low-order ANOVA truncation is sufficient to capture the response
statistics). Another advantage offered by the AAPG formulation is that it is a non-iterative
scheme in contrast to the GSD approach considered in Chapter 3.
65
In the next chapter, we will extend the AAPG scheme to nonlinear stochastic structural
dynamics. A few high-dimensional test cases of spring-mass system with nonlinear terms
will be used to demonstrate that the AAPG scheme is able to tackle not only linear but also
(multi-dof) nonlinear systems in a very efficient way. So far we have been experimenting
with different schemes in Matlab. Its inefficiency has limited our capacity to test larger
examples. We will move the numerical testing to C++ from this point onwards in order to
enable the parallel solution of large-scale test problems.
66
Chapter 5
Anchored ANOVA Petrov-Galerkin
scheme for nonlinear stochastic
structural dynamics
In this chapter we extend the AAPG scheme to dynamic analysis of geometrically nonlinear
stochastic structures. Such analysis can be useful in design and reliability analysis of dynamic
systems subject to large deformations. Fish et al. [208] suggested that nonlinear analysis
should be applied to deformations that are of the order 10−2 of the dimensions of a body,
which implies that the error due to the assumption of linearity are of the order of 10−2. On the
other hand, stochastic nonlinear systems are much more complicated to model and compute
than linear systems. Most of the reported approaches are applicable to certain nonlinear
systems with limitations with respect to the nature of the excitation, the type of nonlinearity
and/or the number of dof. In this chapter we propose a new numerical scheme based on
AAPG for the analysis of nonlinear structural dynamic problems subject to randomness in
initial conditions, excitation, nonlinearity parameter and damping ratio. It is shown via
numerical studies that AAPG scheme is able to alleviate the curse of dimensionality by
solving the independent subproblems in parallel and reach the same level of accuracy as
gPC with much lower computational cost.
67
The following nonlinear SODE system arising in the dynamic analysis of structural sys-
tems with geometrical nonlinearity were introduced in section 2.2 and reproduced here for
convenience:
M(ξ)u(t; ξ) + C(ξ)u(t; ξ) + K(ξ)u(t; ξ) + γ(u(t; ξ); ξ) = f(t; ξ) a.s. in [0, T ]× ΓM , (5.1)
where M(ξ),C(ξ),K(ξ) ∈ Rn×n denote the stochastic mass, damping and stiffness matrices,
respectively. γ(u; ξ) ∈ Rn denotes the nonlinear restoring force, which is a deviation from
the linear restoring force vector K(ξ)u(t; ξ). The external force f(t; ξ) ∈ Rn is assumed to
be a time-dependent stochastic process. u(t; ξ) ∈ Rn is the displacement vector, t ∈ [0, T ]
denotes time (T < ∞) and n is the total number of dof. We denote the probability space
by the triplet (Ω,F ,P), where Ω ⊂ Rq is the sample space, F is the σ-algebra associated
with Ω and P : F → [0, 1] is a probability measure. The components of the vector ξ =
(ξ1, ξ2, . . . , ξM)T : Ω → RM are assumed to be a set of i.i.d. random variables whose joint
pdf can be written as the product of its marginal densities, i.e. ρ(ξ) =∏M
i=1 ρi(ξi). We
denote by Γ = Γ1 × · · · × ΓM the joint image of ξ. The governing equation is supplemented
by the following stochastic initial conditions
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (5.2)
To facilitate numerical studies in the chapters that follow, we choose to use the Duffing
oscillator that represents a variety of physical nonlinear systems [209, 196, 210, 211, 212, 213].
For a system of n coupled Duffing oscillator the nonlinear term is of the the form
γ(u(t; ξ); ξ) = K(u(t; ξ))u(t; ξ),
where K(u(t; ξ)) =
k1 + k2 −k2 0 . . . 0
−k2 k2 + k3 −k3 . . . 0
· · · . . . ·0 . . . 0 −kn kn
, ki = ηiki (ui − ui−1)2 ,(5.3)
for i = 1, . . . , n, u0 = 0. Here ηi ≥ 0 is the nonlinearity parameter at the ith dof. The
68
stochastic mass, damping and stiffness matrices are defined as
M(ξ) =
m1 0 0 . . . 0
0 m2 0 . . . 0
· · · . . . ·0 . . . 0 0 mn
,C(ξ) =
c1 + c2 −c2 0 . . . 0
−c2 c2 + c3 −c3 . . . 0
· · · . . . ·0 . . . 0 −cn cn
,
and K(ξ) =
k1 + k2 −k2 0 . . . 0
−k2 k2 + k3 −k3 . . . 0
· · · . . . ·0 . . . 0 −kn kn
,(5.4)
respectively. Here mi, ci, ki, i = 1, . . . , n are random variables whose dependencies on ξ are
not explicitly written to simplify notations.
The rest of this chapter is organized as follows: Section 5.1 provides mathematical deriva-
tion of the AAPG scheme applied to the nonlinear SODE (5.1). The solution is approximated
with the same ANOVA decomposition we introduced in Chapter 4 for linear SODEs. By
applying the same set of specially designed test functions in the weighted residual form, we
will prove Theorem 5.1.1, i.e. ANOVA component functions can be post-processed from
solutions of low-dimensional nonlinear stochastic subproblems. Section 5.2 demonstrates the
application of the gPC Galerkin scheme to nonlinear SODE (5.1). Section 5.3 introduces the
single-dof version of (5.1). With this simpler governing equation, deeper insights into the
application of the gPC Galerkin scheme can be obtained. It is shown that in such nonlin-
ear systems, the direct product between multiple stochastic variables requires computation
and storage of large matrices and the pseudo-spectral approach is utilized to speed up this
process. In the end, we will apply the AAPG scheme to solve a few single-dof test cases in
Section 5.3.3 and multi-dof test cases in Section 5.4.
5.1 Mathematical derivation
To solve (5.1), we use the L-th order truncated anchored ANOVA approximation (4.2) for
the solution and introduce the following Petrov-Galerkin weighted residual form
Find uL(t; ·) ∈ ULANOV A such that⟨
v,M(ξ)uL + C(ξ)uL + K(ξ)uL + γ(uL(t; ξ); ξ)− f(t; ξ)⟩
= 0, ∀v ∈ V L, (5.5)
69
where the test (V L) and trial (ULANOVA) spaces are identical to the ones in the linear weighted
residual form (4.17) and defined in (4.16) and (4.15), respectively. Initial conditions for uL
and uL can be expressed in terms of Z0 and Z1 evaluated at the anchor point as follows
uL(0; ξ) = Z0(ξa) +M∑j1=1
(Z0(ξaj1)− Z0(ξa)
)+ · · · (5.6)
uL(0; ξ) = Z1(ξa) +M∑j1=1
(Z1(ξaj1)− Z1(ξa)
)+ · · · (5.7)
Next, we shall show that an L-th order anchored functional ANOVA decomposition of u(ξ, t)
along with the test space defined earlier leads to a system of decoupled low-dimensional
nonlinear subproblems.
Theorem 5.1.1. Consider the L-th order truncated anchored ANOVA approximation uL
for the solution of the weighted residual form (5.5). Let u0,uj1 , . . . ,uj1...jL be the component
functions of the anchored ANOVA decomposition (4.2) that are subject to the null integral
constraints (4.8). If the test functions are chosen from the space V L defined in (4.16) then
the zero-order component function u0 satisfies the deterministic system of ODEs
M(ξa)u0 + C(ξa)u0 + K(ξa)u0 + γ(u0; ξa) = f(t; ξa), (5.8)
with the initial conditions u0(0) = Z0(ξa) and u0(0) = Z1(ξa), where Z0,Z1 are defined in
(5.2).
The higher order ANOVA component functions uj1...jk , k = 1, 2, . . . , L, are given by
uj1...jk = uj1...jk − u0 −∑l1∈Ik
ul1 −∑
l1<l2,li∈Ik
ul1l2 − · · · −∑
l1<l2···<lk−1,li∈Ik
ul1l2...lk−1 , (5.9)
where Ik = j1, j2, . . . , jk, and the auxiliary variable uj1...jk is the solution of the following
low-dimensional system of SODEs (with k random variables)
M(ξaj1...jk)¨uj1...jk
+ C(ξaj1...jk)˙uj1...jk
+ K(ξaj1...jk)uj1...jk + γ(uj1...jk ; ξaj1...jk) = f(t; ξaj1...jk),
(5.10)
with the initial conditions uj1...jk(0; ξj1 , . . . , ξjk) = Z0(ξaj1...jk) and ˙uj1...jk
(0; ξj1 , . . . , ξjk) =
Z1(ξaj1...jk).
70
Proof. The proof uses ideas from Theorem 1 in [52] and from Theorem 4.2.1 in section 4.2,
which presents similar results for linear parabolic SPDEs and linear SODEs, respectively.
For simplicity of notation, we first rewrite (5.5) in the compact form
Find uL(t; ·) ∈ ULANOVA such that
⟨v, a(uL, uL, uL; ξ) + γ(uL; ξ)− f(t; ξ)
⟩= 0,∀v ∈ V L,
(5.11)
where a(uL, uL, uL; ξ) = M(ξ)uL + C(ξ)uL + K(ξ)uL. Expanding uL and by definition of
a, we have
a(uL, uL, uL; ξ) = a(u0, u0, u0; ξ) +L∑k=1
M∑j1<···<jk
a(uj1...jk , uj1...jk , uj1...jk ; ξ). (5.12)
Hence, the weighted residual form (5.11) can be written as
⟨v, a(u0, u0, u0; ξ)
⟩+
L∑k=1
M∑j1<···<jk
⟨v, a(uj1...jk , uj1...jk , uj1...jk ; ξ)
⟩+
⟨v,γ(u0 +
L∑k=1
M∑j1<···<jk
uj1...jk ; ξ)
⟩= 〈v, f(t; ξ)〉 , ∀v ∈ V L.
(5.13)
To prove (5.8) we consider test-functions v ∈ V L ∩ V0 = V0 of the form v(ξ) = wδ(ξ − ξa),
with deterministic vectors w ∈ Rn. Since the ANOVA component functions satisfy the
null integral property (4.8), we have uj1...jk |ξ=ξa = 0,∀k ≥ 1 and similarly uj1...jk |ξ=ξa =
0, uj1...jk |ξ=ξa = 0, ∀k ≥ 1. As a result, (5.13) becomes the deterministic weighted residual
equation
wT(a(u0, u0, u0; ξa) + γ(u0; ξa)− f(t; ξa)
)= 0,∀w ∈ Rn, (5.14)
which implies a(u0, u0, u0; ξa) + γ(u0; ξa) = f(t; ξa), i.e., equation (5.8). We shall next
consider test-functions v ∈ V L ∩ Vj1...jk = Vj1...jk that can be written as
v(ξ) = w(ξj1 , . . . , ξjk)M∏i 6∈Ik
δ(ξi − ξai ), (5.15)
with Ik = j1, . . . , jk, w ∈ L2(Γj1 × · · · × Γjk)n. We expand the first term in (5.13) as
⟨v, a(u0, u0, u0; ξ)
⟩= cj1...jk
∫Γj1×···×Γjk
wTa(u0, u0, u0; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk , (5.16)
71
with cj1...jk =∏
i 6∈Ik ρi(ξai ) > 0. The first-order terms in (5.13) can be written as
M∑j′1=1
⟨v, a(uj
′1 , uj
′1 , uj
′1 ; ξ)
⟩
= cj1...jk
M∑j′1=1
∫Γj1×···×Γjk
wTa(uj′1 , uj
′1 , uj
′1 ; ξaj1...jk)
∏i∈Ik
ρi(ξi)dξj1 . . . dξjk .
(5.17)
From the null integral constraints (4.8), we have uj′1|ξaj1...jk = 0 for j′1 6∈ Ik, uj
′1 |ξaj1...jk = uj
′1
for j′1 ∈ Ik, and similar conditions hold for the derivatives uj′1 and uj
′1 . Hence (5.17) reduces
to the following summation with k terms
cj1...jk∑l1∈Ik
∫Γj1×···×Γjk
wTa(ul1 , ul1 , ul1 ; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk . (5.18)
Similarly, the second-order terms in (5.13) are given by
M∑j′1<j
′2
⟨v, a(uj
′1j′2 , uj
′1j′2 , uj
′1j′2 ; ξ)
⟩
= cj1...jk
M∑j′1<j
′2
∫Γj1×···×Γjk
wTa(uj′1j′2 , uj
′1j′2 , uj
′1j′2 ; ξaj1...jk)
∏i∈Ik
ρi(ξi)dξj1 . . . dξjk ,
(5.19)
which simplify to the following summation with k(k−1)2
terms
cj1...jk∑
l1<l2,li∈Ik
∫Γj1×···×Γjk
wTa(ul1l2 , ul1l2 , ul1l2 ; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk , (5.20)
since the null integral constraints (4.8) lead to the conditions: uj′1j′2|ξaj1...jk = 0 for j′1 6∈ Ik or
j′2 6∈ Ik, uj′1j′2|ξaj1...jk = uj
′1j′2 for j′1, j
′2 ∈ Ik (the same conditions hold for the derivatives uj
′1j′2
and uj′1j′2). Similar arguments can be used when considering higher order terms in (5.13).
The nonlinear term in (5.13) can be written as⟨v,γ(u0 +
L∑k′=1
M∑j′1<···<j′k′
uj′1...j
′k′ ; ξ)
⟩
= cj1...jk
∫Γj1×···×Γjk
wTγ(u0 +L∑
k′=1
M∑j′1<···<j′k′
uj′1...j
′k′ ; ξaj1...jk)
∏i∈Ik
ρi(ξi)dξj1 . . . dξjk ,
(5.21)
72
which simplify to the following integration due to the null integral constraints (4.8)
cj1...jk
∫Γj1×···×Γjk
wTγ(u0 +L∑
k′=1
∑l1<···<lk′ ,li∈Ik
ul1...lk′ ; ξaj1...jk)∏i∈Ik
ρi(ξi)dξj1 . . . dξjk . (5.22)
Gathering all the terms obtained by expanding (5.13) such as (5.16), (5.18), (5.20) and
(5.22), using the fact that cj1...jk 6= 0 and introducing the auxiliary variable uj1...jk defined in
(5.9), we recover the following low-dimensional stochastic weighted residual form
Find uj1...jk ∈ L2(Γj1 × · · · × Γjk)n such that⟨
w, a(uj1...jk , ˙uj1...jk
, ¨uj1...jk
; ξaj1...jk) + γ(uj1...jk ; ξaj1...jk)− f(t; ξaj1...jk)⟩
= 0, (5.23)
∀w ∈ L2(Γj1 × · · · × Γjk)n, which implies the strong form (5.10). The initial conditions for
u0, u0 and uj1...jk , ˙uj1...jk
, k = 1, . . . , L, follow from the combination of (4.8), (5.9) and (5.40).
For example, we have
u0(0) = u(0; ξa) = Z0(ξa), (5.24)
u0(0) = u(0; ξa) = Z1(ξa), (5.25)
uj1(0; ξj1) = uj1(0; ξj1) + u0(0) = u(0; ξaj1)− u0(0) + u0(0) = Z0(ξaj1), (5.26)
˙uj1
(0; ξj1) = uj1(0; ξj1) + u0(0) = u(0; ξaj1)− u0(0) + u0(0) = Z1(ξaj1). (5.27)
This completes the proof.
It is worth noting that the low-dimensional stochastic subproblems governing the auxil-
iary variables uj1...jk , k = 1, 2, . . . , L, can be solved in parallel, independently of each other.
After solving the subproblems (5.10) in parallel, the resulting auxiliary variables are post-
processed using (5.9) to compute the ANOVA component functions and steps are taken to
ensure that the component functions uj1 ,uj2 , . . . ,uj1...jL are orthogonal with respect to the
Dirac product measure (4.5) by construction. The approximate solution uL can be post-
processed for its statistical moments or other metrics of interest [214]. This completes the
derivation of the AAPG scheme when considering nonlinear SODEs of the form (5.1).
73
5.2 Application of gPC Galerkin scheme
We will demonstrate the application of the gPC Galerkin scheme to solve the nonlinear SODE
(5.1) in this section. The same procedure is also applied to solve the AAPG subproblems.
The gPC approximation of the solution can be written as
u(t; ξ) ≈ u(t; ξ) =
Nξ∑i=1
ui(t)ϕi(ξ), (5.28)
where ui(t) ∈ Rn are undetermined vector functions of time and ϕi(ξ), i = 1, 2, . . . , Nξ de-
notes a set of orthonormal gPC basis functions. Substituting (5.28) into (5.1), the coefficients
ui(t) can be computed by applying the stochastic Galerkin projection scheme which involves
the enforcement of the following orthogonality conditionsM(ξ)¨u(t; ξ) + C(ξ) ˙u(t; ξ) + K(ξ)u(t; ξ) + γ(u(t; ξ); ξ)− f(t; ξ) ⊥ ϕi(ξ),
u(ξ, 0)− Z0(ξ) ⊥ ϕi(ξ),˙u(ξ, 0)− Z1(ξ) ⊥ ϕi(ξ),
(5.29)
where i = 1, 2, . . . , Nξ. We assume that the nonlinear term takes the form of γ(u(t; ξ); ξ) =
K(u(t; ξ))u(t; ξ), where K(u(t; ξ)) is defined in (5.3). The random matrices M(ξ),C(ξ),K(ξ)
and the random vector f(t; ξ) can be expanded using gPC basis functions as follows
M(ξ) ≈NM∑m=0
Mmϕm(ξ),C(ξ) ≈NC∑m=0
Cmϕm(ξ),K(ξ) ≈NK∑m=0
Kmϕm(ξ), (5.30)
with Mm,Cm,Km ∈ Rn×n, and f(ξ, t) ≈∑Nf
m=0 fm(t)ϕm(ξ) with fm(t) ∈ Rn. The orthonor-
mal gPC basis functions are chosen from the Askey family [30]. The resulting second order
nonlinear governing equation for the PC expansion coefficients is of the form
MU(t) + CU(t) + KU(t) + Θ(u(t; ξ))U(t) = F(t), (5.31)
where U = (u1, . . . ,uNξ)T ∈ RnNξ denotes the vector of undetermined coefficients in the
gPC expansion (5.28). The coefficient matrices M,C,K and Θ(u(t; ξ)) are all matrices with
Nξ ×Nξ blocks and each block can be defined as:
Mj,i =
NM∑m=0
Mm 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ, (5.32)
74
Cj,i =
NC∑m=0
Cm 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ. (5.33)
Kj,i =
NK∑m=0
Km 〈ϕmϕiϕj〉 , i, j = 1, 2...Nξ, (5.34)
Θj,i(u(t; ξ)) =⟨K(u(t; ξ))ϕiϕj
⟩, i, j = 1, 2...Nξ, (5.35)
where M,C,K are deterministic. The nonlinear term Θj,i(u(t; ξ)) can be evaluated by sub-
stituting in the expression of K(u(t; ξ)) in (5.3). Substituting the gPC expansion of um and
assuming deterministic nonlinearity coefficient ηm, (5.35) can be converted to a deterministic
matrix where the entries in block Θj,i can be evaluated as
⟨kmϕiϕj
⟩= ηmkm
Nξ∑k
Nξ∑l
(ukm − ukm−1)(ulm − ulm−1) 〈ϕiϕjϕkϕl〉 , (5.36)
with i, j, k, l = 1, 2, . . . , Nξ,m = 1, . . . , n. Note that the assumption that ηm is deterministic
is made to simplify (5.36). The evaluation and storage of 〈ϕiϕjϕkϕl〉 can be challenging if Nξ
is large and we will discuss this topic in detail in section 5.3.2. If no deterministic assumption
about ηm is made, we can expect terms of the form 〈ϕiϕjϕkϕlϕm〉, i, j, k, l,m = 1, . . . , Nξ
in (5.36) that poses a even greater computation and storage challenge. F(t) in (5.31) is a
vector with dimension of nNξ:
F(t) =(F1(t),F2(t), . . .FNξ(t)
)T,Fj(t) =
fj(t)
⟨ϕ2j
⟩when j = 1, 2...Nf
0 when j = Nf + 1, ...Nξ
(5.37)
Note that (5.31) is deterministic and can be solved using a time-marching schemes such as
the Newmark integration scheme [179].
5.3 Single-dof Duffing oscillator
In this section we introduce the scalar version of (5.1) for the nonlinear structural dynamic
system. The nonlinear restoring force term γ(u(t; ξ); ξ) is in the form of (5.3). This simpler
governing equation helps to reveal more details on the application of gPC Galerkin scheme
to nonlinear SODEs, which lead us to the pseudo-spectral approach to calculate the product
of multiple random variables. Extensive numerical studies based on this single-dof SODE
75
will be provided later in Section 5.3.3. The single-dof SODE is written as
u(τ ; ξ) + 2ζ(ξ)u(τ ; ξ) + u(τ ; ξ) + η(ξ)u3(τ ; ξ) = p(τ ; ξ) a.s. in [0, w0T ]× ΓM . (5.38)
Note that the governing equation has been normalized with regards to the mass m and we
use an unitless time τ = w0t, where w0 =√k/m is the undamped natural frequency of
the system. ζ(ξ) and η(ξ) denote the damping ratio and the nonlinearity parameter in the
system, respectively, and can be represented as
ζ(ξ) =
M1∑i=0
ζiξi, η(ξ) =
M2∑i=0
ηiξi. (5.39)
We further specify the form of initial conditions as
u(0; ξ) = Z0(ξ) = u0 +
M3∑i=1
aiξi, u(0; ξ) = Z1(ξ) = v0 +
M4∑i=1
biξi. (5.40)
Here ζi, ηi, ai, bi are constants. The forcing p(τ ; ξ) has units of acceleration and is a time-
dependent random process applied over [0, w0T ] characterized by the autocorrelation function
Rpp(τ1, τ2) = σ2pe− |τ1−τ2|
Aω0 , A > 0, (5.41)
where A is the correlation length and σp is the standard deviation of the process. This
random process can be decomposed using a truncated Karhunen-Loeve (KL) expansion up
to the M5-th random dimension as follows [84]
p(τ ; ξ) = p(τ) + σp
M5∑i=1
√λiΨi(τ)ξi = p(τ) +
M5∑i=1
pi(τ)ξi. (5.42)
Here p(t) is the mean of the process. λi and Ψi(t) are the eigenvalues and eigenfunctions of
a Fredholm integral equation of the second kind given by∫ T
0Rpp(t1, t2)Ψi(t1)dt1 = λiΨi(t2).
λi are ordered such that λ1 ≥ λ2 ≥ · · · ≥ λM5 . The total number of random dof is M =∑5i=1 Mi. In practice, it is convenient to replace Mi, i = 1, . . . , 5 in (5.39), (5.40) and (5.42)
with M , by adding zero values at added stochastic dofs.
