nicholas zabaras and xiang ma

11

CCOORRNNEELLLL U N I V E R S I T Y


Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

An adaptive hierarchical sparse grid An adaptive hierarchical sparse grid collocation algorithm for the solution of collocation algorithm for the solution of

stochastic differential equationsstochastic differential equations

Nicholas Zabaras and Xiang MaMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering101 Frank H. T. Rhodes Hall

Cornell University Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/

Symposium on `Numerical Methods for Stochastic Computation and Uncertainty Quantification' in the 2008 SIAM Annual Meeting, San Diego CA, July 7-11, 2008

22




Outline of the presentation

o Stochastic analysis

o Issues with gPC and ME-gPc

o Collocation based approach

o Basics of stochastic collocation

o Issues with existing stochastic sparse grid collocation method (Smolyak algorithm)

o Adaptive sparse grid collocation method

o Some numerical examples

33




Motivation

All physical systems have an inherent associated randomness

SOURCES OF UNCERTAINTIES

•Multiscale material information – inherently statistical in nature.

•Uncertainties in process conditions

•Input data

•Model formulation – approximations, assumptions.

Why uncertainty modeling ?

Assess product and process reliability.

Estimate confidence level in model predictions.

Identify relative sources of randomness.

Provide robust design solutions.

Engineering component

Heterogeneous random

Microstructural features

Process

Control?

44




Problem definition

Define a complete probability space . We are interested to find a stochastic function such that for P-almost everywhere (a.e.) , the following equation hold:

, ,F P

:u D

, ; , ,

, ,

L u f D

B u g D

x x x

x x x

where are the coordinates in , L is (linear/nonlinear) differential operator, and B is a boundary operators.

1 2, , , dx x xx d

In the most general case, the operator L and B as well as the driving terms f and g, can be assumed random.

In general, we require an infinite number of random variables to completely characterize a stochastic process. This poses a numerical challenge in modeling uncertainty in physical quantities that have spatio-temporal variations, hence

necessitating the need for a reduced-order representation.

55




Representing Randomness

S

Sample space of elementary events

Real line

Random

variable

MAP

Collection of all possible outcomes

Each outcome is mapped to a

corresponding real value

Interpreting random variables as functions

A general stochastic process is a random field with variations along space and time – A

function with domain (Ω, Τ, S)

1. Interpreting random variables

2. Distribution of the random variable

E.g. inlet velocity, inlet temperature

1 0.1o

3. Correlated data

Ex. Presence of impurities, porosity

Usually represented with a correlation function

We specifically concentrate on this.

66




Representing Randomness

1. Representation of random process

- Karhunen-Loeve, Polynomial Chaos expansions

2. Infinite dimensions to finite dimensions

- depends on the covariance

Karhunen-Loèvè expansion

Based on the spectral decomposition of the covariance kernel of the stochastic process

Random process Mean

Set of random variables to

be found

Eigenpairs of covariance

kernel

• Need to know covariance

• Converges uniformly to any second order process

Set the number of stochastic dimensions, N

Dependence of variables

Pose the (N+d) dimensional problem

77




Karhunen-Loeve expansion

1

( , , ) ( , ) ( , ) ( )i ii

X x t X x t X x t Y

Stochastic process

Mean function

ON random variablesDeterministic functions

Deterministic functions ~ eigen-values, eigenvectors of the covariance function

Orthonormal random variables ~ type of stochastic process

In practice, we truncate (KL) to first N terms

1( , , ) fn( , , , , )NX x t x t Y Y

88




The finite-dimensional noise assumption

By using the Doob-Dynkin lemma, the solution of the problem can be described by the same set of random variables, i.e.

1, , , , Nu u Y Y x x

So the original problem can be restated as: Find the stochastic function such that

:u D

, ; , ,

, ; , ,

L u f D

B u g D

x Y x Y x

x Y x Y x

In this work, we assume that are independent random variables with probability density function . Let be the image of . Then

1

, N

i ii

Y

Y Y

1

N

i iY

i i iY

is the joint probability density of with support 1, , NY YY

1

NN

ii

99




Uncertainty analysis techniques

Monte-Carlo : Simple to implement, computationally expensive

Perturbation, Neumann expansions : Limited to small fluctuations, tedious for higher order statistics

Sensitivity analysis, method of moments : Probabilistic information is indirect, small fluctuations