76
5.3.1 Application of gPC Galerkin scheme
As an extension to section 5.2, we include in this section the application of the gPC Galerkin
scheme to the single-dof nonlinear SODE (5.38). The following scalar form of gPC expansion
of the solution is used
u(t; ξ) ≈ u(t; ξ) =
Nξ∑i=1
ui(t)ϕi(ξ), (5.43)
where ui(t) ∈ Rn are undetermined functions of time and ϕi(ξ), i = 1, 2, . . . , Nξ denotes a set
of orthonormal gPC basis functions. Substituting (5.43) and (5.42) into (5.38) and applying
the stochastic Galerkin condition (5.29) result in
Nξ∑i=1
ui(τ)ϕi(ξ) + 2ζ
Nξ∑i=1
ui(τ)ϕi(ξ) +
Nξ∑i=1
ui(τ)ϕi(ξ)
+ η
Nξ∑i=1
Nξ∑j=1
Nξ∑k=1
ui(τ)uj(τ)uk(τ)ϕi(ξ)ϕj(ξ)ϕk(ξ) =M∑i=0
pi(τ)ξi,
(5.44)
with initial conditions:u0(0) = u0, u0(0) = v0,
ui(0) = ai, ui(0) = bi, i = 1, . . . ,M,
ui(0) = 0, ui(0) = 0, i = M + 1, . . . , Nξ.
(5.45)
We have defined ξ0 = 1 and p0(τ) = p(τ) to make the expressions more compact. Also, we
assumed ζ and η to be deterministic for simplicity. Now the above equation is projected
onto the random space spanned by the orthogonal polynomial bases, which leads to the Nξ
coupled deterministic nonlinear ODEs of the form
ul(τ)+2ζul(τ)+ul(τ)+η
〈ϕ2l 〉
Nξ∑i=1
Nξ∑j=1
Nξ∑k=1
ui(τ)uj(τ)uk(τ)Eijkl =1
〈ϕ2l 〉
M∑i=0
pi(τ)ξiϕl(ξ), (5.46)
∀l = 1, 2 . . . , Nξ. Here Eijkl = 〈ϕiϕjϕkϕl〉. Since the first order gPC basis take the form
ϕi = ξi,∀i = 1, . . . ,M , the right side of the above equation can be simplified as follows
ul(τ)+2ζul(τ)+ul(τ)+η
〈ϕ2l 〉
Nξ∑i=1
Nξ∑j=1
Nξ∑k=1
ui(τ)uj(τ)uk(τ)Eijkl =
pi(τ), ∀l = 0, 1, . . . ,M,
0, ∀l = M + 1, . . . , Nξ.
77
The coefficients 〈ϕ2l 〉 and Eijkl can be determined analytically or numerically using multi-
dimensional numerical quadratures. Once ul(τ) are computed, the statistical moments of
the result can be assembled using (5.43). Note that although we assumed ζ and η to be
deterministic for simplicity, a more general case of stochastic ζ(ξ) and η(ξ) can be dealt
with following the same methodology. In that case we will have to use even more complex
coefficients such as 〈ϕiϕjϕkϕlϕm〉 in the nonlinear term, which lead us to the next section
on a more efficient way to compute and store these coefficients.
5.3.2 Pseudo-spectral approach
In the previous section we have used the following form of gPC approximation of the nonlinear
term γ = u3(τ):
γ ≈Nξ∑i=1
ui(τ)ϕi
Nξ∑j=1
uj(τ)ϕj
Nξ∑k=1
uk(τ)ϕk, (5.47)
here Nξ is the number of terms in the p-th order gPC expansion. The gPC coefficients of
the nonlinear term can be calculated using the full spectral approach as
γl =
Nξ∑i=1
Nξ∑j=1
Nξ∑k=1
ui(τ)uj(τ)uk(τ)Eijkl,∀l ∈ 1, . . . , Nξ, (5.48)
where Eijkl = 〈ϕiϕjϕkϕl〉. In (5.47) all the coefficient terms up to order 3p are kept to the
last step, when the expansion is truncated to the first p-th order. Computation of higher
order terms can be very demanding and so is the evaluation and storage of Eijkl, since it
is a tensor with N4ξ elements. With even more random variables, it is hard to apply such
full spectral approach because the cost grows exponentially. In practice, the pseudo-spectral
method [215] is more efficient and commonly used. The central idea of the pseudo-spectral
approach is to regroup the product of M random variables such that we only have to work
with products of two random variables at a time. In other words, we repeatedly apply a
formula for the product of two PC expansions. To illustrate the pseudo-spectral approach,
consider the case when we wish to approximate the PC expansion of γ = u3(τ). We can
rewrite this product of three random variables as follows:
γ ≈ γ = u2(τ) · u(τ) ≈Nξ∑i=1
u2i (τ)ϕi
Nξ∑i=1
uj(τ)ϕj, (5.49)
78
where u2i (τ) and ui(τ), i = 1, . . . , Nξ are the coefficients in the p-th order gPC expansion of
u2(τ) and u(τ), respectively. The gPC expansion coefficients of γ are computed as
γk =
Nξ∑i=1
Nξ∑i=1
u2i (τ)uj(τ)Dijk,∀k ∈ 1, . . . , Nξ, (5.50)
where Dijk = 〈ϕiϕjϕk〉. For the terms corresponding to u2(τ), the full spectral approach has
terms up to 2p-th order while pseudo-spectral approach truncate the gPC approximation
to the first p-th order. As a result, in later steps there are less terms to compute and the
same Dijk can be reused. The same process can be adopted to approximate the product of
any number of random variables. It is to be noted that the pseudo-spectral approach would
introduce additional truncation error that is negligible if p is chosen sufficiently high, but
need to be considered when performing repeated multiplications [215].
5.3.3 Numerical studies
Now we are ready to apply the AAPG scheme to the single-dof Duffing oscillator (5.38).
There are three test cases we present in the following. The first test case has M = 2 and
preliminary comparisons are made between AAPG and other methods. The second case
features M = 15 and includes a convergence study of the AAPG scheme. The last test
case has M = 100, representing a high dimensional problem where implementing second or
higher order gPC scheme becomes impractical, while AAPG schemes provide more accurate
results than the first-order gPC scheme and effectively alleviate the curse of dimensionality.
In all test cases, we include studies on how the performance of different schemes depend on
the level of randomness and the nonlinearity coefficient. The mean and standard deviation
are calculated using MCS with sample size 106 to serve as benchmark against which other
methods are compared. All the numerical studies from this point beyond are conducted
using a software library written in C++ and carried out on an IBM Power 755 Server with
4x 8core 3.3GHz Power7 CPUs and 128GB RAM. OpenMP is utilized whenever parallel
implementation is possible, i.e. for MCS and parallel computation of AAPG subproblems.
The anchor point (ξa in (4.5)) is set to ξai = 〈ξi〉 = 0, i = 1, . . . ,M in all three test cases.
79
Stochastic initial conditions (M = 2)
In this first test case we set the external forcing p(τ ; ξ), damping coefficient ζ(ξ) and non-
linearity coefficient η(ξ) to be deterministic. The initial conditions are
u(0; ξ) = x0 + σξ1, and u(0; ξ) = v0 + σξ2, (5.51)
where σ = 0.5, ξ1, ξ2 are normally distributed random variables. The settings of other
variables are shown in Table 5.1. We apply first- and second-order AAPG (referred to as
AAPG1 and AAPG2 ) and compare its performance to the first- and second-order gPC
method (referred to as gPC1 and gPC2 ). The AAPG subproblems are solved using gPC2.
The fourth-order Runge-Kutta method (RK4) with time marching step ∆t = 0.01 is used to
numerically solve the SODE.
x0 v0 ζ η w0T p(τ)0 0 0.1 1.0 10.0 2.0[1−sin(2πτ) exp(−0.3τ)]
Table 5.1: Settings of the coefficients in the nonlinear Duffing oscillator (5.38) and (5.40).
Figure 5.1 shows the mean and standard deviation of u(τ ; ξ) for τ ∈ [0, 10]. The relative
error of gPC and AAPG schemes compared to MCS are also included. Higher order moments
of results are presented in Figure 5.4 in the form of probability density function (pdf) on the
two-dimensional state space u(τ ; ξ) vs u(τ ; ξ) at τ = 0, 3, 7. We can see that:
1. All methods we tested (gPC1, gPC2, AAPG1 and AAPG2) provide very good approx-
imation of the mean in this problem up to τ = 10.
2. gPC2 and AAPG2 provide almost identical results. This is expected since there are
only two random variables, meaning that there is no truncation error setting L = 2
for AAPG, and subproblems in AAPG are solved using the same accuracy (p = 2) to
gPC2.
3. gPC2 and AAPG2 provide good approximation to the PDF in the given time scale. In
comparison, gPC1 and AAPG1 provide worse approximation of the PDF.
To study how the randomness and the nonlinearity coefficient connect to the performance
of different methods, we vary the value of σ (the level of randomness in the initial conditions)
and η (the nonlinearity coefficient) separately and include results in Figure 5.2 and Figure
80
Mean Standard Deviationu
(τ;ξ
)
0 1 2 3 4 5 6 7 8 9 10−0.5
0
0.5
1
1.5
2
2.5
τ
0 1 2 3 4 5 6 7 8 9 10
0.1
0.2
0.3
0.4
0.5
MCS
gPC1
gPC2AAPG1(p
ξ=2)
AAPG2(pξ=2)
τ
Err
or
ofu
(τ;ξ
)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
gPC2
gPC1
AAPG1
AAPG2
gPC AAPG(pξ=2)
0
0.02
0.04
0.06
0.08
0.1
gPC1
gPC2
AAPG1
AAPG2
gPC AAPG(pξ=2)
Figure 5.1: Upper row: Mean and standard deviation of u(τ ; ξ) for τ ∈ [0, 10]. Black dashlines represent instants when the pdf of solution will be included in Figure 5.4.Lower row: Error of mean and standard deviation of u(τ ; ξ) computed using gPC1/gPC2,AAPG1/AAPG2 compared to MCS (sample size 1 × 106). Subproblems in AAPG aresolved using gPC2. Error of the mean of the gPC results are defined as EgPC
µ =∫ w0T0 |µgPC(τ)−µMCS(τ)|dτ∫ w0T
0 |µMCS(τ)|dτ, where µgPC(τ) and µMCS(τ) are the time-dependent mean of u(τ ; ξ)
computed using gPC and MCS respectively. Error of the standard deviation of gPC andAAPG results are defined in a similar fashion.
5.3. Results confirm the conclusion from the previous test case (η = 1, σ = 0.5) that for
system with relatively low level of randomness and nonlinearity gPC1 and AAPG1 behave
similarly, while gPC2 and AAPG2 have almost identical results that have lower level of error
compared to gPC1 and AAPG1. Additionally, we notice that AAPG1 has slightly lower
level of error compared to gPC1 for the range of η and σ we tested. When η or σ gets bigger,
the error of standard deviation from gPC2/AAPG2 grow quickly and at some point even
higher than error from gPC1/AAPG1. To understand this, we include the pdf of solution
at τ = 0, 3, 5, 7 for σ = 1, η = 1 and σ = 0.5, η = 10 in Figure 5.5 and 5.6, respectively. It
is clear that when high level of randomness or nonlinearity exists, the pdf of the solution
becomes very complicated and the first two statistical moments of the results may not suffice
as indicator of the schemes’ performance.
81
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-7
10-6
10-5
10-4
10-3
10-2
10-1
(a) Error in mean.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
10-3
10-2
10-1
gPC1
gPC2/AAPG2(p=2)
AAPG1
(b) Error in standard deviation.
Figure 5.2: Error in mean and standard deviation of different schemes for different value ofσ ∈ [0.02, 1], η = 1. pdf of solution for σ = 1, η = 1 are included in Figure 5.5.
0 1 2 3 4 5 6 7 8 9 10
10-3
10-2
10-1
(a) Error in mean.
0 1 2 3 4 5 6 7 8 9 10
10-2
10-1
gPC1
gPC2/AAPG2(p=2)
AAPG1
(b) Error in standard deviation.
Figure 5.3: Error in mean and standard deviation of different schemes for different value ofη ∈ [0.2, 10], σ = 0.5. pdf of solution for σ = 0.5, η = 10 are included in Figure 5.6.
Stochastic forcing p(τ ; ξ) (M = 15).
The previous test case provides some comparison between gPC and AAPG methods in a
simple M = 2 setting and proved AAPG scheme to be effective. This test case features
convergence study of the AAPG scheme for a system with stochastic dof M = 15. The
values in Table 5.1 are used as the default settings if not otherwise specified. The coefficients
σp = 2.5, A = 0.05 and p = 4.0 − 2.0 sin(2πτ) exp(−0.3τ) are supplemented to define the
stochastic force in (5.41), (5.42) and is plotted in Figure 5.7. ξi ∈ [−1, 1], i = 1, . . . ,M , are
uniformly distributed random variables.
82
τ = 0 τ = 3 τ = 7
MC
S
−1 0 1
−1
0
1
−0.5 0 0.5 1 1.5
−2
−1
0
1
0 0.5 1−1
−0.5
0
0.5
1
gP
C1
−1 0 1
−1
0
1
−0.5 0 0.5 1 1.5
−2
−1
0
1
0 0.5 1−1
−0.5
0
0.5
1
gP
C2
−1 0 1
−1
0
1
−0.5 0 0.5 1 1.5
−2
−1
0
1
0 0.5 1−1
−0.5
0
0.5
1
AA
PG
1
−1 0 1
−1
0
1
−0.5 0 0.5 1 1.5
−2
−1
0
1
0 0.5 1−1
−0.5
0
0.5
1
AA
PG
2
−1 0 1
−1
0
1
−0.5 0 0.5 1 1.5
−2
−1
0
1
0 0.5 1−1
−0.5
0
0.5
1
Figure 5.4: Probability density function of u(τ ; ξ) (x-axis) vs. u(τ ; ξ) (y-axis) at τ = 0, 3, 7computed using MCS, gPC1, gPC2, AAPG1 and AAPG2. σ = 0.5, η = 1.
Before we proceed, it is helpful to outline the sources of errors in the AAPG scheme:
1. Temporal discretization error when solving AAPG subproblems that depends on the
time-marching step ∆t.
2. Stochastic discretization error when solving AAPG subproblems using gPC schemes
that depends on the gPC order p.
3. Truncation error in the ANOVA expansion of the solution that depends on the ANOVA
truncation order L.
This incomplete list excludes the error associated with approximating the product of more
than two random variables using the pseudo-spectral approach introduced in section 5.3.2.
A discussion on the error associated with the pseudo-spectral approach can be found in [215].
83
τ = 0 τ = 3 τ = 5 τ = 7M
CS
−2 0 2
−2
0
2
−2 −1 0 1 2
−2
0
2
0 1 2
−2
0
2
−1 0 1 2−2
−1
0
1
2
3
gP
C1
−2 0 2
−2
0
2
−2 −1 0 1 2
−2
0
2
0 1 2
−2
0
2
−1 0 1 2−2
−1
0
1
2
3
gP
C2
−2 0 2
−2
0
2
−2 −1 0 1 2
−2
0
2
0 1 2
−2
0
2
−1 0 1 2−2
−1
0
1
2
3
AA
PG
1
−2 0 2
−2
0
2
−2 −1 0 1 2
−2
0
2
0 1 2
−2
0
2
−1 0 1 2−2
−1
0
1
2
3
AA
PG
2
−2 0 2
−2
0
2
−2 −1 0 1 2
−2
0
2
0 1 2
−2
0
2
−1 0 1 2−2
−1
0
1
2
3
Figure 5.5: Probability density function of u(τ ; ξ) (x-axis) vs. u(τ ; ξ) (y-axis) at τ = 0, 3, 5, 7computed using MCS (sample size 1×106), gPC1, gPC2, AAPG1 and AAPG2. σ = 1, η = 1.
Mean Standard deviation∆t 0.1 0.05 0.01 0.005 0.1 0.05 0.01 0.005
gPC1 0.00245 0.00245 0.00244 0.00243 0.03275 0.03256 0.03242 0.03240gPC2 0.00041 0.00040 0.00040 0.00040 0.00760 0.00761 0.00760 0.00760AAPG1 0.00629 0.00630 0.00632 0.00632 0.05584 0.05553 0.05532 0.05529AAPG2 0.00105 0.00105 0.00105 0.00105 0.01681 0.01670 0.01664 0.01663AAPG3 0.00035 0.00035 0.00034 0.00034 0.00477 0.00476 0.00474 0.00473
Table 5.2: Error in the mean and standard deviation computed using gPC1/gPC2,AAPG1/AAPG2/AAPG3 for different ∆t. gPC3 is used to solve AAPG subproblems. Wecan conclude that all schemes converge at ∆t = 0.01.
84
τ = 0 τ = 3 τ = 5 τ = 7M
CS
−1 0 1
−1
0
1
−0.5 0 0.5 1−2
−1
0
1
2
−0.5 0 0.5 1−2
−1
0
1
2
0 0.5 1
−1
0
1
2
gP
C1
−1 0 1
−1
0
1
−0.5 0 0.5 1−2
−1
0
1
2
−0.5 0 0.5 1−2
−1
0
1
2
0 0.5 1
−1
0
1
2
gP
C2
−1 0 1
−1
0
1
−0.5 0 0.5 1−2
−1
0
1
2
−0.5 0 0.5 1−2
−1
0
1
2
0 0.5 1
−1
0
1
2
AA
PG
1
−1 0 1
−1
0
1
−0.5 0 0.5 1−2
−1
0
1
2
−0.5 0 0.5 1−2
−1
0
1
2
0 0.5 1
−1
0
1
2
AA
PG
2
−1 0 1
−1
0
1
−0.5 0 0.5 1−2
−1
0
1
2
−0.5 0 0.5 1−2
−1
0
1
2
0 0.5 1
−1
0
1
2
Figure 5.6: Probability density function of u(τ ; ξ) (x-axis) vs. u(τ ; ξ) (y-axis) at τ = 0, 3, 5, 7computed using MCS (sample size 1 × 106), gPC1, gPC2, AAPG1 and AAPG2. σ = 0.5,η = 10.
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
τ
Figure 5.7: Stochastic forcing with σp = 2.5, A = 0.05 and p = 4.0−2.0 sin(2πτ) exp(−0.3τ).Solid line is the mean and grey area is mean ± standard deviation.
85
Table 5.2 reports the error associated with different values of ∆t for gPC1/gPC2 and
AAPG1/AAPG2/AAPG3 whose subproblems are solved using gPC with p = 3. RK4 is
used to solve the resulting ODE. We can conclude that all schemes converge at ∆t = 0.01
and will continue using this setting in this chapter.
When solving the AAPG subproblems using gPC schemes, the assembled solution would
inherit the stochastic discretization error from all subproblems. This error accumulates to a
level that can not be ignored when the number of AAPG subproblems is large, i.e. M 1.
In this test case, we present the level of error for different values of p in Figure 5.8. We can
conclude that
1. AAPG1 converges at p = 1, AAPG2 converges at p = 2. AAPG3 is not converged for
p = 4. To get similar level of error to gPC2, AAPG3 with p = 3 is needed.
2. By observing the level of error for different AAPG schemes at p = 4, it is clear that
when p is big enough so that stochastic discretization error is negligible, the truncation
error in the ANOVA decomposition decreases with increasing order L. Note that ∆t
is small enough for all schemes to converge (Table 5.2).
In Figure 5.9 we present the mean and standard deviation of the displacement u(t; ξ)
together with the error for gPC1/gPC2 and AAPG1/AAPG2/AAPG3 schemes compared
to MCS (sample size 1 × 106). Subproblems in AAPG are solved using gPC scheme with
p = 3. Figure 5.9 shows that all methods we tested provide good accuracy in predicting
the mean values (error less than 1 × 10−2). The error in standard deviation can be large
over long integration times but both gPC2 and AAPG3 provide error less than 1× 10−2. In
contrast to the previous test case when AAPG1 has lower level of error compared to gPC1
and gPC2/AAPG2 have almost identical results, the error from AAPG methods is slightly
higher than gPC methods of the same order.
So far we have found that using time-marching step ∆t = 0.01 and p = 3 would lead to
convergence and eliminate the error in the AAPG scheme due to temporal and stochastic
discretization. Results also show that AAPG3 provides the same level of accuracy as gPC2
under the current setting. Similar to the previous test case, we will verify these conclusions
by varying σp and η separately. The error for gPC and AAPG with different values of σp
(measure of randomness in the force ) is included in Figure 5.10 at η = 1.0, σp = [0.2, 10].
86
1 2 3 4p
10-4
10-2
100
102
AAPG1
AAPG2
AAPG3
gPC1
gPC2
(a) Error in mean.
1 2 3 4p
10-3
10-2
10-1
100
101
102
AAPG1
AAPG2
AAPG3
gPC1
gPC2
(b) Error in standard deviation.
Figure 5.8: Error of AAPG1/AAPG2/AAPG3 (dash lines) when the subproblems are solvedusing gPC scheme of order p = 1, 2, 3, 4. Error of gPC1/gPC2 (solid lines) are also includedfor comparison. ∆t = 0.01.
Mean standard deviation
u(τ
;ξ)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
MCS
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
0 1 2 3 4 5 6 7 8 9 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Err
or
ofu
(τ;ξ
)
0
1
2
3
4
5
6
7
10-3
gPC
gPC1
gPC2
AAPG1
AAPG2
AAPG3
AAPG(p=3)0
0.01
0.02
0.03
0.04
0.05
0.06
gPC1
gPC2
gPC
AAPG1
AAPG3
AAPG2
AAPG(p=3)
Figure 5.9: Upper row: Mean and standard deviation of u(τ ; ξ) for τ ∈ [0, 10].Lower row: Error of the results of gPC1/gPC2, AAPG1/AAPG2/AAPG3 compared toMCS (sample size 1× 106). Subproblems in AAPG are solved using gPC3.
87
0 1 2 3 4 5 6 7 8 9 10
10-4
10-3
10-2
10-1
100
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
AAPG3(p=4)
p
(a) Mean.
0 1 2 3 4 5 6 7 8 9 10
10-3
10-2
10-1
100
p
(b) Standard deviation.
Figure 5.10: Error of results of gPC and AAPG schemes when η = 1.0. σp = [0.2, 10].
The main conclusions that can be draw from these numerical studies are summarized
below:
1. For relatively small value of σp (in this test case σp ≤ 1), AAPG2 (p = 3) has similar
level of error compared to gPC2. AAPG3 (p = 3 or p = 4) provides lower level of error
compared to gPC2 for σp ≤ 2.5. For σp > 2.5, error in results of both AAPG2 and
AAPG3 grows faster than gPC2 and approaches gPC1 when σp = 4.
2. Since AAPG2 converges for p = 3 as demonstrated in Figure 5.8, it is sufficient to use
AAPG2 with p = 3 in order to get comparable level of error to gPC2 when σp is small
(in this case σp ≤ 1).
3. By comparing the error of AAPG3 using p = 3 and p = 4, we can see that although
AAPG3 is not fully converged at p = 4 (Figure 5.8), using p > 4 would not improve
performance of AAPG3 significantly.
4. AAPG2 and AAPG3 fail to provide satisfying accuracy for σp ≥ 5. We have tested
the same setting and p = 4 and ∆t = 0.001 and the results are visually identical.
The impact of the nonlinearity coefficient η on the performance of different schemes are
presented in Figure 5.11. Similar to the impact of increasing value of σp, AAPG3 (p = 3) has
lower level of error compared to gPC2 when η ≤ 1. When η > 1, error of AAPG2/AAPG3
grows faster than gPC1/gPC2. And for η ≥ 6, AAPG2/AAPG3 fail to provide satisfying
accuracy in predicting the mean and standard deviation.
88
0 1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
(a) Mean.
0 1 2 3 4 5 6 7 8 9 10
10-3
10-2
10-1
100
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
AAPG3(p=4)
(b) Standard deviation.
Figure 5.11: Error of results of gPC and AAPG schemes when σp = 2.5. η = [0, 10].
The studies conducted so far indicate that AAPG2 and AAPG3 behaves well for relatively
small σp and η. To demonstrate how the performance of gPC and AAPG scheme is affected
jointly by σp and η, we plot contours corresponding to three different level of error in Figure
5.12. Figure 5.12 (a), (b) reveals that AAPG3 (p = 3) is able to provide very small error
(< 0.0001 for mean and < 0.001 for standard deviation) in a broader range of σ and η than
gPC2, while AAPG2 (p = 3) and AAPG1 (p = 3) are slightly more restricted compared to
gPC2 and gPC1. Nevertheless, the error of AAPG3 (p = 3) grows quickly with respect to σp
and η and becomes less favourable when it comes to large value of σp and η. The following
conclusions can be drawn from Figure 5.12:
1. For relatively small σp (level of randomness in the excitation force p(t; ξ)) and η (non-
linearity coefficient), AAPG2 (p = 2) provides comparable level of error to gPC2, and
AAPG3 (p = 3) results in slightly smaller error than gPC2.
2. For cases with high value of σp and η, AAPG2 and AAPG3 fail to provide level of error
less than gPC1. Using higher order gPC scheme to solve subproblems can improve the
performance of AAPG by limited amount (Figure 5.10, 5.11). In these cases, gPC2 is
a better choice than AAPG2/AAPG3.
3. If gPC2 is not feasible for larger systems, AAPG1 is preferable to gPC1 since it provides
a comparable level of error, while AAPG1 can be easily paralleled and will be much
faster.
89
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10p
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error < 0.0001
Error > 0.0001
(a) Error in mean = 0.0001.
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10
p
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error < 0.001
Error > 0.001
(b) Error in standard deviation = 0.001.
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10
p
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error > 0.001
Error < 0.001
(c) Error in mean = 0.001.
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10
p
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error < 0.01
Error > 0.01
(d) Error in standard deviation = 0.01.
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10
p
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error > 0.01
Error < 0.01
(e) Error in mean = 0.01.
0.2 0.4 0.6 1 2 4 100.2
0.4
0.6
1
2
4
10
p
gPC1
AAPG1(p=3)
AAPG2(p=3)
AAPG3(p=3)
Error < 0.1
Error > 0.1
(f) Error in standard deviation = 0.1.
Figure 5.12: Contour of different level of error in mean and standard deviation. Thereis no contour for gPC2 in (f) because in the given range (σp ∈ [0.2, 10] and η ∈ [0.2, 10]),gPC2 always provide results with error in standard deviation less than 0.1. Sample values aregenerated at σp, η ∈ [0.2, 0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 2, 3, 4, 5, 6, 7, 8, 9, 10],Matlab function contourc is used to generate contours.
90
Stochastic ζ(ξ), η(ξ) and p(τ ; ξ) (M = 100).
One of the main drawbacks of the gPC scheme is that the resulting system grows quickly
with M and we purposely set M = 100 in this test case to demonstrate the performance of
AAPG when M 1. The values in Table 5.1 are used as default settings in this test case.
σp = 0.5, A = 0.05 is supplemented to (5.41), (5.42) to define the stochastic force. 98 terms
are retained in the KL expansion (5.42) and two more stochastic dofs ξ1, ξ2 are needed to
define stochastic ζ(ξ), η(ξ) as
ζ(ξ) = 0.1 + 0.05ξ1, η(ξ) = 1.0 + 0.35ξ2. (5.52)
The total number of stochastic dof N = 98 + 2 = 100. Second or higher order gPC scheme
are prohibitively expensive thus not included in this case study. In comparison, we can
easily implement AAPG1 and AAPG2 since the AAPG schemes break the original high-
dimensional problem into smaller, independent subproblems that can be easily parallelized.
Table 5.3 includes the wall time of different schemes.
gPC1 AAPG1(p = 2) AAPG2(p = 2) MCS (sample size 106)0.20 0.08 5.62 114.80
Table 5.3: Wall time (in seconds) of different schemes on an IBM Power 755 server with4x8core 3.3GHz Power 7 CPUs and 128GB RAM.
Subproblems in AAPG1/AAPG2 are solved using gPC (p = 2). ξi ∈ [−1, 1], i = 0, . . . , N ,
are uniformly distributed random variables. The mean/standard deviation and pdf are shown
in Figure 5.13 and 5.14. We can see that in this test case the error in results computed using
gPC1 are much higher than error in results computed using AAPG1 and AAPG2.
Conclusions about the single-dof Duffing oscillator test cases
In the last three test cases, we have demonstrated that for low level of randomness (σ in
the first test case and σp in the second test case) and low level of nonlinearity (η), AAPG2
(subproblems solved using gPC2) provides comparable level of accuracy to gPC2 in this
single-dof Duffing oscillator problem. In some cases, using higher truncation order L in
ANOVA decomposition or p in the solution of subproblems might be helpful to improve the
accuracy. We will present a detailed theoretical analysis later in Chapter 6 to shed more light
91
Mean standard deviation
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5 MCS
gPC1
AAPG1(p=2)
AAPG2(p=2)
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5E
rror
0
0.002
0.004
0.006
0.008
gPC
gPC1
AAPG1
AAPG2
AAPG(p=2)0
0.05
0.1
gPC
gPC1
AAPG1 AAPG2
AAPG(p=2)
Figure 5.13: Upper row: Mean and standard deviation of gPC1, AAPG1/AAPG2 andMCS (with sample size 106). Black dash lines represent instants when the pdf of solutionwill be included in Figure 5.14.Lower row: corresponding error of gPC1, AAPG1, AAPG2 compared to MCS.
on the settings of L and p. In the last test case when M 1, gPC2 becomes too expensive
to implement and the results show that AAPG scheme can effectively alleviate the curse of
dimensionality and offer significant improvement in accuracy. In the next section, we will
test the AAPG scheme on a more challenging multi-dof problem.
92
τ = 0 τ = 4 τ = 7 τ = 10M
CS
−1 0 1
−1
0
1
0 0.5 1
0
0.5
1
1.5
0 0.5 1
−1
0
1
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
gP
C1
−1 0 1
−1
0
1
0 0.5 1
0
0.5
1
1.5
0 0.5 1
−1
0
1
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
AA
PG
1
−1 0 1
−1
0
1
0 0.5 1
0
0.5
1
1.5
0 0.5 1
−1
0
1
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
AA
PG
2
−1 0 1
−1
0
1
0 0.5 1
0
0.5
1
1.5
0 0.5 1
−1
0
1
0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
Figure 5.14: Probability density function of u(τ ; ξ) (x-axis) vs. u(τ ; ξ) (y-axis) at τ =0, 4, 7, 10 computed using MCS (with sample size 1 × 106), gPC1/gPC2, AAPG1/AAPG2(p = 2).
5.4 Multi-dof Duffing oscillator
In this section we will apply different schemes to the multi-dof Duffing oscillator model (5.1)
and compare their performance. Two test cases will be included: the first has relatively
small spatial dof n = 2 while the second has n = 10. The nonlinear restoring force term
γ(u(t; ξ); ξ) is in the form of (5.3). Recall that mi, ci and ki, i = 1, . . . , n are the mass,
damping and stiffness factors on the i-th dof defined in (5.4). Z0i (ξ), Z1
i (ξ) are the initial
conditions of ui and ui respectively. ηi is the nonlinearity coefficient on the i-th dof. The
same time-marching step ∆t = 0.01 from last section is used in this section.
The random force f(t; ξ) at the i-th dof is fi(t; ξ) = f(t; ξ)mi, i = 1, . . . , n. σp =
0.4 and A = 0.1 is supplemented to define the autocorrelation function (5.41) of f(t; ξ),
which is expanded using KL expansion (3.28) as f(t; ξ) = f +∑Nf
i=1 fi(t)ξi. Here f(t) =
2 − 2 sin(2πt) exp(−0.1t), ξi (i = 1, . . . , Nf ) are independent random variables uniformly
distributed in [-1,1]. A plot of the random force is included in Figure 5.15. We study the
93
dynamic response of the stochastic system for t ∈ [0, 10] using standard RK4 time marching
method. MCS sample size is 105.
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
t
Figure 5.15: Stochastic forcing with σf = 0.4, A = 0.1 and f = 2.0−2.0 sin(2πτ) exp(−0.1τ).Solid line is the mean and grey area is mean ± standard deviation.
Test case 1: n = 2
We will implement the AAPG scheme in this small multi-dof test case. Ma = 12, Mf = 3
and total stochastic dof is M = Ma +Mf = 15. The solution error is defined as:
e =
∫ T0
(∑n
i=1 ∆2i (t))
1/2dt∫ T
0(∑n
i=1 M2i (t))
1/2dt, (5.53)
here Mi(t) is the MCS solution and ∆i(t) is the difference between gPC/AAPG solution and
MCS solution on the i-th dof. This definition of error has been normalized with regards to
the MCS solution and can be applied to both mean and standard deviation of the result.
The parameter settings in Table 5.4 are used in this test case.
mi ci ki Z0i Z1
i ηiµ 1.0 0.2 1.0 0 0 1.0σ 0.1 0.02 0.1 0.5 0.1 0.1
Table 5.4: Mean and standard deviation for parameters in the numerical studies of themulti-dof Duffing oscillator. All random scalars in the table can be expressed in the formof µ + σξi, where ξi, i = 1, . . . ,Ma are uniformly distributed random variables in [-1,1]. Formi, ci, ki, ηi, σ < µ is needed to guarantee positive value of these random variables. Ma = 6nbecause there are 6 random variables for each dof.
94
1 2 3 4p
10-4
10-3
10-2
gPC1
gPC2
AAPG1
AAPG2
(a) Error in mean.
1 2 3 4p
10-3
10-2
10-1
100
gPC1
gPC2
AAPG1
AAPG2
(b) Error in standard deviation.
Figure 5.16: Error of AAPG1/AAPG2 (dash lines) when the subproblems are solved usinggPC scheme of order p = 1, 2, 3, 4. Error of gPC1/gPC2 (solid lines) are also included forcomparison. ∆t = 0.01.
Similar to the test case in 5.3.3, we will present the error of the AAPG scheme with
different order p in the solution of its subproblems in Figure 5.16. We can conclude from
Figure 5.16 that AAPG1/AAPG2 converge at p = 3 and will continue using this setting
in the rest of this test case. Figure 5.17 shows the mean and standard deviation of the
displacement on the two dofs computed using gPC1/gPC2 and AAPG1/AAPG2 with p = 3.
The associated errors, defined in (5.53), are also included. pdf of u1(τ ; ξ) vs u1(τ ; ξ) at
t = [0, 7.2, 7.8, 9.4] are included in Figure 5.18. We can conclude that in this test case,
AAPG1 results in similar level of error to gPC1, and AAPG2 (p = 3) results in similar level
of error to gPC2.
Test case 2: n = 10
In this test case we will continue to use the same order of gPC solver in AAPG subproblems
from convergence study in the previous test case, i.e. p = 3. Settings of the random
parameters are summarized in Table 5.5. Note that the standard deviation for mi, ci, ki and
ηi are larger than in the previous test case. In the KL expansion 20 stochastic modes are
retained, making the total stochastic dof M = Ma +Mf = 6× 10 + 20 = 80. These settings
are implemented to make this test case a high-dimensional system with large uncertainty
and large nonlinear coefficient (highly uncertain too).
95
Mean standard deviationu
1(τ
;ξ)
0 1 2 3 4 5 6 7.2 7.8 9.4 10
t
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7.2 7.8 9.4 10t
0.05
0.1
0.15
0.2
0.25
0.3
MCS
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
u2(τ
;ξ)
0 1 2 3 4 5 6 7.2 7.8 9.4 10
t
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7.2 7.8 9.4 10t
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
MCS
gPC1
gPC2
AAPG1(p=3)
AAPG2(p=3)
e
0
0.2
0.4
0.6
0.8
1
10-3
gPC1
gPC2
gPC
AAPG1
AAPG2
AAPG(p=3)0
0.005
0.01
0.015
0.02
0.025
0.03
gPC
gPC2 AAPG2
gPC1 AAPG1
AAPG(p=3)
Figure 5.17: Upper row: Mean and standard deviation of u1(t; ξ) for t ∈ [0, 10]. Blackdash line mark the instants when the pdf of displacement vs velocity at the first dof will bepresented in Figure 5.18.Middle row: Mean and standard deviation of u2(t; ξ) for t ∈ [0, 10].Lower row: Error defined in (5.53) of the results of gPC1/gPC2, AAPG1/AAPG2 comparedto MCS (sample size 1 × 105) across the two dofs. Subproblems in AAPG are solved usinggPC scheme of order p = 3.
96
t = 0 t = 7.2 t = 7.8 t = 9.4M
CS
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
1.7 1.8 1.9 2 2.1 2.2−2
−1
0
1
2
1 1.5 2−1.5
−1
−0.5
0
0.6 0.7 0.8 0.9 1 1.1
−0.6
−0.4
−0.2
0
0.2
0.4
gP
C1
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
1.7 1.8 1.9 2 2.1 2.2−2
−1
0
1
2
1 1.5 2−1.5
−1
−0.5
0
0.6 0.7 0.8 0.9 1 1.1
−0.6
−0.4
−0.2
0
0.2
0.4
gP
C2
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
1.7 1.8 1.9 2 2.1 2.2−2
−1
0
1
2
1 1.5 2−1.5
−1
−0.5
0
0.6 0.7 0.8 0.9 1 1.1
−0.6
−0.4
−0.2
0
0.2
0.4
AA
PG
1
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
1.7 1.8 1.9 2 2.1 2.2−2
−1
0
1
2
1 1.5 2−1.5
−1
−0.5
0
0.6 0.7 0.8 0.9 1 1.1
−0.6
−0.4
−0.2
0
0.2
0.4
AA
PG
2
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
1.7 1.8 1.9 2 2.1 2.2−2
−1
0
1
2
1 1.5 2−1.5
−1
−0.5
0
0.6 0.7 0.8 0.9 1 1.1
−0.6
−0.4
−0.2
0
0.2
0.4
Figure 5.18: pdf of u1(τ ; ξ) (x-axis) vs. u1(t; ξ) (y-axis) at t = [0, 7.2, 7.8, 9.4] computedusing MCS (sample size 1× 105), gPC1/gPC2, AAPG1/AAPG2 (p = 3).
The wall time for different methods is presented in Table 5.6. It can be noted that
AAPG1 is much faster than the gPC methods, and gPC2 is much more computationally
expensive than all other methods. It is also worth to mention that the computational cost
of AAPG2 is as comparitively low as in previous test cases. Indeed, we will discuss the
theoretical cost of AAPG2 (and potentially higher order AAPGs) in the next chapter, which
directly lead us to the adaptive AAPG method.
Figure 5.19 includes the mean and standard deviation of u1(τ ; ξ) computed using gPC1,
gPC2 and AAPG1/AAPG2 with p = 3. The associated errors across all dofs, defined
in (5.53), are also included. The observed error of AAPG1 is much smaller than gPC1
while taking much less time to run. gPC2 provides improved accuracy at much higher cost.
AAPG2 scheme provides more accurate results than AAPG1 but is computationally expen-
sive because of its numerous second-order subproblems (80×79/2=3160 of them). We will
97
mi ci ki Z0i Z1
i ηiµ 1.0 0.2 1.0 0 0 1.0σ 0.15 0.03 0.15 0.5 0.1 0.2
Table 5.5: Settings in the numerical studies of the multi-dof Duffing oscillator.
gPC1 gPC2 AAPG1(p = 3) AAPG2(p = 3) MCS (sample size 105)72.55 4755.39 6.66 672.22 129.47
Table 5.6: Wall time (in seconds) of different schemes on an IBM Power 755 server with 4x8core 3.3GHz Power7 CPUs and 128GB RAM.
demonstrate in the numeric studies next chapter that the adaptive AAPG scheme effectively
reduces the number of subproblems to solve, resulting in accuracy similar to the AAPG2
and gPC2 scheme at a fraction of their cost.
Conclusions about the multi-dof Duffing oscillator test cases
Two test cases are examined in this section. The first is relatively simple (n = 2, M = 15)
and we can draw similar conclusions as in the single-dof Duffing oscillator test cases, i.e.
the level of accuracy in mean and standard deviation of the result is similar in the group
of gPC1/AAPG1 (p = 3) and gPC2/AAPG2 (p = 3), respectively. The second test case
features n = 10, M = 80 and we have demonstrated that the AAPG scheme is much faster
than the gPC scheme at similar level of accuracy.
5.5 Concluding remarks
In this chapter, we extended the AAPG scheme to solve nonlinear SODEs arising in dynamic
analysis of structural systems with geometrical nonlinearity. It was shown that similar to
linear SODEs, high-dimensional nonlinear SODEs can be decoupled into low-dimensional
subproblems that can be solved independently of each other. Such subproblems can be
solved using the gPC-based stochastic Galerkin projection scheme that was discussed in
single-dof and multi-dof Duffing oscillator systems. Some numerical challenges associated
with the application of the gPC scheme to solve nonlinear systems are discussed and the
pseudo-spectral approach is introduced as a more efficient alternative to the full-spectral
approach.
With a few test cases of the Duffing oscillator, we have shown that under low randomness
98
Mean standard deviationu
1(τ
;ξ)
0 1 2 3 4 5 6 7 8 9 9.5 10
t
-5
0
5
10
15
20
MCS
gPC1
gPC2
AAPG1(p=3)
0 1 2 3 4 5 6 7 8 9 9.5 10
t
0
0.5
1
1.5
2
2.5
3
MCS
gPC1
gPC2
AAPG1(p=3)
e
0
0.002
0.004
0.006
0.008
0.01
0.012
gPC
gPC1
gPC2
AAPG1
AAPG2
AAPG(p=3)0
0.05
0.1
0.15
0.2gPC1
gPC2
gPC AAPG(p=3)
AAPG1
AAPG2
Figure 5.19: Upper row: Mean and standard deviation of u1(t; ξ) for t ∈ [0, 10]. Dashline mark the instants when the pdf of displacement vs velocity at this dof will be presentedin Figure 5.20. Lower row: Error defined in (5.53) of the results of gPC1/gPC2, AAPG1compared to MCS (sample size 1×105). Subproblems in AAPG are solved using gPC schemeof order p = 3.
and stochasticity conditions, AAPG2 (subproblems solved using gPC2 or gPC3) generally
provides similar level of accuracy to gPC2, given both schemes are feasible. In single-
dof test case 3 (M = 100), gPC2 has large number of expansion terms and the resulting
deterministic system after applying the stochastic Galerkin projection becomes too large to
be solved numerically using our system. AAPG2 is advantageous in this test case since it
results in better level of accuracy compared to gPC1 and AAPG1. In multi-dof test case 2
(n = 10, M = 80), AAPG2 becomes computationally expensive because of the numerous
subproblems to solve. A solution to this problem will be provided in the next chapter,
when adaptive AAPG scheme is applied to selectively solve subproblems corresponding to
the more important ANOVA decomposition terms. The resulting adaptive AAPG2 scheme
is shown to provide comparable level of accuracy to the full AAPG at significantly lower
computational cost.
99
t = 0 t = 5.0 t = 8.0 t = 9.5
MC
S
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
12 14 16 18 20−2
0
2
4
10 12 14 16 18 20 22 24−7
−6
−5
−4
−3
−2
4 6 8 10 12 14 16
−6
−4
−2
0
gP
C1
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
12 14 16 18 20−2
0
2
4
10 12 14 16 18 20 22 24−7
−6
−5
−4
−3
−2
4 6 8 10 12 14 16
−6
−4
−2
0
gP
C2
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
12 14 16 18 20−2
0
2
4
10 12 14 16 18 20 22 24−7
−6
−5
−4
−3
−2
4 6 8 10 12 14 16
−6
−4
−2
0
AA
PG
1
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
0
0.5
12 14 16 18 20−2
0
2
4
10 12 14 16 18 20 22 24−7
−6
−5
−4
−3
−2
4 6 8 10 12 14 16
−6
−4
−2
0
Figure 5.20: pdf of u5(τ ; ξ) (x-axis) vs. u5(t; ξ) (y-axis) when t = [0, 7.5, 8.1, 8.8] computedusing MCS (sample size 1× 105), gPC1, gPC2 and AAPG1 (p = 3).
100
Chapter 6
Theoretical analysis and adaptive
AAPG scheme for structural
dynamics
Following the mathematical derivation and numerical evaluation in the previous chapters,
this chapter is dedicated to the theoretical analysis of the AAPG scheme with an emphasis
on a priori error estimation and the computation cost. This present analysis shows that both
the error and the computational cost grow quickly with larger number of random variables
especially when the ANOVA approximation order is high. Although the subproblems in
AAPG can be solved in parallel, it will be helpful to reduce the total number of subproblems.
The adaptive AAPG scheme exploits the fact that not all terms in the ANOVA expansion
are equally important and only the more important ones need to be computed. We include
a numerical study of a multi-dof Duffing oscillator with n = 10 spatial dofs and M = 80
stochastic dofs. While the first-order AAPG provides unsatisfying accuracy and the full
second-order AAPG is too costly, the adaptive second-order AAPG is affordable and provides
more accurate results compared to the first-order AAPG scheme. A few selection criteria
used in the adaptive ANOVA scheme are discussed.
6.1 A priori error estimation
In this section we will derive a priori error estimates for AAPG schemes. To begin with, it
is helpful to identify the different sources of errors, namely
101
1. Truncation error in the ANOVA approximation of the solution.
2. Stochastic discretization error when solving AAPG subproblems using gPC schemes.
3. Temporal discretization error when solving AAPG subproblems.
We first review a few key references on a priori error estimates for the AAPG scheme when
applied to different systems. For completeness, the strong and weighted residual form of
the governing equation are included in 6.1.2. The key conclusions from [216, 217] on the
stochastic and temporal discretization errors are summarized in 6.1.3 and 6.1.4. Adding
the ANOVA truncation error we then obtain the complete a priori error estimates in 6.1.5.
Some discussion on how the error constants scale with M are provided in 6.1.6. Finally, we
will extend the a priori error estimates to the nonlinear SODE system in 6.1.7.
6.1.1 Background and strategy
Recently, Audouze and Nair proposed a priori error estimates for Legendre polynomial
chaos based finite element approximation of elliptic and parabolic linear SPDEs [218] and
gPC based stochastic Galerkin methods for linear and nonlinear SODEs [217]. These results
can be used to obtain error estimates for gPC approximations of each AAPG subproblem
under appropriate stochastic regularity assumptions, since the AAPG subproblems are solved
using the gPC method. The ANOVA truncation error is estimated using a similar approach
adopted from [52], where a priori error estimates are provided for AAPG schemes applied to
high-dimensional parabolic linear SPDEs. Theorem 6.1.1 presents a priori error estimates for
the AAPG scheme when applied to high-dimensional SODEs encountered in linear stochastic
structural dynamics and can be extended to nonlinear systems as shown in 6.1.7.