Spectral stochastic uncertainty representation: Basis in probability and functional analysis, can address second order stochastic processes, can handle large fluctuations, derivations are general

Stochastic collocation: Results in decoupled equations

1010




Spectral stochastic presentation

: ( , , )X x t

A stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE

PROBABILITY SPACE

HYPERGEOMETRIC ASKEY POLYNOMIALS

PIECEWISE POLYNOMIALS (FE TYPE)

SPECTRAL DECOMPOSITION

COLLOCATION, MC (DELTA FUNCTIONS)

GENERALIZED POLYNOMIAL CHAOS EXPANSION

SUPPORT-SPACE REPRESENTATION

KARHUNEN-LOÈVE EXPANSION

SMOLYAK QUADRATURE, CUBATURE, LH

1111




Generalized polynomial chaos (gPC)

Generalized polynomial chaos expansion is used to represent the stochastic output in terms of the input

0

( , , ) ( , ) ( ( ))P

i ii

Z x t Z x t Y

Stochastic output

Askey polynomials in inputDeterministic functions

Stochastic input

1( , , ) fn( , , , , )NX x t x t Y Y

Askey polynomials ~ type of input stochastic process

Usually, Hermite, Legendre, Jacobi etc.

1212





Advantages:1) Fast convergence1.

2) Very successful in solving various problems in solid and fluid mechanics – well understood 2,3.

3) Applicable to most random distributions 4,5.

4) Theoretical basis easily extendable to other phenomena

1. P Spanos and R Ghanem, Stochastic finite elements: A spectral approach, Springer-Verlag (1991)

2. D. Xiu and G. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg. 187 (2003) 137--167.

3. S. Acharjee and N. Zabaras, Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach, Comput. Methods Appl. Mech. Engrg. 195 (2006) 2289--2312.

4. D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comp. 24 (2002) 619-644.

5. X. Wan and G.E. Karniadakis, Beyond Wiener-Askey expansions: Handling arbitrary PDFs, SIAM J Sci Comp 28(3) (2006) 455-464.

1313





Disadvantages:1) Coupled nature of the formulation makes implementation nontrivial1.

2) Substantial effort into coding the stochastics: Intrusive

3) As the order of expansion (p) is increased, the number of unknowns (P) increases factorially with the stochastic dimension (N): combinatorial explosion2,3.

4) Furthermore, the curse of dimensionality makes this approach particularly difficult for large stochastic dimensions

1. B. Ganapathysubramaniana and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp Physics, 2007, doi:10.1016/j.jcp.2006.12.014

2. D Xiu, D Lucor, C.-H. Su, and G Karniadakis, Performance evaluation of generalized polynomial chaos, P.M.A. Sloot et al. (Eds.): ICCS 2003, LNCS 2660 (2003) 346-354

3. B. J. Debusschere, H. N. Najm, P. P. Pebray, O. M. Knio, R. G. Ghanem and O. P. Le Maître, Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes, SIAM Journal of Scientific Computing 26(2) (2004) 698-719.

!1

! !

N pP

N p

1414




Failure of gPC

Gibbs phenomenon: Due to its global support, it fails to capture the

stochastic discontinuity in the stochastic space.

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

19.2E-07 5.7E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 3.0E-05 3.4E-05

X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1-6.4E-08 4.3E-07 9.3E-07 1.4E-06 1.9E-06 2.4E-06 2.9E-06 3.4E-06

X-v

eloc

ity

Y-v

eloc

ityCapture unsteady equilibrium in

natural convection

Therefore, we need some method which can resolve the discontinuity locally. This motivates the following Multi - element gPC method.

1515




Multi-element generalized polynomial chaos (ME-gPC)

Basic concept:

Decompose the space of random inputs into small elements.

In each element of the support space, we generate a new random variable and apply gPCE.

The global PDF is also simultaneously discretized. Thus, the global PC

basis is not orthogonal with the local PDF in a random element. We

need to reconstruct the orthogonal basis numerically in each random

element. The only exception is the uniform distribution whose PDF is a

constant.

We can also decompose the random space adaptively to increase its efficiency.