6.1.2 Mathematical background and notations
The following second-order SODE system encountered in linear stochastic structural dynam-
ics is reproduced here for convenience
Mu(t; ξ) + K(ξ)u(t; ξ) + C(ξ)u(t; ξ) = f(t; ξ) a.s. in [0, T ]× ΓM , (6.1)
with initial conditions
u(0; ξ) = Z0(ξ), u(0; ξ) = Z1(ξ), where Z0(ξ),Z1(ξ) ∈ Rn. (6.2)
102
Here ξ = (ξ1, ξ2, · · · , ξM)T ∈ RM is a set of M independent and identically distributed (i.i.d)
random variables with joint pdf ρ(ξ) =∏M
i=1 ρi(ξi) and joint image ΓM = Γ1×· · ·×ΓM . ρ(ξ)
is assumed to be continuous on ΓM and its moments of all orders are supposed to be finite,
i.e.,⟨|ξ|k⟩< +∞,∀k ∈ N. We use the notation 〈·〉 to denote the expectation operator with
respect to ρ, i.e., 〈·〉 =∫
ΓM·ρ(ξ)dξ. u ∈ Rn is the displacement vector, t ∈ [0, T ] denotes
time (T < +∞) and n is the total number of dof. M ∈ Rn×n is an SPD deterministic
matrix while K(ξ) ∈ Rn×n is an SPD random matrix a.s. in ΓM . Under the assumption of
proportional/Rayleigh damping, i.e. C(ξ) = γ1K(ξ) + γ2M where γ1, γ2 ∈ R+, it follows
that C(ξ) is SPD a.s. in ΓM .
In practice, it is convenient to convert (6.1) into a first-order ODE system to take advan-
tage of the error analysis from [217] on gPC schemes applied to first-order random ODEs
with various temporal discretization schemes. We reproduce the first-order ODE system
(2.3) defined earlier in Chapter 2 below for convenience
U = F(t,U; ξ), (6.3)
where
U(t; ξ) =
(u(t; ξ)
u(t; ξ)
),F(t,U; ξ) = −A(ξ)U +
(M−1f(t; ξ)
0
),
A(ξ) =
(M−1C(ξ) M−1K(ξ)
−I 0
),
with U(t; ξ),F(t,U; ξ) ∈ R2n and initial conditions
U(0; ξ) = Z(ξ), where Z(ξ) ≡
(Z1(ξ)
Z0(ξ)
). (6.4)
The solution of (6.3)-(6.4) lies in the weighted Sobolev space L2(0, T ;L2
ρ(ΓM)2n
), where
L2ρ(Γ
M)2n = U : (ΓM)2n → R measurable such that ||U||L2ρ(ΓM )2n < +∞, where
||U(t; ·)||L2ρ(ΓM )2n =
(2n∑i=0
∫ΓM
U2i (t; ξ)ρ(ξ)dξ
) 12
. (6.5)
In the following sections we are concerned with a finite-dimensional random space and the
following classical tensor product Sobolev norms || · ||Hkρ (ΓM )2n will be used in the discussion
that follows:
103
||U(t; ·)||Hkρ (ΓM )2n =
2n∑i=1
k∑|j|=0
∫ΓM
(∂|j|Ui(t; ξ)
∂ξj11 · · · ∂ξjMM
)2
ρ(ξ)dξ
12
, (6.6)
where j = (j1, j2, . . . , jM) is an M-dimensional multi-index and |j| = j1 + j2 + · · ·+ jM .
We are interested in approximating U using the following truncated ANOVA approxi-
mation space (see (4.13) in Chap. 4.2.1)
ULANOVA = V2n
ξ,0 ⊕
(M⊕j1=1
V2nξ,j1
)⊕
(M⊕
j1<j2
V2nξ,j1j2
)⊕ · · · ⊕
(M⊕
j1<···<jL
V2nξ,j1...jL
), (6.7)
where the ANOVA subspaces are assembled using tensor products of multivariate polynomi-
als of total degree smaller than p as shown below
V2nξ = span
ψα(ξ) =
M∏j=1
ψαj(ξj), |α| = α1 + · · ·+ αM ≤ p
. (6.8)
where ψαj(ξ)αj∈N0 , j = 1, . . . ,M are the sequence of one-dimensional gPC basis functions.
The sequence ψα is dense in the Hilbert space L2ρ(Γ
M)2n if and only if the moment problem
is uniquely solvable for each random variable ξm,m = 1, . . . ,M , or equivalently, if ρ is
determinate (see [178] for more details). Hence, if ρ is determinate (e.g. normal or uniform
pdf) then any random process U(t; ξ) ∈ L2ρ(Γ
M)2n can be expanded as
U(t; ξ) =+∞∑k=0
βk(t)ψk(ξ), with βk(t) = 〈U(t; ·)ψk(·)〉 . (6.9)
The above expansion is convergent in a mean-square sense, i.e., with respect to || · ||L2ρ(ΓM )2n
(see [178]).
With the ANOVA approximation space (6.7), the weighted residual form associated with
(6.3)-(6.4) can be written as
Find U(t; ξ) ∈ ULANOVA such that for any V ∈ V L
⟨VT
(U(t; ξ)− F(t,U; ξ)
)⟩= 0,⟨
VT (U(0; ξ)−Z(ξ))⟩
= 0.(6.10)
104
Here V L is the space of test functions corresponding to the L-th order truncated ANOVA
decomposition defined as (see also (4.16) in Chap. 4.2.1)
V L = V0 ⊕
(M⊕j1=1
Vj1
)⊕
(M⊕
j1<j2
Vj1j2
)⊕ · · · ⊕
(M⊕
j1<···<jL
Vj1...jL
), (6.11)
with
V0 =wδ(ξ − ξa),w ∈ R2n, ξa ∈ Γ
,
Vj1 =
w(ξj1)
M∏i 6=j1
δ(ξi − ξai ),w ∈ L2(Γj1)2n, ξai ∈ Γi
,
...
Vj1...jL =
w(ξj1 . . . ξjL)
M∏i 6∈IL
δ(ξi − ξai ),w ∈ L2(Γj1 × · · · × ΓjL)2n, ξai ∈ Γi, IL = j1, j2, . . . , jL
,
where δ(ξ − ξa) =∏N
i=1 δ(ξi − ξai ).
The weighted residual form (6.10) can then be solved using standard first-order temporal
discretization methods. As an example, using the explicit Euler method would result in the
following expression⟨VT
(UL,m+1p −UL,m
p
∆t− F(tm,UL,m
p ; ξ)
)⟩= 0, ∀V ∈ V L, (6.12)⟨
VT(UL,0p −Z
)⟩= 0, ∀V ∈ V L. (6.13)
Here UL,mp =
∑|α|≤p β
L,mα ψα(ξ) denotes the ANOVA approximate solution computed at time
tm = m∆t for 0 ≤ m ≤ Nt with ∆t = TNt
, and βL,mα is the L-th order ANOVA approximation
of the coefficient vector βα(tm).
6.1.3 Spectral decomposition error
The one-dimensional L2ρj
-orthogonal projector in the j-th variable ξj can be defined as
πjp u =
p+1∑k=1
uk(t)ψk(ξj), uk(t) = (u(t; ·), ψk(·))L2ρ(Γ), (6.14)
105
where p is the highest polynomial degree of ψk(ξj). The corresponding multivariable L2ρ-
orthogonal projector is defined as
ΠMp u = π1
p · · · πMp u =
Nξ∑i=1
ui(t)ψi(ξ), ui(t) = (u(t; ·), ψi(·))L2ρ(ΓM ). (6.15)
The total number of polynomials in the expansion is Nξ = (M+p)!M !p!
. There exists a large volume
of literature on the error bounds for gPC spectral decompositions [219, 220, 216]. Existing
results can be found in [217] for uniform, normal, beta and arcsin pdf. For simplicity, we
summarize the results for uniform and normal distribution from [216] below:
Lemma 1: Let u ∈ Hkρ (ΓM) be a random process depending on M i.i.d random variables
with uniform or normal pdf. The following L2ρ-approximation errors hold
||u− ΠMp u||L2
ρ(ΓM ) ≤ CMϕ(p, k)||u||Hkρ (ΓM ) (6.16)
withϕ(p, k) =
p−k uniform pdf.
p−k/2 normal pdf,(6.17)
where C is a constant independent of p for the uniform [221] and normal distribution [222].
Lemma 1 is a direct extension of one-dimensional spectral error analysis in [221, 222] to
the multivariate case. The spectral approximation errors (6.16) are essentially obtained
by applying tensorization techniques to results for univariate spectral approximation errors
(see [223, 224, 216] for details).
6.1.4 Error estimation for gPC approximations of SODEs
In this section we will use the fact that F, described in (6.3), satisfies a Lipschitz condition
with respect to the second argument (see [217]), i.e.,
||F(t,U; ·)− F(t,V; ·)||2L2ρ(ΓM )2n ≤ L||U−V||L2
ρ(ΓM )2n , (6.18)
where the Lipschitz constant L satisfies
L =
(2D2
(λMmin)2+ 1
)1/2
, with D = max(λKmax, λCmax).
106
Here λKmax, λCmax are the largest eigenvalues of K(ξ) and C(ξ), respectively. λMmin is the
smallest eigenvalue of M. We shall assume that the stochastic regularity of K,C, f(t; ·) and
the initial conditions ensures that U(t; ·) ∈ Hkρ (ΓM)2n and U(t; ·) ∈ L2
ρ(ΓM)2n, and that an
explicit Euler scheme is used for the temporal discretization. Using Theorem 2 from [217],
it holds that
maxm=1,...,Nt
||U(tm; ·)−Ump ||L2
ρ(ΓM )2n ≤ CMϕ(p, k) maxt∈[0,T ]
||U(t; ·)||Hkρ (ΓM )2n︸ ︷︷ ︸
(I)
+ ∆teTL√Nξ − 1
2L√Nξ
maxt∈[0,T ]
||U(t; ·)||L2ρ(ΓM )2n︸ ︷︷ ︸
(II)
(6.19)
Here U(tm; ·) represents the solution of (6.3) and Ump is its gPC approximation computed
using an explicit Euler scheme, Nξ is the total number of terms in the spectral expansion
(6.15) and ϕ(p, k) is defined in (6.17). The first term (I) is the spectral approximation error
that can be estimated using Lemma 1. The temporal discretization error can be estimated
from the term (II) using the classical error analysis techniques derived for first-order deter-
ministic ODEs [217].
Remark 1: The random forcing term f(t; ξ) appears implicitly in (6.19) since
U = F(t,U; ξ) = F + JF(U)F = −A(ξ)U +
(M−1f
0
)+ JF(U)
(−A(ξ)U +
(M−1f
0
))
= (−A(ξ) + JF(U))
(−A(ξ)U +
(M−1f
0
))+
(M−1f
0
),
where JF(U) = dFdU
is the Jacobian matrix.
Remark 2: The temporal discretization error (II) vanishes when ∆t → 0, that is, the
explicit Euler scheme is convergent. It is worth mentioning that ∆t needs to satisfy certain
restrictions to ensure absolute stability (see Appendix B in [217] for a detailed discussion).
In particular, for stiff SODEs the Lipschitz constant L will be large which implies that the
time step ∆t needs to be very small to ensure absolute stability conditions.
107
Remark 3: The error estimate (6.19) can be rewritten in terms of the temporal partial
derivative of u as
maxm=1,...,Nt
||u(tm; ·)− ump ||L2ρ(ΓM )n ≤ CMϕ(p, k) max
t∈[0,T ]
(||u(t; ·)||Hk
ρ (ΓM )n + ||u(t; ·)||Hkρ (ΓM )n
)+ ∆t
eTL√Nξ − 1
2L√Nξ
maxt∈[0,T ]
(||u(t; ·)||Hk
ρ (ΓM )n + ||...u(t; ·)||Hkρ (ΓM )n
)
6.1.5 Main results
We provide here an a priori error estimate for the AAPG projection scheme applied to the
linear SODE system (6.1).
Theorem 6.1.1. Let U ∈ L2(0, T ;Hk
ρ (ΓM)2n)
with k > M/2 denote the solution of (6.3)-
(6.4) with U ∈ L2(0, T ;L2
ρ(ΓM)2n
)and initial value U(0; ξ) ∈ L2
ρ(ΓM)2n. Let UL,m
p be
the AAPG approximate solution from (6.12)-(6.13), where gPC scheme with PC order p
and explicit Euler time marching method is used for solving each subproblem. Assume all
mixed derivatives of U including no more than one differentiation with respect to each ξi are
piecewise continuous. Then, for 1 ≤ L < M , the following a priori error estimate holds:
maxm=1,...,Nt
||U(tm; ·)−UL,mp ||L2
ρ(ΓM )2n ≤ E1ϕ(p, k) + E2∆t
+ maxm=1,...,Nt
M∑s=L+1
∑i1<···<is
γi1...is
∣∣∣∣∣∣∣∣ ∂sU(tm; ·)∂ξi1 . . . ∂ξis
∣∣∣∣∣∣∣∣L∞(ΓM )2n
,
(6.20)
where E1 and E2 are constants independent of ∆t and p. The weights γi1...is > 0 are defined
as γi1...is =∏s
l=1
⟨(ξil − ξail)
2⟩1/2
.
Proof. The proof closely follows the arguments in [52], Theorem 2. First, we split the
approximation error at time tm as
eL,mp =(U(tm; ·)−UL(tm; ·)
)+(UL(tm; ·)−UL,m
p
), (6.21)
where UL(tm; ·) ∈ ULANOV A is the solution of (6.10). Since k > M/2, the Sobolev embedding
Hk(ΓM) → C0(ΓM) holds [225], which ensures the uniqueness of the anchored ANOVA
decomposition for U. Hence we have U(tm; ·) = UM(tm; ·). Using the triangle equality we
108
get
||eL,mp ||L2ρ(ΓM )2n ≤ ||UM(tm; ·)−UL(tm; ·)||L2
ρ(ΓM )2n︸ ︷︷ ︸(I)
+ ||UL(tm; ·)−UL,mp ||L2
ρ(ΓM )2n︸ ︷︷ ︸(II)
.(6.22)
The term (I) denotes the ANOVA truncation error and (II) is the error related to the
stochastic and temporal discretization incurred in each AAPG subproblem. The term (II)
can be expanded as
(II) = ||UL(tm; ·)−UL,mp ||L2
ρ(ΓM )2n
≤ ||U0(tm; ·)−U0,m||L2ρ(ΓM )2n +
L∑s=1
∑i1<···<is
||Ui1...is(tm; ·)−Ui1...is,mp ||L2
ρ(ΓM )2n
(6.23)
Since each AAPG subproblem is solved using the gPC scheme, we can use estimates of the
form (6.19) for each auxiliary variables Ui1...isp from which the AAPG component functions
Ui1...isp can be recovered. We then obtain the following inequality
maxm=1,...,Nt
||Ui1...is(tm; ·)−Ui1...is,mp ||L2
ρ(ΓM )2n ≤ Ei1...is1 ϕ(p, k) + Ei1...is
2 ∆t, (6.24)
where the constants Ei1...is1 and Ei1...is
2 are independent of p and ∆t. Since the zeroth order
subproblem governing the term U0 is deterministic, we can apply (6.19) with E01 = 0.
Substituting (6.24) into (6.23) leads to
maxm=1,...,Nt
||UL(tm; ·)−UL,mp ||L2
ρ(ΓM )2n ≤ E1ϕ(p, k) + E2∆t, (6.25)
where
E1 =L∑s=1
∑i1<···<is
Ei1...is1 ,
E2 = E02 +
L∑s=1
∑i1<···<is
Ei1...is2 .
Note that applying (6.19) to each auxiliary problem in AAPG result in constants Ei1...is1
109
and Ei1...is2 , which can be used to recover constants Ei1...is
1 and Ei1...is2 . As an example, error
constants associated with first-order AAPG solutions can be estimated as
maxm=1,...,Nt
||Ui1(tm; ·)−Ui1,mp ||L2
ρ(ΓM )2n
≤ maxm=1,...,Nt
||(Ui1(tm; ·)−U0(tm; ·)
)−(Ui1,mp −U0,m
)||L2
ρ(ΓM )2n
≤ maxm=1,...,Nt
||Ui1(tm; ·)− Ui1,mp ||L2
ρ(ΓM )2n + maxm=1,...,Nt
||U0(tm; ·)−U0,m||L2ρ(ΓM )2n
≤Ei11 ϕ(p, k) + Ei1
2 ∆t+ E02∆t = Ei1
1 ϕ(p, k) + (Ei12 + E0
2)∆t.
(6.26)
To estimate the bound for (I) due to ANOVA truncation, we introduce the following repre-
sentation of UL(tm; ξ) [226]
UL(tm; ξ) = U(tm; ξa) +L∑s=1
∑i1<···<is
∆i1 . . .∆isU(tm; ξa), (6.27)
where ∆j with ηj = ξj − ξaj denotes the finite difference operator
∆jU(tm; ξa) = U(tm; ξa1 , . . . , ξaj−1, ξ
aj + ηj, ξ
aj+1, . . . , ξ
aM)−U(tm; ξa). (6.28)
Note that (6.27) coincides with the expression for the so-called finite difference-HDMR (or
cut HDMR) expansion [45, 207, 226], which is due to the fact that Dirac product measure is
implemented to enforce orthogonality of the ANOVA component functions (see section 4.1).
The resulting ANOVA truncation error is
UM(tm; ξ)−UL(tm; ξ) =M∑
s=L+1
∑i1<···<is
∆i1 . . .∆isU(tm; ξa). (6.29)
Using integral representation for mixed finite differences
∆i1 . . .∆isUj(tm; ξa) =
∫ ξi1−ξai1
0
· · ·∫ ξis−ξais
0
∂sUj(tm; ξa + η)
∂ξi1 . . . ∂ξisdηi1 . . . dηis , j = 1, . . . , 2n,
where Uj is the j-th component of U, it follows that
|∆i1 . . .∆isUj(tm; ξa)| ≤ sup
ξ∈ΓM
∣∣∣∣∂sUj(tm; ξ)
∂ξi1 . . . ∂ξis
∣∣∣∣ s∏l=1
∣∣ξil − ξail∣∣ , j = 1, . . . , 2n. (6.30)
110
The mixed partial derivatives in (6.30) are bounded since we assumed they are piecewise
continuous. As a result we have
||∆i1 . . .∆isUj(tm; ξa)||2L2
ρ(ΓM ) ≤∣∣∣∣∣∣∣∣∂sUj(t
m; ·)∂ξi1 . . . ∂ξis
∣∣∣∣∣∣∣∣2L∞(ΓM )
s∏l=1
⟨(ξil − ξail)
2⟩
︸ ︷︷ ︸γ2i1...is
, j = 1, . . . , 2n.
Since ||∆i1 . . .∆isU(tm; ξa)||2L2ρ(ΓM )2n =
∑2nj=1 ||∆i1 . . .∆isUj(t
m; ξa)||2L2ρ(ΓM ), we get
||∆i1 . . .∆isU(tm; ξa)||2L2ρ(ΓM )2n ≤ γ2
i1...is
∣∣∣∣∣∣∣∣ ∂sU(tm; ·)∂ξi1 . . . ∂ξis
∣∣∣∣∣∣∣∣2L∞(ΓM )2n
. (6.31)
Substituting (6.31) into (6.29) we deduce
(I) ≤M∑
s=L+1
∑i1<···<is
||∆i1 . . .∆isU(tm; ξa)||L2ρ(ΓM )2n
≤M∑
s=L+1
∑i1<···<is
γi1...is
∣∣∣∣∣∣∣∣ ∂sU(tm; ·)∂ξi1 . . . ∂ξis
∣∣∣∣∣∣∣∣L∞ρ (ΓM )2n
.
(6.32)
The combination of (6.25) and (6.32) with (6.22) finally gives the error estimate (6.20).
Remark 1: The value of γj1...js in (6.32) depends on the choice of the anchor point ξa.
For example, if the random variables are uniformly distributed, setting |ξa1 | < 0.8165,∀i =
1, . . . ,M would result in γi1...is < 1 and γi1...is will decrease as s goes to M . The minimum
value of γi1...is =(
13
)s/2is obtained for ξai = 〈ξi〉 ,∀i = 1, . . . ,M (see [52]).
Remark 2: Other error estimates can be obtained when using different temporal dis-
cretization schemes in the gPC resolution. We refer the reader to [217] for temporal dis-
cretization errors provided for the θ-weighted scheme and explicit one-step schemes including
a two-stage Runge-Kutta method.
Remark 3: There exist time-step restrictions due to the explicit Euler scheme used in
gPC (see the discussion about absolute stability conditions for stochastic Galerkin schemes
based on explicit Euler scheme provided in [217]).
Remark 4: The constants E1, E2 depend on M and L which we will explore further in
the next section.
111
6.1.6 Discussion on error constants E1 and E2 in Theorem 6.1.1.
In this section we discuss how the error constants E1 and E2 in Theorem 6.1.1 grow with
respect to the number of random variables (M) and the ANOVA truncation order L. Recall
that the ANOVA component functions Ui1...is are recovered from auxiliary solutions Ui1...is
using the following steps
Ui1 = Ui1 −U0, i1 = 1, . . . ,M,
Ui1j2 = Ui1i2 −Ui1 −Ui2 −U0, 1 ≤ i1 < i2 ≤M,
Ui1i2i3 = Ui1i2i3 −Ui1i2 −Ui1i3 −Ui2i3 −Ui1 −Ui2 −Ui3 −U0, 1 ≤ i1 < i2 < i3 ≤M,
...
Ui1...is = Ui1...is −∑
l1<···<ls−1,lk∈Is
Ul1...ls−1 −∑
l1<···<ls−2,lk∈Is
Ul1...ls−2 − · · · −∑l1∈Is
Ul1 −U0,
(6.33)
where Is = i1, . . . , is. As a result, an L-th order ANOVA expansion of the solution can be
written in terms of auxiliary functions for L = 1, 2, 3 as follows
U1AAPG =
M∑i1
Ui1 − (M − 1)U0,
U2AAPG =
M∑i1<i2
Ui1i2 − (M − 2)M∑i1
Ui1 +(M − 1)(M − 2)
2U0,
U3AAPG =
M∑i1<i2<i3
Ui1i2i3 − (M − 3)M∑
i1<i2
Ui1i2 +(M − 2)(M − 3)
2
M∑i1
Ui1
− (M − 1)(M − 2)(M − 3)
6U0.
(6.34)
Since all the AAPG subproblems are solved using the gPC scheme, we can apply an error
estimate of the form (6.19) to each auxiliary variables Ui1...is and get
maxm=1,...,Nt
||Ui1...is(tm; ·)− Ui1...is,mp ||L2
ρ(ΓM )2n ≤ Ei1...is1 ϕ(p, k) + Ei1...is
2 ∆t, (6.35)
where the constants Ei1...is1 , Ei1...is
2 are independent of p and ∆t. (6.19) can also be applied
to the zeroth order subproblem of the solution U0 with E01 = 0. Substituting (6.35) into
112
(6.34) leads to
maxm=1,...,Nt
||UL(tm; ·)−UL,mp ||L2
ρ(ΓM )2n ≤ E1ϕ(p, k) + E2∆t, (6.36)
where ∀j = 1, 2
L = 1 : Ej =M∑i1
Ei1j + (M − 1)E0
j ,
L = 2 : Ej =M∑
i1<i2
Ei1i2j + (M − 2)
M∑i1
Ei1j +
(M − 1)(M − 2)
2E0j ,
L = 3 : Ej =M∑
i1<i2<i3
Ei1i2i3j + (M − 3)
M∑i1<i2
Ei1i2j +
(M − 2)(M − 3)
2
M∑i1
Ei1j
+(M − 1)(M − 2)(M − 3)
6E0j ,
(6.37)
with
Ei1...is1 = Cs max
t∈[0,T ]||U(t; ·)||Hk
ρ (ΓM )2n ∝ s, (6.38)
Ei1...is2 =
eTL√Nξ − 1
2L√Nξ
maxt∈[0,T ]
||U(t; ·)||L2ρ(ΓM )2n ∝
eTL√Nξ − 1
2L√Nξ
=T
2+T 2L
√Nξ
4+ · · · , (6.39)
and Nξ = (s+p)!s!p!
. Since we are are interested in the proportional relationship between the
constants E1 and E2 for different values of M and L rather than finding their exact rela-
tionship, we assume the norms maxt∈[0,T ] ||U(t; ·)||Hkρ (ΓM )2n and maxt∈[0,T ] ||U(t; ·)||L2
ρ(ΓM )2n
are approximately the same for all auxiliary solutions and independent of M and L. Note
that this assumption is only valid when the randomness in the system is relatively small.
In (6.39), Taylor expansion is applied and under the assumption that TL√Nξ is small, we
can conclude that Ei1...is2 is independent of the number of gPC expansion terms Nξ (or the
number of random variables s in subproblems). Note that (6.38) is also valid for the zeroth
order subproblem since E01 = 0. Substituting (6.38)-(6.39) into (6.37), we can evaluate how
E1 and E2 increases with M and L and the results obtained are shown in Table 6.1.
Note that the coefficients in Table 6.1 would be different if the assumptions we made on the
norms in (6.38)-(6.39) and TL√Nξ no longer holds. Nevertheless, the growth of E1 and E2
is dominated by the number of necessary auxiliary problems to solve in order to compute
113
Table 6.1: Error constants E1, E2 for L = 1, 2, 3.
L E1 (in units of Ei11 ) E2 (in units of E0
2)1 M 2M − 12 2M2 − 3M 2M2 − 4M + 13 2M3 − 8M2 + 7M 4
3M3−6M2 + 20
3M −1
the ANOVA components as shown in (6.37). In summary, E1, E2 ∼ O(ML), L = 1, 2, 3.
This means caution must be exercised when increasing the order of AAPG scheme, since the
increase in summation of error related to the stochastic and temporal discretization incurred
in each AAPG subproblem (II) may out-weigh the decrease of the ANOVA truncation error
(I) in (6.22).
6.1.7 A priori error estimate for the nonlinear SODE system
Recall that under the assumptions we made in Chapter 5, the governing equation for the
nonlinear Duffing oscillator can also be written in the form of (6.1) by replacing the linear
stiffness matrix K(ξ) with the following form of the nonlinear stiffness matrix K(u; ξ)
K(u; ξ) = K(ξ) + K(u(t; ξ)),
where K(u; ξ) =
k1 + k2 −k2 0 . . . 0
−k2 k2 + k3 −k3 . . . 0
· · · . . . ·0 . . . 0 −kn kn
, ki = εi(ξ)ki (ui − ui−1)2 , u0 = 0,
for i = 1, . . . , n. Here εi ≥ 0 is the nonlinearity parameter at the ith dof. The resulting
governing system can be converted to the following form of first-order SODE system (6.3)
U = F(t,U; ξ), (6.40)
where
U(t; ξ) =
(u(t; ξ)
u(t; ξ)
),F(t,U; ξ) = −A(U; ξ)U +
(M−1f(t; ξ)
0
),
A(U; ξ) =
(M−1C(ξ) M−1K(u; ξ)
−I 0
),
(6.41)
114
with U(t; ξ),F(t,U; ξ) ∈ R2n and initial conditions specified in (6.4). (6.40) is in the same
form as the first-order SODE system (6.3) for the linear system. The only difference as a
result of the nonlinear stiffness term is that A now explicitly depends on U.
Earlier we examined the error estimation of gPC approximation of random linear ODEs
in (6.19) when the explicit Euler scheme is used for temporal discretization. To extend the
results in section 6.1.4 to the nonlinear case we need to verify that a Lipschitz condition,
similar to (6.18) in the linear case, is in place for the nonlinear case. From (6.41) we have
||F(t,U; ξ)− F(t,V; ξ)||2L2ρ(ΓM )2n =
∣∣∣∣∣∣M−1C(ξ)(u− v) + M−1(K(u; ξ)u− K(v; ξ)v
)∣∣∣∣∣∣2L2ρ(ΓM )n
+ ||u− v||2L2ρ(ΓM )n . (6.42)
Assuming that the eigenvalues of M, C(ξ) are such that
0 < λMmin ≤ λMi ≤ λMmax, i = 1, . . . , n (6.43)
0 < λCmin ≤ λCi (ξ) ≤ λCmax, i = 1, . . . , n, ∀ξ ∈ ΓM , (6.44)
it follows that∣∣∣∣∣∣M−1C(ξ)(u− v) + M−1(K(u; ξ)u− K(v; ξ)v
)∣∣∣∣∣∣L2ρ(ΓM )n
≤ (λMmin)−1
(λCmax||u− v||L2
ρ(ΓM )n +∣∣∣∣∣∣K(u; ξ)u− K(v; ξ)v
∣∣∣∣∣∣L2ρ(ΓM )n
),
(6.45)
using the basic property ||Ax||L2ρ(ΓM )n ≤ λAmax ||x||L2
ρ(ΓM )n for any diagonalizable matrix A(ξ).
Let us now focus on estimating
∣∣∣∣∣∣∣∣∣∣∣∣∣∣K(u; ξ)u︸ ︷︷ ︸
g(u;ξ)
− K(v; ξ)v︸ ︷︷ ︸g(v;ξ)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣L2ρ(ΓM )n
. Using the mean value theo-
rem, there exist w(t; ξ) ∈ [u,v] := u(1− t) + tv : t ∈ [0, 1] such that
g(u; ξ)− g(v; ξ)
u− v= Jg(w), (6.46)
where Jg(·) is the Jacobian matrix of g(·; ξ). Assuming that the eigenvalues of Jg(w) are
such that ∣∣∣λJgi (t; ξ)∣∣∣ ≤ λJgmax, t ∈ [0, T ], (6.47)
115
we get
||g(u; ξ)− g(v; ξ)||L2ρ(ΓM )n ≤ λJgmax ||u− v||L2
ρ(ΓM )n . (6.48)
Substituting (6.48), (6.45) and the following equation
||U−V||2L2ρ(ΓM )2n = ||u− v||2L2
ρ(ΓM )n + ||u− v||2L2ρ(ΓM )n , (6.49)
into (6.42), we have
||F(t,U; ξ)− F(t,V; ξ)||2L2ρ(ΓM )2n
≤ 2
(λMmin)2
((λCmax
)2 ||u− v||2L2ρ(ΓM )n +
(λJgmax
)2 ||u− v||2L2ρ(ΓM )n
)+ ||u− v||2L2
ρ(ΓM )n
≤
(2
(D
λMmin
)2
+ 1
)||U−V||2L2
ρ(ΓM )2n ,
(6.50)
where D = max(λCmax, λJgmax).
In conclusion, the Lipschitz condition is indeed satisfied by the excitation term of the
nonlinear governing equation (6.40) with Lipschitz constant L =
(2(
DλMmin
)2
+ 1
)1/2
. As a
result, the error estimate provided in section 6.1.4 can be extended to the nonlinear case
under the condition that eigenvalues of Jg are bounded, as stated in (6.47). The same
condition applies to the main results (6.20), i.e., the largest eigenvalues of Jg in all of AAPG
subproblems over [0,T ] are bounded.
6.2 Analysis of the computational cost
In this section, we shall provide an estimation of the computational cost of gPC and AAPG
schemes for a linear and nonlinear model problem. As discussed in section 6.1.7, the same
form of first-order SODE (6.3) can be used in linear and nonlinear structural dynamic prob-
lems. For simplicity, we discuss the single-dof (n = 1) case, i.e.,
U = F(t,U; ξ), (6.51)
116
with U = (u, u)T , ξ ∈ RM . We shall first study the cost associated with the gPC scheme
and later apply the results to analyze the cost of the AAPG scheme, since the subproblems
are solved in parallel using the gPC scheme. Applying the gPC expansion of the solution
U =∑Nξ
i=1 Uiϕi(ξ) and the stochastic Galerkin scheme to (6.51), we have the following
deterministic system of SODEs
Ui
⟨ϕ2i
⟩=
⟨F(t,
Nξ∑i=1
Uiϕi; ·)ϕi
⟩, i = 1, 2, . . . , Nξ, (6.52)
where Nξ is the number of terms in the p-th order gPC expansion of u(ξ, t). The SODEs in
(6.52) are coupled and can be written collectively as
U = F(t,U; ξ), U,F ∈ R2Nξ , (6.53)
where
U =
U1 〈ϕ2
1〉U2 〈ϕ2
2〉...
UNξ
⟨ϕ2Nξ
⟩
,F(t,U; ξ) =
⟨F(t,
∑Nξi=1 Uiϕi; ·)ϕ1
⟩⟨F(t,
∑Nξi=1 Uiϕi; ·)ϕ2
⟩...⟨
F(t,∑Nξ
i=1 Uiϕi; ·)ϕNξ⟩
.
Assuming the fourth-order Runge-Kutta (RK4) method is used for solving (6.53), we focus
on the computational cost at each time step. The RK4 method can be written as:
Um+1 = Um +h
6(I1 + 2I2 + 2I3 + I4), (6.54)
where Um is the solution at step m, h is the time-marching step. Ii are intermediate incre-
ments that can be computed as:
I1 = F(tm,Um; ξ), (6.55)
I2 = F(tm +
h
2,Um +
h
2I1; ξ
), (6.56)
I3 = F(tm +
h
2,Um +
h
2I2; ξ
), (6.57)
I4 = F (tm + h,Um + hI3; ξ) . (6.58)
117
We can decompose the computational cost of each step of RK4 into the following three parts:
1. Evaluation of F(t,U; ξ) at intermediate points (four times). Let’s look at the model
single-dof structural dynamic problem
u(t; ξ) + 2ζu(t; ξ) + u(t; ξ) = p(t; ξ) a.s. in [0, w0T ]× ΓM , (6.59)
which has been normalized with regards to the mass and characteristic frequency w0.
Furthermore, we have assumed that the damping coefficient ζ is deterministic for sim-
plicity of notation. The governing equation can be converted to a first-order ODE of
the form of (6.51) with
U(t; ξ) =
(u(t; ξ)
u(t; ξ)
), and F(t,U; ξ) =
(p(t; ξ)− 2ζu(t; ξ)− u(t; ξ)
u(t; ξ)
). (6.60)
After applying the gPC expansion of the solution and the stochastic Galerkin condi-
tions, the deterministic ODE system is in the form of (6.53). In practice, the last part
u(t; ξ) in F(t,U; ξ) can be copied from U(t; ξ) computed during the previous step and
only the first part of F(t,U; ξ) needs to be computed as
1
〈ϕ2i 〉
⟨F1(t,
Nξ∑i=1
Uiϕi; ξ)ϕi
⟩= pi(t)− 2ζui(t)− ui(t), (6.61)
for i = 1, . . . , Nξ. Here we used the KL expansion of p(t; ξ) =∑M
i=1 pi(t)ϕi, and set
pi(t) = 0 for M < i ≤ Nξ. Denoting by C1 the number of floating-point operations
associated with evaluating (6.61), it follows that
C1 = 3Nξ, (6.62)
The terms 〈ϕ2i 〉 in (6.61) were moved to the left-hand-side because they also appear
on the other side of (6.53) and will cancel out. We count one multiplication and two
additions in (6.61) and there are Nξ of them. The cost associated with multiplication
and addition are assumed to be the same for simplicity.
2. Advance Um from previous step to intermediate points as inputs to intermediate func-
tion evaluations (6.56)-(6.58). As an example, let us examine the cost (denoted by C2)
to advance Um to Um + h2I1 as input to (6.56). We count one multiplication and one
118
addition of size 2Nξ, thus
C2 = 4Nξ. (6.63)
3. Update Um to Um+1 using (6.54) with the increments Ii, i = 1, 2, 3, 4 (one time). We
count two multiplications and four additions in (6.54) of size 2Nξ, thus
C3 = 12Nξ. (6.64)
The total cost at every RK4 time step can be estimated as
C = 4C1 + 3C2 + C3. (6.65)
Substituting in (6.62)-(6.64), we have
CgPC = 36Nξ = 36(N + p)!
N !p!. (6.66)
Using p=1 and p=2 in (6.66) leads to
C1gPC = 36(N + 1), (6.67)
C2gPC = 18(N + 1)(N + 2). (6.68)
Note that N = M when the gPC scheme is directly applied to solve (6.51). When the gPC
scheme is used to solve subproblems in the AAPG scheme, N is the number of random
variables in the subproblems. If we assume the subproblems in AAPG are solved using
the gPC scheme with p=2, the total number of operations in the AAPG scheme of order
L = 1, 2, 3 (without accounting for the cost associated with the statistical post-processing
procedure) are
C1AAPG = M(18× 2× 3) = 108M, (6.69)
using (6.68) with N = 1. Similarly
C2AAPG = C1
AAPG +1
2M(M − 1)(18× 3× 4) = 108M2, (6.70)
119
using (6.68) with N = 2, and
C3AAPG = C2
AAPG +1
6M(M − 1)(M − 2)(18× 4× 5)
= 60M3 − 72M2 + 120M.(6.71)
using (6.68) with N = 3. Next we consider a stochastic single-dof nonlinear Duffing oscillator
whose governing equation is reproduced here
u(ξ, t) + 2ζu(ξ, t) + u(ξ, t) + ε(ξ)u3(ξ) = p(ξ, t) a.s. in Γ× [0, w0T ]. (6.72)
Compared to the linear governing equation (6.59) there is an extra nonlinear term ε(ξ)u3(ξ).
The cost associated with its evaluation is
Cnl = 3ND, (6.73)
where ND is the number of non-zero terms in Dijk = 〈ϕiϕjϕk〉, i, j, k ∈ [1, Nξ]. By inspecting
the pattern of Dijk, the number of non-zero terms corresponding to p=1 and p=2 are given
by
N1D = 3M + 1, (6.74)
N2D = M3 + 4.5M(M + 1) + 1. (6.75)
Details on how to derive the above number of non-zero terms can be found in the Appendix.
Adding the additional cost associated with the nonlinear term to (6.65), the total cost at
each RK4 time-step for solving the nonlinear system (6.72) is
C = 4(C1 + Cnl) + 3C2 + C3. (6.76)
The resulting cost for the nonlinear Duffing oscillator example solved with gPC1/gPC2 and
AAPG1/AAPG2/AAPG3 where subproblems are solved using gPC2 are included in Table
6.2 together with the linear results (6.67)-(6.71).
These results are shown in Figure 6.1 for 1 ≤ M ≤ 100. Note that the total wall time
for AAPG can be significantly reduced when using a parallel implementation.
Remark 1: In the linear case, the number of operations in a single RK4 time step scales
120
Table 6.2: Number of floating-point operations for a single RK4 time step in the solution ofsingle-dof (n = 1) linear and nonlinear structural dynamics problems. M is total number ofrandom variables. gPC (p=2) is used for solving the AAPG subproblems. The computationalcost associated with statistical post-processing of the AAPG solution is not included in theestimate.
Linear NonlineargPC1 36M + 36 72M + 48gPC2 18M2 + 54M + 36 12M3 + 72M2 + 108M + 48AAPG1 (p=2) 108M 240MAAPG2 (p=2) 108M2 324M2 − 84MAAPG3 (p=2) 60M3−72M2 + 120M 224M3 − 350M2 + 364M
0 20 40 60 80 100
M
102
104
106
108
gPC1
gPC2
AAPG1(p=2)
AAPG2(p=2)
AAPG3(p=2)
(a) Linear
0 20 40 60 80 100
M
102
104
106
108
gPC1
gPC2
AAPG1(p=2)
AAPG2(p=2)
AAPG3(p=2)
(b) Nonlinear
Figure 6.1: Representation of computational cost of gPC and AAPG for 1 ≤M ≤ 100.
as O(Mp) for the gPC scheme, and O(ML) for the AAPG scheme. Although in Table 6.2
and Figure 6.1 we assumed gPC (p=2) is used to solve subproblems in the AAPG scheme,
a closer look at (6.69)-(6.71) reveals that the cost of AAPG is dominated by the number of
subproblems and scales as O(ML) regardless of the order of gPC used in the resolution of
subproblems. The numerical studies in Chapter 4 and 5 show that for problems with low
effective dimensions, using the same order gPC and AAPG scheme (p=L) leads to compara-
ble level of error. Based on the theoretical analysis of cost (number of operations) presented
here, it is clear that the serial AAPG is more expensive than the same order gPC scheme.
121
Remark 2: Most of the schemes have the same order of number of operations in the
nonlinear case compared to the linear case, except for gPC (p=2) which scales as O(M3) in
the nonlinear case compared to O(M2) in the linear case. We can see from Figure 6.1 (b)
that the serial AAPG (L = 2) becomes cheaper than gPC (p=2) for M ≥ 20.
Remark 3: Using a parallel implementation for AAPG will significantly lower the total
wall time. To demonstrate that, we include the wall time when different schemes are applied
to solve the single-dof SODE (5.38) in the following Table 6.3. The force is stochastic with
M = 35. An IBM Power 755 server with 4×8 core 3.3 GHz Power7 CPUs and 128GB RAM
is used. With 128 threads used in the parallel implementation of AAPG, the wall time for
AAPG3 (p=2) is very close to gPC2. And with more threads the wall time for AAPG scheme
can be further reduced.
Table 6.3: Wall time of different schemes applied to solve the single-dof SODE with M = 35.128 threads are used in the parallel implementation of AAPG (p=2).
gPC1 gPC2 AAPG1 AAPG2 AAPG3Wall time(s) 0.156 11.568 0.058 0.840 11.780
To demonstrate the trade-off between accuracy and cost for different schemes, we include
results when M = 15, 25 and 35 in Figure 6.2. To fully exploit the potential of the AAPG
scheme, results of AAPG (p=3) are also included. Note that the level of accuracy of different
schemes depends on the value of σ and ε (see Figure 5.12 in section 5.3.3). With σ = 2.0,
ε = 1.0 and otherwise identical settings to section 5.3.3, we can draw the following conclusions
from Figure 6.2:
1. Solving AAPG2/AAPG3 subproblems with gPC (p=3) would reduce the level of error
with additional computational cost.
2. Linear regression of all data points are plotted as straight lines in each figure. We can
conclude that gPC and AAPG (128 parallel threads), with the current problem setting
of σ = 2.0, ε = 1.0, offer similar level of error for the same wall time.
122
Mean Standard DeviationM
=15
10-1
100
101
Wall time (s)
10-4
10-3
10-2
Err
or
AAPG(p=2)
AAPG(p=3)
gPC1
AAPG1
AAPG2
gPC2
AAPG3
10-1
100
101
Wall time (s)
10-3
10-2
10-1
Err
or
AAPG(p=2)
AAPG(p=3)
AAPG1
AAPG2
gPC2
gPC1
AAPG3
M=
25
10-1
100
101
102
Wall time (s)
10-4
10-3
10-2
Err
or
AAPG(p=2)
AAPG(p=3)
gPC1
gPC2
AAPG2
AAPG1
AAPG3
10-1
100
101
102
Wall time (s)
10-3
10-2
10-1
Err
or
AAPG(p=2)
AAPG(p=3)AAPG1
gPC1
AAPG2
gPC2
AAPG3
M=
35
10-1
100
101
102
Wall time (s)
10-4
10-3
10-2
Err
or
AAPG(p=2)
AAPG(p=3)
AAPG1
gPC2
AAPG3AAPG2
gPC1
10-1
100
101
102
Wall time (s)
10-3
10-2
10-1
Err
or
AAPG(p=2)
AAPG(p=3)
AAPG1
AAPG2
gPC1
gPC2
AAPG3
Figure 6.2: Error vs wall time when different methods are applied to the nonlinear single-dof SODE (5.38), where the force is stochastic. M = 15, 25, 35. Straight lines are linearregression of all data points in each figure.
123
6.3 Adaptive AAPG Schemes
We have demonstrated in the previous section that the AAPG scheme can be potentially
much faster than the classical gPC scheme, especially when the subproblems are solved in
parallel. In this section we formulate a novel adaptive strategy to further reduce the cost of
AAPG scheme by exploiting the fact that not all random variables (or their combinations)
have equal impact to the system’s dynamic response. The resulting adaptive AAPG scheme
can be based on a variety of adaptivity criteria. Numerical studies implementing one of the
criteria are presented to validate the adaptive AAPG method.
Let u(ξ, t) be a random process that is function of a set of random variables ξ =
ξ1, . . . , ξN. Full ANOVA decomposition of u(ξ, t) is
u(ξ, t) =u0(t) +N∑j1
uj1(ξj1 , t) +N∑
j1<j2
uj1j2(ξj1 , ξj2 , t) + · · ·
+ uj1···jN (ξj1 , · · · , ξjN , t), ξ ∈ Γ ⊂ RN .
(6.77)
The preceding equation can be compactly rewritten as follows
u(ξ, t) =∑A⊆D
uA(ξA, t), (6.78)
where D = 1, . . . , N and the cardinality of A ⊆ D is expressed as |A|. The ANOVA term
uA(ξA, t) of order |A| is the cooperative contribution to the process from a group of active
random variables ξA = ξi, i ∈ A. Since the ANOVA decomposition is orthogonal with
respect to the Dirac product measure dµ(ξ) (see (4.5)), we have [227]
σ2 (u) =∑∅6=A⊆D
σ2(uA), (6.79)
where σ2 (u) is the variance of u(ξ, t) and σ2(uA) is the variance of uA(ξA, t). Inspired by
the work of Caflisch et al. [228] on effective dimensions to explain the success of quasi-Monte
Carlo method in certain applications, similar adaptivity criteria were proposed in [49, 47] for
the adaptive ANOVA decomposition scheme for time-dependent functions. Note that this
was studied in the literature in the non-intrusive setting. In contrast, the present research
focuses on leveraging ANOVA decomposition to accelerate stochastic Galerkin projection
schemes. The adaptivity conditions proposed in [49, 47] are summarized below.
124
Truncation sense: Effective dimension in the truncation sense is dt = |A|, where Aroughly corresponds to the important variables that satisfies∫ T
0
∑∅6=B⊆A
σ2(uB) dt ≥ α1
∫ T
0
σ2(u) dt. (6.80)
Here α1 ∈ (0, 1) is a pre-defined constant very close to 1.
Superposition sense: Effective dimension in the superposition sense is ds, which sat-
isfies ∫ T
0
∑0<|A|≤ds
σ2(uA) dt ≥ α2
∫ T
0
σ2(u) dt, (6.81)
where ds is similar to the concept of ANOVA decomposition order L in the previous chapter.