1. X. Wan and G.E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, SIAM J Sci Comp 209 (2006) 901-928.

1616




Decomposition of the random space

B

1 2

,1 ,1 ,2 ,2 , ,

1

[ , ) [ , ) [ , ]k k k k k k d k d

Mk k

k k

a b a b a b

B

where 1 2, 1, 2 ,k k M

We define a decomposition of as

Based on the decomposition , we define the following indicator random variables:

1

0kI

,kY

otherwise1,2 ,k M

B

Then a local random vector is defined in each element subject to a conditional PDF

iY i

( )ˆ ( | 1)Pr( 1)k

k

kk k

ff I

I

YY

where Pr( 1) 0kk BJ I

1717




An adaptive procedure Let us assume that the gPC expansion of random field in element is

, ,0

ˆ ˆ( ) ( )pN

k k k i k i ki

u u

ξ ξ

k

From the orthogonality of gPC, the local variance approximated by PC with order is given byp

2 2 2, ,

1

ˆpN

k p k i ii

u

The approximate global mean and variance can be expressed by u 2

2 2 2,0 , ,0

1 1

ˆ ˆPr( 1), [ ( ) ]Pr( 1)k k

N N

k B k p k Bk k

u u I u u I

Let denote the error of local variance. We can write the exact global variance as

k

2 2

1

Pr( 1)k

N

k Bk

I

1818




An adaptive procedure We define the local decay rate of relative error of the gPC approximation

in each element as follows:

1

2 22 2,1, , 1

2 2, ,

ˆp

p

N

k i ii Nk p k pk

k p k p

u

Based on and the scaled parameter , a random element will be selected for potential refinement when the following condition is

satisfied

k Pr( 1)kB

I

1Pr( 1) , 0 1kk BI

where and are prescribed constants. 1 Furthermore, in the selected random element , we use another threshold parameter to choose the most sensitive random dimension. We define the sensitivity of each random dimension as

2

1

2 2, ,

2 2

1

ˆ( ), 1, 2, , ,

ˆp

p

i p i p

i N

j jj N

ur i d

u

where we drop the subscript for clarity and the subscript denotes the mode consisting only of random dimension with polynomial order

k ,i p

iY

p

k

1919




An adaptive procedure

All random dimensions which satisfy

2 21,

max , 0 1, 1, 2, , ,i jj d

r r i d

will be split into two equal random elements in the next time step while all other random dimensions will remain unchanged.

Through this adaptive procedure, we can reduce the total element number while gaining the same accuracy and efficiency.

If corresponds to element in the original random space , the element and will be generated in the next level if the two criteria are both satisfied.

kY [ , ]a b [ 1,1][ , ]

2

a ba

[ , ]

2

a bb

2020




An Example: K-O problem

So it can successfully resolve the discontinuity in stochastic space. However, due to its gPC nature in each element and the tree like adaptive refinement, this method still suffers from the curse of dimensionality.

2121




First summary

The traditional gPC method fails when there are steep gradients/finite discontinuities in stochastic space.

gPC related method suffers from the so called curse of dimensionality when the input stochastic dimension is large. High CPU cost for large scale problems.

Although the h-adaptive ME-gPC can successfully resolve discontinuity in stochastic space, the number of element grows fast at the order . So the method is still a dimension-dependent method.

Therefore, an adaptive framework utilizing local basis functions that scales linearly with increasing dimensionality and has the decoupled nature of the MC method, offers greater promise in efficiently and accurately representing high-dimensional non-smooth stochastic functions.

(2 )NO

2222




Collocation based framework

The stochastic variable at a point in the domain is an N dimensional function.

The gPC method approximates this N dimensional function using a spectral element discretization (which causes the coupled set of equations)

: Nf

: Nf

Spectral descretization represents the function as a linear combination of sets of orthogonal polynomials.

Instead of a spectral discretization, use other kinds of polynomial approximations

Use simple interpolation!

That is, sample the function at some points and construct a polynomial approximation

2323




Collocation based framework

Function value at any point is simply

Stochastic function in 2 dimensions

Need to represent this function

Sample the function at a finite set of points

Use polynomials (Lagrange polynomials) to get a approximate representation

: Nf ( )

1 i MM iΘ Y

( ) ( )

1

( ) ( ) ( )M

i ii

i

f f a

Y Y YI

( )f ISpatial domain is approximated using a FE, FD, or FV discretization.