α2 ∈ (0, 1) is a pre-defined constant very clost to 1. It does not depend on the relative
importance between different variables and is thus more useful when all the variables are
almost equally important. As an example, as discussed in Section 2.3, the KL expansion of
the Young’s modulus in a heterogeneous material are high-dimensional, and the weight of
stochastic terms decrease very slowly.
Inspired by the two definitions of effective dimensions, we are ready to introduce the
following L-th order adaptive ANOVA decomposition
u(ξ, t) ≈ uL(ξ, t) =u0(t) +∑j1∈Φ1
uj1(ξj1 , t) +∑
(j1,j2)∈Φ2
uj1j2(ξj1 , ξj2 , t) + · · ·
+∑
(j1,··· ,jL)∈ΦL
uj1···jL(ξj1 , · · · , ξjL , t), ξ ∈ Γ ⊂ RN ,(6.82)
where Φi, i = 1, . . . , L are the collections of effective dimensions in the i-th order ANOVA
components. Since decomposition (6.82) contain components up to the L-th order in the
ANOVA decomposition, this is already adaptive in the superposition sense. The following
criterion is then used to select Φi, i = 2, . . . , L depending on the low-order statistics, i.e.,
variance of the first-order terms in the ANOVA decomposition. This is comparable to the
truncation sense (6.80) where relatively important variables are selected based on their con-
tribution to the total variance.
125
Criterion 1. The criteria for first-order active dimensions Φ1 is
∑j1∈Φ1
∫ T
0
σ2(uj1) dt ≥ θ1
N∑j1=1
∫ T
0
σ2(uj1) dt, (6.83)
here θ1 is a pre-defined threshold very close to 1. Second-order active dimensions (j1, j2) ∈Φ2, j1 < j2 satisfies j1, j2 ∈ Φ1. Higher order terms are selected similarly with different
threshold level. It can be noted that (6.83) is similar to (6.80) in the sense that it selects
the important variables by their contribution to the total variance. However, (6.83) contains
only first order ANOVA components while (6.80) contain all orders of ANOVA components.
Using (6.83) instead of (6.80) is necessary as we rarely know the total variance σ2(u) a priori.
In practice, we almost always keep all first-order ANOVA components, i.e., Φ1 = 1, . . . , N.In that case, (6.83) is still needed to select higher-order active dimensions.
Using the adaptive ANOVA decomposition (6.82) will reduce the number of auxiliary
subproblems to solve in the AAPG scheme. In theory, since all the auxiliary problems in
AAPG can be solved in parallel, this would not reduce the computation cost. But in practice
we rarely have access to unlimited number of computational nodes and reducing the number
of auxiliary problems is useful.
Criterion 2. There are other adaptive criteria used in the non-intrusive ANOVA de-
composition literature [47]. For an example, a criterion based on the mean of the component
functions uA was proposed by Ma and Zabaras [49] in the frequency analysis of SPDEs in
a non-intrusive function approximation setting. The following criterion is an extension to
the time-variant SODE system under study. For |A| ≥ 1, a component function is deemed
important if
γA =
∫ T0
⟨uA⟩dt∫ T
0
⟨u|A|−1
⟩dt≥ θ2, (6.84)
here u|A|−1 is the (|A| − 1)-th order ANOVA expansion of the solution, θ2 is a pre-defined
threshold very close to 0, first-order active dimensions Φ1 satisfy
γj1 =
∫ T0〈uj1〉 dt∫ T
0〈u0〉 dt
≥ θ12, ∀j1 ∈ Φ1, (6.85)
126
while the second-order active dimensions (j1, j2) ∈ Φ2, j1 < j2 satisfy
j1, j2 ∈ Φ1, (6.86)
γj1j2 =
∫ T0〈uj1j2〉 dt∫ T
0
⟨u0 +
∑Nj=1 u
j⟩dt≥ θ2
2, (6.87)
here θ12, θ
22 are pre-defined constants very close to 0. In practice, all first-order terms are
retained in the adaptive ANOVA decomposition and Φ1 is used in selecting higher-order
components.
6.3.1 Numerical studies
In this section we will apply the adaptive AAPG scheme with Criterion 1 to the problem
considered earlier in section 5.4, which describes a multi-dof Duffing oscillator whose initial
conditions, coefficient matrices and excitation force are uncertain. The mass (mi), damping
(ci), stiffness (ki), initial displacement (Z0i ), initial velocity (Z1
i ) and nonlinearity coefficient
(εi) at dof i in the governing equation (5.38) are defined in the following Table 6.4:
mi ci ki Z0i Z1
i εimean (µ) 1.0 0.2 1.0 0 0 1.0std (σ) 0.15 0.03 0.15 0.5 0.1 0.2
Table 6.4: Settings in the numerical studies of the multi-dof Duffing oscillator.
Table 6.4 includes the mean (µ) and standard deviation (σ) of the corresponding variables.
For an example, εi = µεi + σεjξj with ξj ∈ [−1, 1] being an uniformly distributed random
variable. With six random variables at each dof and 20 modes retained in the KL expansion
of the excitation force, there are in total M = 80 stochastic dofs in this Duffing oscillator
system with n = 10 physical dof.
We already demonstrated in section 5.4 that while AAPG1 has better accuracy and effi-
ciency than gPC1 (error in standard deviation eAAPG1 = 0.072, egPC1 = 0.172 and computa-
tional cost 5.11 s for AAPG1, 72.6 s for gPC1), there is room for improvement in accuracy.
Indeed, using gPC2 leads to higher accuracy in standard deviation (egPC2 = 0.055) but it
takes much longer to finish (4755.39 s). AAPG2 has better accuracy compared to AAPG1
but takes much longer (672.22 s) because of its numerous subproblems (80 × 79/2 = 3160
second-order subproblems). In this section, we will implement the adaptive AAPG scheme
127
to reduce the number of second order subproblems thus making the parallel implementation
much easier.
Figure 6.3 includes error in the mean and standard deviation of u(τ ; ξ) as defined in
(5.53). Results are computed using gPC1, gPC2, AAPG1, AAPG2 and adaptive AAPG2
with adaptivity criterion (6.83) and θ1 = 0.99. The adaptive AAPG2 selects 29 active
dimensions out of 80 and subsequently reduce the number of second-order subproblems from
3160 to 406. Also included in Figure 6.3 is error vs. wall time for each method. It is clear
that adaptive AAPG2 is much faster than AAPG2 and gPC2 while providing the same level
of accuracy.
Mean Standard Deviation
Err
or
of
u(τ
;ξ)
0
0.002
0.004
0.006
0.008
0.01
0.012gPC1
gPC2
gPC AAPG(p=3)
AAPG1
AAPG2 Adaptive
AAPG2
0
0.05
0.1
0.15
0.2
AAPG(p=3)
AAPG1
AAPG2Adaptive
AAPG2
gPC
gPC2
gPC1
Err
or
vs.
wall
tim
e
100
101
102
103
104
Wall time (s)
10-3
10-2
10-1
Err
or
AAPG(p=3)
gPC
AAPG2
gPC2
AAPG1
Adaptive
AAPG2
gPC1
100
101
102
103
104
Wall time (s)
10-2
10-1
100
Err
or
AAPG(p=3)
gPC
gPC2AAPG1
gPC1
Adaptive
AAPG2
AAPG2
Figure 6.3: Upper row: Error of mean and standard deviation of u(τ ; ξ) computed usinggPC1, gPC2, AAPG1, AAPG2 and adaptive AAPG2 schemes with adaptivity constantθ1 = 0.99. Sub-problems in AAPG schemes are solved using gPC3 (p = 3). Benchmarkcomputed using MCS with sample size 1 × 105. Lower row: Error vs wall time fordifferent schemes ran on an IBM Power 755 server with 4x 8core 3.3GHz Power7 CPUs and128GB RAM.
128
6.3.2 Remarks
We have demonstrated in Chapter 4 and 5 that the subproblems in the AAPG scheme can
be solved in parallel. However, we found the AAPG scheme faces problems when higher
order ANOVA decomposition is needed in systems with larger number of random variables.
The adaptive AAPG scheme selects the more important modes to solve based on certain
selection criteria and add another level of efficiency to the AAPG scheme.
129
Chapter 7
Concluding remarks and future work
7.1 Conclusions
The focus of this thesis is to formulate efficient numerical schemes to alleviate the curse of
dimensionality in parametric uncertainty analysis of high-dimensional stochastic linear and
nonlinear SODEs arising in dynamic analysis of stochastic structural problems. The main
contributions of the research are outlined below:
1. In Chapter 3, gPC-based stochastic Galerkin projection scheme and Generalized Spec-
tral Decomposition (GSD) methods are formulated for linear SODEs. Results obtained
for some model problems show that the gPC scheme is subject to the curse of dimen-
sionality. The proposed GSD method results in comparable level of accuracy to the gPC
scheme while incurring higher computational cost. The fact there lacks clear guidance
on the number of expansion terms, and need to iteratively estimate the decomposition
components made GSD a less favourable method. This motivates the development of
non-iterative decomposition schemes with better computational properties.
2. A novel numerical scheme is proposed for the solution of linear SODEs in Chapter
4. It was proved that using a Hoeffding functional ANOVA approximation and a
set of test functions for a stochastic Petrov-Galerkin projection scheme, the weighted
residual form of governing SODE can be decomposed into a sequence of decoupled low-
dimensional stochastic subproblems that can be solved independently of each other.
The resulting AAPG scheme holds significant potential to alleviate the curse of dimen-
sionality and this was confirmed by numerical studies.
130
3. Chapter 5 extends the proposed AAPG scheme to solve nonlinear SODEs. Similar to
the linear SODE system, it was proved that by using a functional ANOVA decom-
position of the solution and an appropriate test space, the original high-dimensional
nonlinear weak form can be decoupled into low-dimensional stochastic subproblems.
Numerical studies were presented for a range of test problems to demonstrate that the
AAPG scheme results in similar level of accuracy to the gPC scheme with much lower
computational cost when the subproblems are solved in parallel.
4. Chapter 6 presents a priori error estimates for the AAPG scheme in linear and non-
linear settings. Theoretical analysis of the computational cost reveals that for systems
with large number of stochastic dofs, there would be numerous subproblems to solve
for second-order AAPG schemes. This is also observed in the multi-dof test case 2
in Chapter 5. To address this issue, we introduced a few adaptive criteria to select
more important ANOVA decomposition terms and the corresponding subproblems to
solve. The resulting number of subproblems is much less in the adaptive AAPG scheme
when applied to the same test case 2 in Chapter 5. The adaptive AAPG scheme is
shown to provide comparable level of accuracy to the second-order gPC at much lower
computational cost.
7.2 Future work
Some directions for future research are outlined below:
Anchor points
In all the numerical studies for the AAPG scheme, we used ξai = 〈ξi〉 = 0, i = 1, . . . ,M as
anchor points. This artificial choice is made simply to facilitate easier numerical studies and
may have negative impact on the performance of the AAPG scheme, as suggested by Remark
1 following the a priori error estimate derived in section 6.1.5. Some studies in the context of
function approximation and nonintrusive uncertainty analysis also confirm the importance
of optimizing the anchor points [229, 230, 227]. [52] points out that the optimum choice
of anchor points would involve the solution of a high-dimensional minimization problem.
Further research is required to understand the impact of the anchor point on the performance
of the AAPG scheme. It may be worthwhile to develop efficient methods to find optimal or
near-optimal anchor points.
131
Adaptivity criteria
We have discussed a few different adaptivity criteria in section 6.3 and implemented one of
them in the test case. Although the results are promising, it will shed more light on the
performance and potential of the adaptive AAPG scheme if alternative adaptivity criteria
are tested and compared.
Other problems in linear stochastic structural dynamics
We focused on time domain analysis of linear and nonlinear stochastic structural dynamics in
this thesis. It is worth exploring the possibility of extending the AAPG scheme to frequency
domain analysis of linear stochastic structural dynamics and random algebraic eigenvalue
problems.
Complex solutions
Complex solutions such as shock formation or an energy cascade, or when the dependence of
the solution on the random input data varies rapidly are challenging cases for UQ methods.
In this thesis we applied the AAPG scheme to linear and nonlinear SODE systems that have
continuous and smooth solutions. The estimate of the ANOVA truncation error in Chapter
6 is based on the assumption of piecewise continuous solutions. It is worthwhile exploring
the mathematical foundation and numerical performance of the AAPG scheme for more
complicated systems, where the continuity and smoothness of the underlying solution take
more complex forms. Furthermore, we have used classical gPC-based stochastic Galerkin
projection scheme to solve AAPG subproblems in this thesis and it is possible to leverage
recent developments in gPC expansion, such as piecewise polynomial basis expansions [167],
wavelet basis expansion [165, 166] and multi-element gPC [168, 164], to address performance
issues in complex systems.
Combine with state vector decomposition methods
While the AAPG scheme focusses on alleviating the curse of dimensionality in the random
parameter space, an alternative approach is to decompose the state vector of the SODE using
non-overlapping Schur complement methods [231, 232] or overlapping Schwarz methods [231,
232, 233, 213]. It was demonstrated that the gPC method can be combined with the domain
decomposition methods to solve linear SPDEs [234] and nonlinear SODEs [235] efficiently.
It is worth exploring the possibility of combining state vector decomposition methods with
the AAPG scheme to achieve better efficiency.
132
Dynamic adaptive AAPG scheme
The adaptive AAPG scheme discussed in section 6.3 selectively computes the important
ANOVA terms and results in improved computational performance. However, the selection
criteria we discussed did not take into consideration the temporal change of ANOVA terms. A
dynamic selection criterion would more accurately reflect the importance of different ANOVA
terms at each time step, and lead to a more accurate adaptive AAPG scheme when used to
solve long-term SODEs.
133
Chapter 8
Appendix
8.1 Non-zero terms in Dijk = 〈φi(ξ)φj(ξ)φk(ξ)〉
In this section we will examine the number of non-zero termsNnnz inDijk = 〈φi(ξ)φj(ξ)φk(ξ)〉,i, j, k = 1, 2, . . . , P . Here ξ = (ξ1, ξ2, . . . , ξM)T is a set of M uniformly distributed random
variables in [-1,1] and φi(ξ), i = 1, 2, . . . , P are terms in Legendre polynomials of ξ up to
order p, where the number of terms P = (M+p)!M !p!
. Dijk is needed whenever the pseudo spectral
approach is used to compute product of gPC expansion of two or more random variables
(see section 5.3.2). We will provide a heuristic formulation for Nnnz when p = 1, 2, i.e. first-
and second-order gPC expansion here. The results for p ≥ 3 are not provided because we
are focusing on the case when M is relatively big, and the cost for gPC scheme becomes
prohibitively high when p ≥ 3.
Our heuristic approach is based on a simple observation that Dijk can be written as
Dijk = 〈φi(ξ)φj(ξ)φk(ξ)〉 = 〈ϕ1(ξ1)ϕ2(ξ2) . . . ϕM(ξM)〉
= 〈ϕ1(ξ1)〉 〈ϕ2(ξ2)〉 . . . 〈ϕM(ξM)〉 ,(8.1)
where φi(ξ), i = 1, 2, . . . , P are products of single-dof Legendre polynomials. They can be
regrouped into ϕl(ξl), l = 1, 2, . . . ,M that depends only on ξl. For Dijk 6= 0, we need to
enforce that 〈ϕl(ξl)〉 6= 0,∀l = 1, 2, . . . ,M . As a result, the order of ϕl(ξl) in terms of ξl
must be even, for all l = 1, 2, . . . ,M . Since p = 2, the order of ϕl(ξl) ∈ 0, 2, 4, 6. As an
example, we list all the non-zero terms in Dijk in the following Table 8.1 for M = 2, p = 2.
Note that we included all types of non-zero terms in second-order gPC expansion except
the type of ξ2i ξ
2j ξ
2k, where i 6= j 6= k. The total number of such terms are M(M − 1)(M − 2).
134
The total number of non-zero terms is:
N2nnz = M3 +
9
2M(M + 1) + 1. (8.2)
When p = 1, the highest order in terms of ξl, l = 1, 2, . . . ,M is 2. Thus
N1nnz = 3M + 1. (8.3)
In Table 8.2 we provide the value of Nnnz for M ∈ [1, 10] calculated using the Uncertainty
Quantification Toolkit by Sandia National Labs [215]. Substitute M ∈ [1, 10], p = 1, 2 to
(8.3) and (8.2), the result matches the value in Table 8.2.
Table 8.1: Non-zero terms in Dijk for M = 2, p = 2. In each category the typical index andthe total number are provided.
Order Dijk Typical (i,j,k)∗ Number of non-zero terms
Zeroth 1 (0,0,0) 1
Secondξ2
1 (0,1,1)3M
ξ22 (0,2,2)
Fourth
(ξ21 − 1)2 (0,3,3)
32M(M + 1)(ξ2
2 − 1)2 (0,5,5)ξ2
1ξ22 (0,4,4)
ξ21ξ
22 (1,4,2) 3M(M-1)
ξ21(ξ2
1 − 1) (1,3,1)3M
ξ22(ξ2
2 − 1) (2,5,2)
Sixth
(ξ21 − 1)3 (3,3,3)
M(ξ2
2 − 1)3 (5,5,5)(ξ2
1 − 1)ξ21ξ
22 (3,4,4)
3M(M-1)(ξ2
2 − 1)ξ21ξ
22 (5,4,4)
* Typical index values usually have a few variants. For an example(3,3,0), (3,0,3) are variants of (0,3,3).
Table 8.2: Number of non-zero terms calculated using the Uncertainty Quantification Toolkitby Sandia National Labs.
M 1 2 3 4 5 6 7 8 9 10p = 1 4 7 10 13 16 19 22 25 28 31p = 2 11 36 82 155 261 406 596 837 1135 1496
135
8.2 Examples applying the AAPG scheme to a linear
SODE system
To demonstrate the decoupling of AAPG sub-problems and shed light on the choice of specific
anchor points, we include two examples applying first-order AAPG scheme to
1. a single-dof linear system with stochastic initial conditions, and
2. a two-dof linear undamped system with stochastic stiffness coefficients.
8.2.1 Single-dof linear system with stochastic initial conditions
Consider the following SODE for a single-dof linear system
u(τ ; ξ) + 2ζu(τ ; ξ) + u(τ ; ξ) = p(τ) a.s. in [0, ω0T ]× Γ. (8.4)
Similar to the single-dof Duffing oscillator example in section 5.3, this equation has been
normalized respect to the mass factor m and uses a non-dimensional time τ = ω0t, where
ω0 is the undamped natural frequency of the corresponding linear system. The damping
ratio ζ and the forcing p(τ) are set to be deterministic. The stochastic initial conditions are
specified as
u(0; ξ) = Z0(ξ0) = u0 + aξ0, u(0; ξ) = Z1(ξ1) = v0 + bξ1,
where a, b are constants, ξ = (ξ0, ξ1)T is a two-dimensional random vector, i.e. M = 2.
Analytical solution for (8.4) with underdamped motion (ζ < 1) is
u(τ ; ξ0, ξ1) = e−ζτ[Z1(ξ1) + ζZ0(ξ0)
λsin(λτ) + Z0(ξ0) cos(λτ)
], (8.5)
where λ =√
1− ζ2 [236]. Re-organize the terms on the right-hand-side of (8.5), we have
u(τ ; ξ0, ξ1) = C0(τ)Z0(ξ0) + C1(τ)Z1(ξ1), (8.6)
where the deterministic terms C0(τ) = e−ζτ ζλ
sin(λτ) + cos(λτ), C1(τ) = e−ζτ sin(λτ)/λ. Set
the anchor points on the two stochastic dof to be ξa0 , ξa1 respectively, the zeroth-order ANOVA
term is
u0(τ) = C0(τ)(u0 + aξa0) + C1(τ)(v0 + bξa1). (8.7)
136
The first-order auxiliary functions are
uj1=0(τ ; ξ0) = C0(τ)(u0 + aξ0) + C1(τ)(v0 + bξa1),
uj1=1(τ ; ξ1) = C0(τ)(u0 + aξa0) + C1(τ)(v0 + bξ1).(8.8)
The following form of first-order ANOVA terms can be post-processed from the auxiliary
functions
uj1=0(τ ; ξ0) = uj1=0(τ ; ξ0)− u0(τ) = aC0(τ)(ξ0 − ξa0),
uj1=1(τ ; ξ1) = uj1=1(τ ; ξ0)− u0(τ) = bC1(τ)(ξ1 − ξa1).(8.9)
Combine the ANOVA zeroth-order (8.7) and first-order terms (8.9), we have
u0 + uj1=0 + uj1=1 = C0(τ)Z0(ξ0) + C1(τ)Z1(ξ1), (8.10)
which coincides with the exact value of u(τ ; ξ0, ξ1) in (8.6).
So far we have demonstrated how to apply the first-order AAPG scheme to solve a
single-dof test case with M = 2. In this trivial example ANOVA truncation order L = 1
is enough to capture the stochastic response with no truncation error because there is no
second or higher order interaction between random variables in the solution (8.6). The choice
of anchor points is unimportant in this example, but some studies in the context of function
approximation confirm the importance of optimizing the anchor points [229, 230, 227].
8.2.2 Two-dof linear undamped system with stochastic stiffness
coefficients
Now we consider a two-dof system excited by a deterministic harmonic force F sinwt and
governed by the following equation
Mu(t; ξ) + K(ξ)u(t; ξ) = F(t), t ∈ [0, T ], (8.11)
with initial conditions u(0; ξ) = (0, 0)T , u(0; ξ) = (0, 0)T and
u(t; ξ) =
(u1(t; ξ)
u2(t; ξ)
),M =
[m11 m12
m21 m22
],K(ξ) =
[k11(ξ) k12(ξ)
k21(ξ) k22(ξ)
],F(t) =
(F
0
)sinωt,
(8.12)
137
where ξ = (ξ0, . . . , ξM)T is a set of independent random variables. Assume the stochastic
stiffness coefficients can be described in the form of
kij(ξ) = kij + k0ijξ0 + k1
ijξ1, ∀i, j ∈ 1, 2, (8.13)
where the two random variables are assumed to be uniformly distributed, i.e. ξm ∈ [−1, 1],m =
0, 1 and ξ = (ξ0, ξ1)T . The solution can be written in the form(u1
u2
)=
(U1
U2
)sinωt. (8.14)
Substituting (8.14) and (8.12) in (8.11), we obtain[k11 −m11ω
2 k12 −m12ω2
k21 −m21ω2 k22 −m22ω
2
](U1
U2
)=
(F
0
). (8.15)
The solution is
U1(ξ) =(k22(ξ)−m22ω
2)F
|Z(ω; ξ)|, (8.16)
U2(ξ) =(k21(ξ)−m21ω
2)F
|Z(ω; ξ)|, (8.17)
where
Z(ω; ξ) =
[k11(ξ)−m11ω
2 k12(ξ)−m12ω2
k21(ξ)−m21ω2 k22(ξ)−m22ω
2
]. (8.18)
Applying first-order AAPG scheme to solve (8.11), we would have the following form of
zero-th order ANOVA term for U1(ξ)
U01 (ξa) =
(k22 + k022ξ
a0 + k1
22ξa1 −m22ω
2)F
|Z(ω; ξa)|, (8.19)
138
where ξam,m = 0, 1 are anchor points. First-order auxiliary solutions are
U j=01 (ξaj=0) =
(k22 + k022ξ0 + k1
22ξa1 −m22ω
2)F
|Z(ω; ξaj=0)|, (8.20)
U j=11 (ξaj=1) =
(k22 + k022ξ
a0 + k1
22ξ1 −m22ω2)F
|Z(ω; ξaj=1)|. (8.21)
Following the steps in (4.37), we can post-process (8.20) to get the corresponding first order
ANOVA component function
U j=01 = U j=0
1 (ξaj=0)− U01 (ξa)
=(k22 + k0
22ξ0 + k122ξ
a1 −m22ω
2)F
|Z(ω; ξa)|· |Z(ω; ξa)||Z(ω; ξaj=0)|
− (k22 + k022ξ
a0 + k1
22ξa1 −m22ω
2)F
|Z(ω; ξa)|
=(ξ0 − ξa0)k0
22F
|Z(ω; ξa)|+ (|Z(ω; ξa)||Z(ω; ξaj=0)|
− 1)(k22 + k0
22ξ0 + k122ξ
a1 −m22ω
2)F
|Z(ω; ξa)|(8.22)
The second term on the right-hand-side of (8.29) is comparatively smaller than the first term
since
|Z(ω; ξa)| ≈ |Z(ω; ξaj=0)| (8.23)
thus can be ignored, resulting in the following approximation
U j=01 ≈ (ξ0 − ξa0)k0
22F
|Z(ω; ξa)|. (8.24)
Similarly, we can post-process (8.21) to get
U j=11 ≈ (ξ1 − ξa1)k1
22F
|Z(ω; ξa)|. (8.25)
Combining the ANOVA zeroth-order and first-order terms in (8.19), (8.24), (8.25), we get
U01 + U j=0
1 + U j=11 ≈ (k22 + k0
22ξ0 + k122ξ1 −m22ω
2)F
|Z(ω; ξa)|. (8.26)
While the actual solution is
U1(ξ) =(k22 + k0
22ξ0 + k122ξ1 −m22ω
2)F
|Z(ω; ξ)|. (8.27)
139
Using the approximation
|Z(ω; ξa)| ≈ |Z(ω; ξ)|, (8.28)
we have
U01 + U j=0
1 + U j=11 ≈ U1(ξ). (8.29)
Similar results can be obtained for U2(ξ). Substituteing (8.29) and approximation of U2(ξ)
into (8.14), we can conclude that first-order AAPG provides an effective way to solve the
original system by solving decoupled low-dimensional sub-problems (8.20), (8.21), of which
the solutions can be post-processed to recover the approximation of the solution of (8.11).
In this example, we have made a few approximations in (8.23), (8.28). Take a closer look
at |Z(ω; ξ)| from (8.18)
|Z(ω; ξ)| = (k11 −m11ω2)(k22 −m22ω
2)− (k12 −m12ω2)(k21 −m21ω
2)
= k11k22 − k12k21 − k11m22ω2 − k22m11ω
2 + k12m21ω2 + k21m12ω
2
+m11m22ω4 −m12m21ω
4. (8.30)
If the anchor points are set to be ξa = 〈ξ〉 and assuming k11, k12, k21, k22 are independent
random variables, we will have
〈|Z(ω; ξ)|〉 =⟨|Z(ω; ξaj=0)|
⟩= |Z(ω; ξa)|, (8.31)
and the resulting approximating error in (8.23), (8.28) will be minimized. Unfortunately, k12
and k21 are not independent of each other thus the aforementioned choice of anchor point is
not optimum. In practice, it is typically expensive to calculate the optimum choice of anchor
points and ξa = 〈ξ〉 is usually considered a good choice with minimum cost. In our example,
the correlation between k12 and k21 will only affect the term k12k21 in (8.30) and we might
be able to ignore its effect if the other terms are more important.
140
Bibliography
[1] Andy Keane and Prasanth Nair. Computational approaches for aerospace design: the
pursuit of excellence. John Wiley & Sons, 2005.
[2] Ioannis Karatzas and Steven Shreve. Brownian motion and stochastic calculus, volume
113. Springer Science & Business Media, 2012.
[3] Christian Soize. A nonparametric model of random uncertainties for reduced matrix
models in structural dynamics. Probabilistic Engineering Mechanics, 15(3):277–294,
2000.
[4] Christian Soize. A comprehensive overview of a non-parametric probabilistic approach
of model uncertainties for predictive models in structural dynamics. Journal of Sound
and Vibration, 288(3):623–652, 2005.
[5] Christian Soize. Random matrix theory for modeling uncertainties in computational
mechanics. Computer Methods in Applied Mechanics and Engineering, 194(12):1333–
1366, 2005.
[6] Sondipon Adhikari. Matrix variate distributions for probabilistic structural dynamics.
AIAA Journal, 45(7):1748–1762, 2007.
[7] Sondipon Adhikari. On the quantification of damping model uncertainty. Journal of
Sound and Vibration, 306(1):153–171, 2007.
[8] Bernd Moller and Michael Beer. Fuzzy randomness: uncertainty in civil engineering
and computational mechanics. Springer Science & Business Media, 2013.
[9] David Moens and Dirk Vandepitte. A survey of non-probabilistic uncertainty treatment
in finite element analysis. Computer Methods in Applied Mechanics and Engineering,
194(12):1527–1555, 2005.
141
[10] James L Beck and Lambros S Katafygiotis. Updating models and their uncertainties.
I: Bayesian statistical framework. Journal of Engineering Mechanics, 124(4):455–461,
1998.
[11] James L Beck and Siu-Kui Au. Bayesian updating of structural models and reliabil-
ity using markov chain Monte Carlo simulation. Journal of Engineering Mechanics,
128(4):380–391, 2002.
[12] David Moens and Michael Hanss. Non-probabilistic finite element analysis for para-
metric uncertainty treatment in applied mechanics: Recent advances. Finite Elements
in Analysis and Design, 47(1):4–16, 2011.
[13] Brian R Mace, Dirk VH Vandepitte, and Pascal Lardeur. Uncertainty in structural
dynamics. Finite Elements in Analysis and Design, 47(1):1–3, 2011.
[14] Gerhart I Schueller. A state-of-the-art report on computational stochastic mechanics.
Probabilistic Engineering Mechanics, 12(4):197–321, 1997.
[15] Hermann G Matthies, Christoph E Brenner, Christian G Bucher, and C Guedes Soares.
Uncertainties in probabilistic numerical analysis of structures and solids-stochastic
finite elements. Structural Safety, 19(3):283–336, 1997.
[16] CS Manohar and RA Ibrahim. Progress in structural dynamics with stochastic param-
eter variations: 1987-1998. Applied Mechanics Reviews, 52(5):177–197, 1999.
[17] Gerhart I Schueller. Computational stochastic mechanics-recent advances. Computers
& Structures, 79(22):2225–2234, 2001.
[18] Gerhart I Schueller and HJ Pradlwarter. Uncertain linear systems in dynamics: Ret-
rospective and recent developments by stochastic approaches. Engineering Structures,
31(11):2507–2517, 2009.
[19] George Stefanou. The stochastic finite element method: past, present and future.
Computer Methods in Applied Mechanics and Engineering, 198(9):1031–1051, 2009.
[20] Michael B Giles. Multilevel Monte Carlo path simulation. Operations Research,
56(3):607–617, 2008.
[21] Michael Kleiber and Tran D Hien. The stochastic finite element method: for use on
IBM PC/XT. Wiley, 1992.
142
[22] Marcin Kaminski. Stochastic second-order perturbation approach to the stress-based
finite element method. International Journal of Solids and Structures, 38(21):3831–
3852, 2001.
[23] Fumio Yamazaki, Masanobu Shinozuka, and Gautam Dasgupta. Neumann expansion
for stochastic finite element analysis. Journal of Engineering Mechanics, 114(8):1335–
1354, 1988.
[24] Dongbin Xiu and George Em Karniadakis. Supersensitivity due to uncertain boundary
conditions. International Journal for Numerical Methods in Engineering, 61(12):2114–
2138, 2004.
[25] Prasanth B Nair, Arindam Choudhury, and Andy J Keane. Some greedy learning
algorithms for sparse regression and classification with mercer kernels. Journal of
Machine Learning Research, 3(Dec):781–801, 2002.
[26] Daniel M Tartakovsky, Alberto Guadagnini, and Monica Riva. Stochastic averaging of
nonlinear flows in heterogeneous porous media. Journal of Fluid Mechanics, 492:47–62,
2003.
[27] Daniel M Tartakovsky and Alberto Guadagnini. Prior mapping for nonlinear flows in
random environments. Physical Review E, 64(3):035302, 2001.
[28] Liyong Li, Hamdi A Tchelepi, and Dongxiao Zhang. Perturbation-based moment
equation approach for flow in heterogeneous porous media: applicability range and
analysis of high-order terms. Journal of Computational Physics, 188(1):296–317, 2003.
[29] Roger Ghanem and Pol D Spanos. Stochastic finite elements: a spectral approach.
Springer-Verlag New York, Inc., 1991.
[30] D. Xiu and G.E. Karniadakis. The Wiener-Askey polynomial chaos for stochastic
differential equations. SIAM Journal on Scientific Computing, 24(2):619–644, 2002.
[31] Abhijit Sarkar and Roger Ghanem. Mid-frequency structural dynamics with parameter
uncertainty. Computer Methods in Applied Mechanics and Engineering, 191(47):5499–
5513, 2002.
[32] Roger Ghanem and Debraj Ghosh. Efficient characterization of the random eigenvalue
problem in a polynomial chaos decomposition. International Journal for Numerical
Methods in Engineering, 72(4):486–504, 2007.
143
[33] K Sepahvand and S Marburg. Stochastic FEM to structural vibration with parametric
uncertainty. In Multiscale Modeling and Uncertainty Quantification of Materials and
Structures, pages 299–306. Springer, 2014.
[34] Jerome Didier, Beatrice Faverjon, and Jean-Jacques Sinou. Analyzing the dynamic
response of a rotor system under uncertain parameters by polynomial chaos expansion.
Journal of Vibration and Control, 18(5):712–732, 2012.
[35] Geraud Blatman and Bruno Sudret. Adaptive sparse polynomial chaos expansion
based on least angle regression. Journal of Computational Physics, 230(6):2345–2367,
2011.
[36] Anthony Nouy. A generalized spectral decomposition technique to solve a class of linear
stochastic partial differential equations. Computer Methods in Applied Mechanics and
Engineering, 196(45-48):4521–4537, 2007.
[37] Anthony Nouy. Generalized spectral decomposition method for solving stochastic finite
element equations: invariant subspace problem and dedicated algorithms. Computer
Methods in Applied Mechanics and Engineering, 197(51-52):4718–4736, 2008.
[38] Anthony Nouy. Recent developments in spectral stochastic methods for the numerical
solution of stochastic partial differential equations. Archives of Computational Methods
in Engineering, 16(3):251–285, 2009.
[39] Christophe Audouze and Prasanth B Nair. Galerkin reduced-order modeling scheme
for time-dependent randomly parametrized linear partial differential equations. Inter-
national Journal for Numerical Methods in Engineering, 92(4):370–398, 2012.
[40] Themistoklis P Sapsis and Pierre FJ Lermusiaux. Dynamically orthogonal field equa-
tions for continuous stochastic dynamical systems. Physica D: Nonlinear Phenomena,
238(23):2347–2360, 2009.
[41] Mulin Cheng, Thomas Y Hou, and Zhiwen Zhang. A dynamically bi-orthogonal
method for time-dependent stochastic partial differential equations I: Derivation and
algorithms. Journal of Computational Physics, 242:843–868, 2013.
[42] M. Chevreuil and A. Nouy. Model order reduction based on proper generalized de-
composition for the propagation of uncertainties in structural dynamics. International
Journal for Numerical Methods in Engineering, 89(2):241–268, 2012.
144
[43] Wassily Hoeffding. A class of statistics with asymptotically normal distribution. The
Annals of Mathematical Statistics, 19(3):293–325, 1948.
[44] Bradley Efron and Charles Stein. The jackknife estimate of variance. The Annals of
Statistics, 9(3):586–596, 1981.
[45] Herschel Rabitz and Omer F Alis. General foundations of high-dimensional model
representations. Journal of Mathematical Chemistry, 25(2-3):197–233, 1999.
[46] Michael Griebel. Sparse grids and related approximation schemes for higher dimen-
sional problems. In Luis M. Pardo, Allan Pinkus, Endre Suli, and Michael J. Todd,
editors, Foundations of Computational Mathematics (FoCM05), Santander, pages 106–
161. Cambridge University Press, 2006.
[47] Xiu Yang, Minseok Choi, Guang Lin, and George Em Karniadakis. Adaptive ANOVA
decomposition of stochastic incompressible and compressible flows. Journal of Com-
putational Physics, 231(4):1587–1614, 2012.
[48] Jasmine Foo and George Em Karniadakis. Multi-element probabilistic collocation
method in high dimensions. Journal of Computational Physics, 229(5):1536–1557,
2010.
[49] Xiang Ma and Nicholas Zabaras. An adaptive high-dimensional stochastic model repre-
sentation technique for the solution of stochastic partial differential equations. Journal
of Computational Physics, 229(10):3884–3915, 2010.
[50] Sharif Rahman. A polynomial dimensional decomposition for stochastic computing.
International Journal for Numerical Methods in Engineering, 76(13):2091–2116, 2008.
[51] Prasanth B Nair, Par Hakansson, and Christophe Audouze. Prospects for overcoming
the curse of dimensionality in polynomial chaos based stochastic projection schemes.
In Proceedings of the 4th European Conference on Computational Mechanics, Paris,
France, 2010.
[52] Christophe Audouze and Prasanth B Nair. Anchored ANOVA Petrov-Galerkin projec-
tion schemes for parabolic stochastic partial differential equations. Computer Methods
in Applied Mechanics and Engineering, 276:362–395, 2014.
145
[53] Dongxiao Zhang. Stochastic methods for flow in porous media: coping with uncertain-
ties. Academic Press, 2001.
[54] Michele M Putko, Arthur C Taylor, Perry A Newman, and Lawrence L Green. Ap-
proach for input uncertainty propagation and robust design in cfd using sensitivity
derivatives. Journal of Fluids Engineering, 124(1):60–69, 2002.
[55] TK Caughey and Fai Ma. The exact steady-state solution of a class of non-linear
stochastic systems. International Journal of Non-linear Mechanics, 17(3):137–142,
1982.
[56] Yu-Kweng Lin and Guo-Qiang Cai. Probabilistic structural dynamics: advanced theory
and applications. Mcgraw-Hill, Inc., New York, 1995.
[57] Bernt Ø ksendal. Stochastic Differential Equations. An introduction with Applications.
Springer-Verlag, Berlin, sixth edition, 2003.
[58] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential
equations. Springer, Berlin, 1995.
[59] Paul Kree and Christian Soize. Mathematics of random phenomena: random vibrations
of mechanical structures. Reidel Pub. Co., 1986.
[60] Yuri Rozanov. Random fields and stochastic partial differential equations. Springer
Science & Business Media, 1998.
[61] John Walsh. An introduction to stochastic partial differential equations. In Ecole d’Ete
de Probabilites de Saint Flour XIV, pages 265–439. Springer, 1984.
[62] R Singh and C Lee. Frequency response of linear systems with parameter uncertainties.
Journal of Sound and Vibration, 168(1):71–92, 1993.
[63] Su-Huan Chen, Zhong-Sheng Liu, and Zong-Fen Zhang. Random vibration analysis for
large-scale structures with random parameters. Computers & structures, 43(4):681–
685, 1992.
[64] L Fryba, S Nakagiri, and N Yoshikawa. Stochastic finite elements for a beam on a
random foundation with uncertain damping under a moving force. Journal of Sound
and Vibration, 163(1):31–45, 1993.
146
[65] Prasanth B Nair and Andrew J Keane. An approximate solution scheme for the al-
gebraic random eigenvalue problem. Journal of Sound and Vibration, 260(1):45–65,
2003.
[66] Roger Ghanem and Abhijit Sarkar. Reduced models for the medium-frequency dynam-
ics of stochastic systems. The Journal of the Acoustical Society of America, 113(2):834–
846, 2003.
[67] S Anantha Ramu and R Ganesan. A galerkin finite element technique for stochastic
field problems. Computer Methods in Applied Mechanics and Engineering, 105(3):315–
331, 1993.
[68] T. S. Sankar, S. A. Ramu, and R. Ganesan. Stochastic finite element analysis for high
speed rotors. Journal of Vibration and Acoustics, 115:59, 1993.
[69] Masanobu Shinozuka and Clifford J Astill. Random eigenvalue problems in structural
analysis. AIAA Journal, 10(4):456–462, 1972.
[70] Jurgen Vom Scheidt and Walter Purkert. Random eigenvalue problems. North Holland:
akademie-verlag, 1983.
[71] Madan Lal Mehta. Random matrices, volume 142. Academic Press, 2004.
[72] A Nayfeh. H., Mook, D., T. John Wiley & Sons, New York, 1979.
[73] DE Panayotoukanos, ND Panayotounakou, and AF Vakakis. On the solution of the
unforced damped duffing oscillator with no linear stiffness term. Nonlinear Dynamics,
28(1):1–16, 2002.
[74] Thomas K Caughey. Equivalent linearization techniques. The Journal of the Acoustical
Society of America, 35(11):1706–1711, 1963.
[75] Pol D Spanos. Stochastic linearization in structural dynamics. Applied Mechanics
Reviews, 34, 1981.
[76] John Brian Roberts and Pol D Spanos. Random vibration and statistical linearization.
John Wiley & Sons New York, 1990.
[77] T.K. Caughey. On the response of non-linear oscillators to stochastic excitation. Prob-
abilistic Engineering Mechanics, 1(1):2 – 4, 1986.
147
[78] Michael A Tognarelli, Jun Zhao, K Balaji Rao, and Ahsan Kareem. Equivalent sta-
tistical quadratization and cubicization for nonlinear systems. Journal of Engineering
Mechanics, 123(5):512–523, 1997.
[79] Michael A Tognarelli, Jun Zhao, and Ahsan Kareem. Equivalent statistical cubicization
for system and forcing nonlinearities. Journal of Engineering Mechanics, 123(8):890–
893, 1997.
[80] PD Spanos, M Di Paola, and G Failla. A Galerkin approach for power spectrum
determination of nonlinear oscillators. Meccanica, 37(1-2):51–65, 2002.
[81] Giulio Fatica and Claudio Floris. Moment equation analysis of base-isolated buildings
subjected to support motion. Journal of Engineering Mechanics, 129(1):94–106, 2003.
[82] Alexander A Muravyov and Stephen A Rizzi. Determination of nonlinear stiffness with
application to random vibration of geometrically nonlinear structures. Computers &
Structures, 81(15):1513–1523, 2003.
[83] Fumio Yamazaki and Masanobu Shinozuka. Simulation of stochastic fields by statistical
preconditioning. Journal of Engineering Mechanics, 116(2):268–287, 1990.
[84] M. Loeve. Probability Theory. Springer, 1977.
[85] Roger G Ghanem and Pol D Spanos. Spectral stochastic finite-element formulation for
reliability analysis. Journal of Engineering Mechanics, 117(10):2351–2372, 1991.
[86] Robert J Adler. The geometry of random fields. SIAM, 2010.
[87] Harry L Van Trees. Detection, estimation, and modulation theory. John Wiley & Sons,
2004.
[88] Erik Vanmarcke. Random fields: analysis and synthesis. World Scientific, 2010.
[89] Jun Zhang and Bruce Ellingwood. Orthogonal series expansions of random fields in
reliability analysis. Journal of Engineering Mechanics, 120(12):2660–2677, 1994.
[90] Mircea Grigoriu. Applied non-Gaussian processes: Examples, theory, simulation, linear
random vibration, and MATLAB solutions. Prentice Hall, 1995.
148
[91] SP Huang, ST Quek, and KK Phoon. Convergence study of the truncated Karhunen–
Loeve expansion for simulation of stochastic processes. International Journal for Nu-
merical Methods in Engineering, 52(9):1029–1043, 2001.
[92] Harry L Van Trees. Detection, estimation and modulation, part I. John Wiley & Sons,
1968.
[93] Kendall E Atkinson. The numerical solution of integral equations of the second kind,
volume 4. Cambridge University Press, 1997.
[94] William H Press. Numerical recipes 3rd edition: The art of scientific computing. Cam-
bridge University Press, 2007.
[95] Richard B Lehoucq, Danny C Sorensen, and Chao Yang. ARPACK users’ guide:
solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods,
volume 6. SIAM, 1998.
[96] K Maschhoff and D.C. Sorensen. Parpack: Parallel version of arpack for solving large
scale eigenvalue problems. Department of Computational and Applied Mathematics,
Rice University.
[97] Christoph Schwab and Radu Alexandru Todor. Karhunen–Loeve approximation of ran-
dom fields by generalized fast multipole methods. Journal of Computational Physics,
217(1):100–122, 2006.
[98] Catherine E Powell and Howard C Elman. Block-diagonal preconditioning for spectral
stochastic finite-element systems. IMA Journal of Numerical Analysis, 29(2):350–375,
2009.
[99] JE Hurtado and AH Barbat. Monte Carlo techniques in computational stochastic
mechanics. Archives of Computational Methods in Engineering, 5(1):3–29, 1998.
[100] Manolis Papadrakakis and Vissarion Papadopoulos. Robust and efficient methods for
stochastic finite element analysis using Monte Carlo simulation. Computer Methods in
Applied Mechanics and Engineering, 134(3):325–340, 1996.
[101] Harald Niederreiter. Quasi-Monte Carlo methods and pseudo-random numbers. Bul-
letin of the American Mathematical Society, 84(6):957–1041, 1978.
149
[102] William J Morokoff and Russel E Caflisch. Quasi-random sequences and their discrep-
ancies. SIAM Journal on Scientific Computing, 15(6):1251–1279, 1994.
[103] Russel E Caflisch. Monte Carlo and quasi-Monte Carlo methods. Acta Numerica,
7:1–49, 1998.
[104] George Fishman. Monte Carlo: concepts, algorithms, and applications. Springer Sci-
ence & Business Media, 2013.
[105] Michael D McKay, Richard J Beckman, and William J Conover. Comparison of three
methods for selecting values of input variables in the analysis of output from a computer
code. Technometrics, 21(2):239–245, 1979.
[106] Gerhart I Schueller and Reinhard Stix. A critical appraisal of methods to determine
failure probabilities. Structural Safety, 4(4):293–309, 1987.
[107] PS Koutsourelakis, HJ Pradlwarter, and GI Schueller. Reliability of structures in high
dimensions, part I: algorithms and applications. Probabilistic Engineering Mechanics,
19(4):409–417, 2004.
[108] Siu-Kui Au and James L Beck. Estimation of small failure probabilities in high di-
mensions by subset simulation. Probabilistic Engineering Mechanics, 16(4):263–277,
2001.
[109] John Hammersley. Monte Carlo methods. Springer Science & Business Media, 2013.
[110] William H Press and Glennys R Farrar. Recursive stratified sampling for multidimen-
sional Monte Carlo integration. Computers in Physics, 4(2):190–195, 1990.
[111] Tore Dalenius and Joseph L Hodges Jr. The choice of stratification points. Scandina-
vian Actuarial Journal, 1957(3-4):198–203, 1957.
[112] Robert L Cook, Thomas Porter, and Loren Carpenter. Distributed ray tracing. Com-
puter Graphics, 18(3):137–145, 1984.
[113] Art B Owen. Monte Carlo variance of scrambled net quadrature. SIAM Journal on
Numerical Analysis, 34(5):1884–1910, 1997.
[114] Michael Stein. Large sample properties of simulations using latin hypercube sampling.
Technometrics, 29(2):143–151, 1987.
150
[115] Reuven Y Rubinstein and Dirk P Kroese. Simulation and the Monte Carlo method,
volume 707. John Wiley & Sons, 2011.
[116] GI Schueller, HJ Pradlwarter, and PS Koutsourelakis. A critical appraisal of reliabil-
ity estimation procedures for high dimensions. Probabilistic Engineering Mechanics,
19(4):463–474, 2004.
[117] Chun-Ching Li and Armen Der Kiureghian. Mean out-crossing rate of nonlinear re-
sponse to stochastic input. Proceedings of ICASP-7, Balkema, Rotterdam, pages 295–
302, 1995.
[118] SK Au and James L Beck. First excursion probabilities for linear systems by very
efficient importance sampling. Probabilistic Engineering Mechanics, 16(3):193–207,
2001.
[119] Josef Dick, Frances Y Kuo, and Ian H Sloan. High-dimensional integration: the quasi-
Monte Carlo way. Acta Numerica, 22:133–288, 2013.
[120] Edmund Hlawka. Funktionen von beschrankter variatiou in der theorie der gle-
ichverteilung. Annali di Matematica Pura ed Applicata, 54(1):325–333, 1961.
[121] Harald Niederreiter. Quasi-Monte Carlo Methods. Wiley Online Library, 2010.
[122] John H Halton. On the efficiency of certain quasi-random sequences of points in
evaluating multi-dimensional integrals. Numerische Mathematik, 2(1):84–90, 1960.
[123] I.M. Sobol. The distribution of points in a cube and the accurate evaluation of integrals
(in Russian). Zh. Vychisl. Mat. i Mat. Phys., (7):784–802, 1967.
[124] Harald Niederreiter. Random number generation and quasi-Monte Carlo methods.
SIAM, 1992.
[125] Henri Faure. Discrepance de suites associees a un systeme de numeration (en dimension
s). Acta Arithmetica, 41(4):337–351, 1982.
[126] NM Korobov. The approximate computation of multiple integrals. Dokl. Akad. Nauk
SSSR, 124(6):1207–1210, 1959.
151
[127] Fred J Hickernell, Hee Sun Hong, Pierre L’Ecuyer, and Christiane Lemieux. Extensi-
ble lattice sequences for quasi-Monte Carlo quadrature. SIAM Journal on Scientific
Computing, 22(3):1117–1138, 2000.
[128] Art B Owen. Quasi-Monte Carlo sampling, 2003.
[129] Pierre L’Ecuyer and Christiane Lemieux. Recent advances in randomized quasi-Monte
Carlo methods. Modeling Uncertainty, pages 419–474, 2005.
[130] Art B Owen. Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo. In
H Niederreiter and J Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods,
pages 86–97. 1998.
[131] William L Briggs, Van Emden Henson, and Steve F McCormick. A multigrid tutorial.
SIAM, 2000.
[132] Pieter Wesseling. An Introduction To Multigrid Methods. John Wiley, Chichester, UK,
1992.
[133] Michael B Giles. Multilevel Monte Carlo methods. Acta Numerica, 24:259–328, 2015.
[134] K Andrew Cliffe, Mike B Giles, Robert Scheichl, and Aretha L Teckentrup. Multi-
level Monte Carlo methods and applications to elliptic pdes with random coefficients.
Computing and Visualization in Science, 14(1):3, 2011.
[135] Vincent Lemaire, Gilles Pages, et al. Multilevel Richardson-Romberg extrapolation.
Bernoulli, 23(4A):2643–2692, 2017.
[136] Abdul-Lateef Haji-Ali, Fabio Nobile, and Raul Tempone. Multi-index Monte Carlo:
when sparsity meets sampling. Numerische Mathematik, 132(4):767–806, 2016.
[137] Charles W Clenshaw and Alan R Curtis. A method for numerical integration on an
automatic computer. Numerische Mathematik, 2(1):197–205, 1960.
[138] T. N. L. Patterson. The optimum addition of points to quadrature formulae. Mathe-
matics of Computation, 22(104):847–856, 1968.
[139] Sergey Smolyak. Quadrature and interpolation formulas for tensor products of certain
classes of functions. In Soviet Math. Dokl., volume 4, pages 240–243, 1963.
152
[140] Thomas Gerstner and Michael Griebel. Numerical integration using sparse grids. Nu-
merical Algorithms, 18(3):209–232, 1998.
[141] Erich Novak and Klaus Ritter. The curse of dimension and a universal method for
numerical integration. In Multivariate approximation and splines, pages 177–187.
Springer, 1997.
[142] Arthur H Stroud. Approximate calculation of multiple integrals. Prentice-Hall, 1971.
[143] Dongbin Xiu. Numerical integration formulas of degree two. Applied Numerical Math-
ematics, 58(10):1515–1520, 2008.
[144] Volker Barthelmann, Erich Novak, and Klaus Ritter. High dimensional polynomial
interpolation on sparse grids. Advances in Computational Mathematics, 12(4):273–
288, 2000.
[145] Fabio Nobile, Raul Tempone, and Clayton G Webster. A sparse grid stochastic colloca-
tion method for partial differential equations with random input data. SIAM Journal
on Numerical Analysis, 46(5):2309–2345, 2008.
[146] Fabio Nobile, Raul Tempone, and Clayton G Webster. An anisotropic sparse grid
stochastic collocation method for partial differential equations with random input data.
SIAM Journal on Numerical Analysis, 46(5):2411–2442, 2008.
[147] Dongbin Xiu and Jan S Hesthaven. High-order collocation methods for differential
equations with random inputs. SIAM Journal on Scientific Computing, 27(3):1118–
1139, 2005.
[148] Ivo Babuska, Fabio Nobile, and Raul Tempone. A stochastic collocation method for
elliptic partial differential equations with random input data. SIAM Journal on Nu-
merical Analysis, 45(3):1005–1034, 2007.
[149] Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica, 13(1):147–
269, 2004.
[150] Philipp Frauenfelder, Christoph Schwab, and Radu Alexandru Todor. Finite elements
for elliptic problems with stochastic coefficients. Computer Methods in Applied Me-
chanics and Engineering, 194(2):205–228, 2005.
153
[151] Maarten De Munck, David Moens, Wim Desmet, and Dirk Vandepitte. A response
surface based optimisation algorithm for the calculation of fuzzy envelope frfs of models
with uncertain properties. Computers & Structures, 86(10):1080–1092, 2008.
[152] Christian G Bucher and U Bourgund. A fast and efficient response surface approach
for structural reliability problems. Structural Safety, 7(1):57–66, 1990.
[153] Ozgur Yeniay, Resit Unal, and Roger A Lepsch. Using dual response surfaces to reduce
variability in launch vehicle design: a case study. Reliability Engineering & System
Safety, 91(4):407–412, 2006.
[154] DR Romero and SD Bankston. Application of decoupled Monte Carlo analysis with
finite element/lattice sampling response surface for multimodal test problem. Sandia
National Laboratories, 1997.
[155] BM Rutherford, LP Swiler, TL Paez, and A Urbina. Response surface (meta-model)
methods and applications. Sandia National Laboratories, 2005.
[156] AA Giunta, JM McFarland, LP Swiler, and MS Eldred. The promise and peril of
uncertainty quantification using response surface approximations. Structures and In-
frastructure Engineering, 2(3-4):175–189, 2006.
[157] GM Fadel, MF Riley, and JM Barthelemy. Two point exponential approximation
method for structural optimization. Structural and Multidisciplinary Optimization,
2(2):117–124, 1990.
[158] Suqiang Xu and Ramana V Grandhi. Effective two-point function approximation for
design optimization. AIAA Journal, 36(12), 1998.
[159] Noel Cressie. Statistics for spatial data. John Wiley & Sons, 2015.
[160] Jerome H Friedman. Multivariate adaptive regression splines. Annals of Statistics,
19(1):1–67, 1991.
[161] B. M. Adams, W. J. Bohnhoff, K. R. Dalbey, J. P. Eddy, M. S. Ebeida, M. S. Eldred,
J. R. Frye, G Gerarci, P. D. Hough, K. T. Hu, J. D. Jakeman, M. Khalil, K. A. Maupin,
J. A. Monschke, E. M. Ridgway, Ahmad Rushdi, L. P. Swiler, J. A. Stephens, D. M.
Vigil, and T. M. Wildey. Dakota, a multilevel parallel object-oriented framework for
design optimization, parameter estimation, uncertainty quantification, and sensitivity
154
analysis: Version 6.6 users manual. Technical Report SAND2014-5015, Sandia National
Laboratories, Albuquerque, NM, 2017.
[162] N. Wiener. The homogeneous chaos. American Journal of Mathematics, 60(4):897–936,
1938.
[163] Christian Soize and Roger Ghanem. Physical systems with random uncertainties:
chaos representations with arbitrary probability measure. SIAM Journal on Scientific
Computing, 26(2):395–410, 2004.
[164] Xiaoliang Wan and George Em Karniadakis. Multi-element generalized polynomial
chaos for arbitrary probability measures. SIAM Journal on Scientific Computing,
28(3):901–928, 2006.
[165] OP Le Maıtre, OM Knio, HN Najm, and RG Ghanem. Uncertainty propagation using
Wiener-Haar expansions. Journal of Computational Physics, 197(1):28–57, 2004.
[166] OP Le Maıtre, HN Najm, RG Ghanem, and OM Knio. Multi-resolution analysis
of Wiener-type uncertainty propagation schemes. Journal of Computational Physics,
197(2):502–531, 2004.
[167] Ivo Babuska, Raul Tempone, and Georgios E Zouraris. Galerkin finite element ap-
proximations of stochastic elliptic partial differential equations. SIAM Journal on
Numerical Analysis, 42(2):800–825, 2004.
[168] Xiaoliang Wan and George Em Karniadakis. An adaptive multi-element generalized
polynomial chaos method for stochastic differential equations. Journal of Computa-
tional Physics, 209(2):617–642, 2005.
[169] Robert H Cameron and William T Martin. The orthogonal development of non-linear
functionals in series of fourier-hermite functionals. Annals of Mathematics, pages 385–
392, 1947.
[170] Dan M Ghiocel and Roger G Ghanem. Stochastic finite-element analysis of seismic
soil-structure interaction. Journal of Engineering Mechanics, 128(1):66–77, 2002.
[171] Olivier P Le Maıtre, Matthew T Reagan, Habib N Najm, Roger G Ghanem, and
Omar M Knio. A stochastic projection method for fluid flow: II. random process.
Journal of Computational Physics, 181(1):9–44, 2002.
155
[172] Seung-Kyum Choi, Ramana V Grandhi, Robert A Canfield, and Chris L Pettit. Poly-
nomial chaos expansion with latin hypercube sampling for estimating response vari-
ability. AIAA Journal, 42(6):1191–1198, 2004.
[173] Marc Berveiller, Bruno Sudret, and Maurice Lemaire. Stochastic finite element: a
non intrusive approach by regression. European Journal of Computational Mechan-
ics/Revue Europeenne de Mecanique Numerique, 15(1-3):81–92, 2006.
[174] Michael Scott Eldred. Recent advances in non-intrusive polynomial chaos and
stochastic collocation methods for uncertainty analysis and design. AIAA Paper,
2274(2009):37, 2009.
[175] D Lucor, D Xiu, and G Karniadakis. Spectral representations of uncertainty in simu-
lations: Algorithms and applications. In Proceedings of the International Conference
on Spectral and High Order Methods (ICOSAHOM-01), Uppsala, Sweden, 2001.
[176] Alexandre Joel Chorin. Gaussian fields and random flow. Journal of Fluid Mechanics,
63(1):21–32, 1974.
[177] Steven A Orszag and LR Bissonnette. Dynamical properties of truncated wiener-
hermite expansions. The Physics of Fluids, 10(12):2603–2613, 1967.
[178] Oliver G Ernst, Antje Mugler, Hans-Jorg Starkloff, and Elisabeth Ullmann. On the
convergence of generalized polynomial chaos expansions. ESAIM: Mathematical Mod-
elling and Numerical Analysis, 46(02):317–339, 2012.
[179] M. Petyt. Introduction to finite element vibration analysis. Cambridge University
Press, 2010.
[180] Hermann G Matthies and Andreas Keese. Galerkin methods for linear and nonlinear
elliptic stochastic partial differential equations. Computer Methods in Applied Mechan-
ics and Engineering, 194(12):1295–1331, 2005.
[181] Roger Ghanem and Pol D Spanos. A stochastic galerkin expansion for nonlinear ran-
dom vibration analysis. Probabilistic Engineering Mechanics, 8(3):255–264, 1993.
[182] Wilfred D Iwan and Huang Ching-Tung. On the dynamic response of non-linear sys-
tems with parameter uncertainties. International Journal of Non-linear Mechanics,
31(5):631–645, 1996.
156
[183] Dongbin Xiu and George Em Karniadakis. Modeling uncertainty in flow simulations
via generalized polynomial chaos. Journal of Computational Physics, 187(1):137–167,
2003.
[184] D. Xiu and G. Em Karniadakis. Modeling uncertainty in steady state diffusion prob-
lems via generalized polynomial chaos. Computer Methods in Applied Mechanics and
Engineering, 191(43):4927–4948, 2002.
[185] Dongbin Xiu, Didier Lucor, C-H Su, and George Em Karniadakis. Stochastic modeling
of flow-structure interactions using generalized polynomial chaos. Journal of Fluids
Engineering, 124(1):51–59, 2002.
[186] Dongbin Xiu and Jie Shen. Efficient stochastic galerkin methods for random diffusion
equations. Journal of Computational Physics, 228(2):266–281, 2009.
[187] Doo Bo Chung, Miguel A Gutierrez, Lori L Graham-Brady, and Frederik-Jan Lingen.
Efficient numerical strategies for spectral stochastic finite element models. Interna-
tional Journal for Numerical Methods in Engineering, 64(10):1334–1349, 2005.
[188] Michael Eiermann, Oliver G Ernst, and Elisabeth Ullmann. Computational aspects
of the stochastic finite element method. Computing and Visualization in Science,
10(1):3–15, 2007.
[189] Andreas Keese and Hermann G Matthies. Hierarchical parallelisation for the solution
of stochastic finite element equations. Computers & Structures, 83(14):1033–1047,
2005.
[190] M.F. Pellissetti and R.G. Ghanem. Iterative solution of systems of linear equations
arising in the context of stochastic finite elements. Advances in Engineering Software,
31(8):607–616, 2000.
[191] Maciej Anders and Muneo Hori. Three-dimensional stochastic finite element method
for elasto-plastic bodies. International Journal for Numerical Methods in Engineering,
51(4):449–478, 2001.
[192] Omar M Knio, Habib N Najm, Roger G Ghanem, et al. A stochastic projection method
for fluid flow: I. basic formulation. Journal of Computational Physics, 173(2):481–511,
2001.
157
[193] Roger Ghanem and Bernard Hayek. Probabilistic modeling of flow over rough terrain.
Journal of Fluid Engineering, 124(1):42–50, 2002.
[194] Velamur Asokan Badri Narayanan and Nicholas Zabaras. Variational multiscale sta-
bilized FEM formulations for transport equations: stochastic advection–diffusion and
incompressible stochastic Navier–Stokes equations. Journal of Computational Physics,
202(1):94–133, 2005.
[195] R Li and R Ghanem. Adaptive polynomial chaos expansions applied to statistics of ex-
tremes in nonlinear random vibration. Probabilistic Engineering Mechanics, 13(2):125–
136, 1998.
[196] P. Sekar and S. Narayanan. Periodic and chaotic motions of a square prism in cross-
flow. Journal of Sound and Vibration, 170(1):1–24, 1994.
[197] X. Wan and G.E. Karniadakis. Long-term behavior of polynomial chaos in stochastic
flow simulations. Computer Methods in Applied Mechanics and Engineering, 195(41-
43):5582–5596, 2006.
[198] Dongbin Xiu. Fast numerical methods for stochastic computations: a review. Com-
munications in Computational Physics, 5(2-4):242–272, 2009.
[199] David Gottlieb and Dongbin Xiu. Galerkin method for wave equations with uncertain
coefficients. Communications in Computational Physics, 3(2):505–518, 2008.
[200] Anthony Nouy and Olivier P Le Maıtre. Generalized spectral decomposition for
stochastic nonlinear problems. Journal of Computational Physics, 228(1):202 – 235,
2009.
[201] Anthony Nouy. A priori model reduction through proper generalized decomposition for
solving time-dependent partial differential equations. Computer Methods in Applied
Mechanics and Engineering, 199(23):1603–1626, 2010.
[202] Anthony Nouy. Proper generalized decompositions and separated representations for
the numerical solution of high dimensional stochastic problems. Archives of Computa-
tional Methods in Engineering, 17(4):403–434, 2010.
[203] Anthony Nouy. Proper generalized decompositions for a priori model reduction of
problems formulated in tensor product spaces: Alternative definitions and algorithms.
158
In Proceedings of the Seventh International Conference on Engineering Computational
Technology, Civil-Comp Press, 2010.
[204] Nathan M Newmark. A method of computation for structural dynamics. Journal of
the Engineering Mechanics Division, 85(3):67–94, 1959.
[205] Klaus-Jurgen Bathe. Finite element procedures. Prentice-Hall, Upper Saddle River,
New Jersey, 1996.
[206] B. Sudret and A. Der Kiureghian. Stochastic finite element methods and reliability.
A state-of-the-art report. Structural Engineering, Mechanics and Materials Program,
University of California Berkeley, Report No. UCB/SEMM-2000/08, 2000.
[207] Genyuan Li, Sheng-Wei Wang, Herschel Rabitz, Sookyun Wang, and Peter Jaffe.
Global uncertainty assessments by high dimensional model representations (HDMR).
Chemical Engineering Science, 57(21):4445–4460, 2002.
[208] Jacob Fish and Ted Belytschko. A first course in finite elements. John Wiley & Sons,
2007.
[209] M. Ding, E. Ott, and C. Grebogi. Controlling chaos in a temporally irregular environ-
ment. Physica D: Nonlinear Phenomena, 74(3-4):386–394, 1994.
[210] Francis C Moon. Chaotic and Fractal Dynamics: Introduction for Applied Scientists
and Engineers. John Wiley & Sons, 2008.
[211] John Guckenheimer and Philip Holmes. Nonlinear oscillations, dynamical systems,
and bifurcations of vector fields, volume 42. Springer Science & Business Media, 2013.
[212] Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biol-
ogy, chemistry, and engineering. CRC Press, 2018.
[213] Alfio Quarteroni and Alberto Valli. Domain decomposition methods for partial differen-
tial equations. Numerical Mathematics and Scientific Computation. Oxford University
Press, New York, 1999.
[214] Lin Gao, Christophe Audouze, and Prasanth B Nair. Anchored analysis of variance
Petrov–Galerkin projection schemes for linear stochastic structural dynamics. Proceed-
ings of the Royal Society A, 471(2182), 2015.
159
[215] Bert J Debusschere, Habib N Najm, Philippe P Pebay, Omar M Knio, Roger G
Ghanem, and Olivier P Le Maıtre. Numerical challenges in the use of polynomial
chaos representations for stochastic processes. SIAM Journal on Scientific Computing,
26(2):698–719, 2004.
[216] Y Maday. Analysis of spectral projectors in one-dimensional domains. Mathematics of
Computation, 55(192):537–562, 1990.
[217] Christophe Audouze and Prasanth B Nair. A priori error analysis of stochastic Galerkin
projection schemes for randomly parametrized ordinary differential equations. Inter-
national Journal for Uncertainty Quantification, 6(4), 2016.
[218] Christophe Audouze and Prasanth Nair. Some a priori error estimates for finite element
approximations of elliptic and parabolic linear stochastic partial differential equations.
International Journal for Uncertainty Quantification, 4(5), 2014.
[219] Claudio Canuto, M Youssuff Hussaini, Alfio Quarteroni, and Thomas A Zang. Spectral
methods: fundamentals in single domains. Scientific Computation. Springer, 2006.
[220] Y Maday, B Pernaud-Thomas, and H Vandeven. Une rehabilitation des methodes
spectrales de type Laguerre. La recherche aerospatiale, 6:353–375, 1985.
[221] Claudio Canuto and Alfio Quarteroni. Approximation results for orthogonal polyno-
mials in Sobolev spaces. Mathematics of Computation, 38(157):67–86, 1982.
[222] Ben-Yu Guo. Error estimation of Hermite spectral method for nonlinear partial differ-
ential equations. Mathematics of Computation of the American Mathematical Society,
68(227):1067–1078, 1999.
[223] Christine Bernardi and Yvon Maday. Properties of some weighted Sobolev spaces
and application to spectral approximations. SIAM Journal on Numerical Analysis,
26(4):769–829, 1989.
[224] Christine Bernardi, Yvon Maday, and Franscesca Rapetti. Discretisations variation-
nelles de problemes aux limites elliptiques, volume 45. Springer Science & Business
Media, 2004.
[225] Robert A Adams and John JF Fournier. Sobolev spaces, volume 140. Academic Press,
2003.
160
[226] IM Sobol. Theorems and examples on high dimensional model representation. Relia-
bility Engineering & System Safety, 79(2):187–193, 2003.
[227] Zhongqiang Zhang, Minseok Choi, and George Em Karniadakis. Anchor points matter
in ANOVA decomposition. In Spectral and High Order Methods for Partial Differential
Equations, pages 347–355. Springer, 2011.
[228] Russel E Caflisch, William J Morokoff, and Art B Owen. Valuation of mortgage
backed securities using Brownian bridges to reduce effective dimension. Department of
Mathematics, University of California, Los Angeles, 1997.
[229] Xiaoqun Wang. On the approximation error in high dimensional model representation.
In Proceedings of the 40th Conference on Winter Simulation, pages 453–462. Winter
Simulation Conference, 2008.
[230] Zhen Gao and Jan S Hesthaven. On ANOVA expansions and strategies for choosing
the anchor point. Applied Mathematics and Computation, 217(7):3274–3285, 2010.
[231] Tarek Mathew. Domain decomposition methods for the numerical solution of partial
differential equations, volume 61. Springer Science & Business Media, 2008.
[232] Tony F Chan and Tarek P Mathew. Domain decomposition algorithms. Acta numerica,
3:61–143, 1994.
[233] Andrea Toselli and Olof Widlund. Domain decomposition methods-algorithms and
theory, volume 34. Springer Science & Business Media, 2006.
[234] Abhijit Sarkar, Nabil Benabbou, and Roger Ghanem. Domain decomposition of
stochastic pdes: theoretical formulations. International Journal for Numerical Methods
in Engineering, 77(5):689–701, 2009.
[235] Waad Subber and Abhijit Sarkar. A parallel time integrator for noisy nonlinear oscil-
latory systems. Journal of Computational Physics, 362:190–207, 2018.
[236] William Thomson. Theory of vibration with applications. CRC Press, 1993.
161