Stochastic domain is approximated using multidimensional interpolating functions

2424




From one-dimension to higher dimensions Denote the one dimensional interpolation formula as

with the set of support nodes:

In higher dimension, a simple case is the tensor product formula

For instance, if M=10 dimensions and we use k points in each direction

Number of Number of points in each points in each direction, kdirection, k

Total number Total number of sampling of sampling pointspoints

22 10241024

33 5904959049

44 1.05x101.05x1066

55 9.76x109.76x1066

1010 1x101x101010

This quickly becomes impossible to use.

One idea is only to pick the most important points from the tensor product grid.

2525




Conventional sparse grid collocation (CSGC)

1D sampling

2D sampling

In higher dimensions, not all the points are equally important.

Some points contribute less to the accuracy of the solution (e.g. points where the function is very smooth, regions where the function is linear, etc.). Discard the points that contribute less: SPARSE GRID COLLOCATION

Tensor product

2626




Conventional sparse grid collocation

Both piecewise linear as well as polynomial functions, L, can be used to construct the full tensor product formula above.

The Smolyak construction starts from this full tensor product representation and discards some multi-indices to construct the minimal representation of the function

Define the differential interpolant, ∆ as the difference between two interpolating approximations

with 1i i iU U 0 0U

where , is the multi-index representation of the support nodes. N is the dimension of the function f and q is an integer (q>N). k = q-N is called the depth of the interpolation. As q is increased more and more points are sampled.

1, ( ) ( )( )Niiq N

q

A f f

i

The sparse grid interpolation of the function f is given by

1 di i i

2727




Conventional sparse grid collocation

The interpolant can be represented in a recursive manner as1 1

, 1,( ) ( )( ) ( ) ( )( )N Ni ii iq N q N

q q

A f f A f f

i i

with 1, 0N NA

Can increase the accuracy of the interpolant based on Smolyak’s construction WITHOUT having to discard previous results.

Can build on previous interpolants due to the recursive formulation

To compute the interpolant , one only needs the function values at the sparse grid point given by:

,q NA

1,

1 | |( ... )Nii

q Nq N q

H X X

i

One could select the sets , in a nested fashion such that iX1i iX X

Similar to the interpolant representation in a recursive form, the sparse grid points can be represented as:

with 1,

| |( ... )Mi i

q Mq

H X X

i and 1\i i iX X X

, 1, ,q M q M q MH H H

2828




Choice of collocation points and nodal basis functions

One of the choice is the Clenshaw-Curtis grid at non-equidistant extrema of the Chebyshev polynomials. ( D.Xiu etc, 2005, F. Nobile etc, 2006)

By doing so, the resulting sets are nested. The corresponding univariate nodal basis functions are Lagrangian characteristic polynomials:

There are two problems with this grid:

(1) The basis function is of global support, so it will fail when there is discontinuity in the stochastic space.

(2) The nodes are pre-determined, so it is not suitable for implementation of adaptivity.

2929




Choice of collocation points and nodal basis functions Therefore, we propose to use the Newton-Cotes grid using equidistant

support nodes and use the linear hat function as the univariate nodal basis.

It is noted that the resulting grid points are also nested and the grid has the same number of points as that of the Clenshaw-Curtis grid.

Comparing with the Lagrangian polynomial, the piecewise linear hat function has a local support, so it is expected to resolve discontinuities in

the stochastic space.

The N-dimensional multi-linear basis functions can be constructed using tensor products:

ijY

12i ijY

12i ijY

1

3030




Comparison of the two grids

Clenshaw-Curtis grid Newton-Cotes grid

3131




From nodal basis to multivariate hierarchical basis We start from the 1D interpolating formula. By definition, we have

with and

we can obtain

since we have

For simplifying the notation, we consecutively number the elements in , and denote the j-th point of as

iX ijY

iX

hierarchical surplus

3232




Nodal basis vs. hierarchical basis

Nodal basis Hierarchical basis

3333




Multi-dimensional hierarchical basis For the multi-dimensional case, we define a new multi-index set

which leads to the following definition of hierarchical basis

Now apply the 1D hierarchical interpolation formula to obtain the sparse grid interpolation formula for the multivariate case in a hierarchical form:

Here we define the hierarchical surplus as:

3434




Error estimate

1. V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000), 273–288

2. E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, On an interpolatory method for high dimensional integration, J. Comp. Appl. Math. 112 (1999), 215–228

Depending on the order of the one-dimensional interpolant, one can come up with error bounds on the approximating quality of the interpolant1,2.

If piecewise linear 1D interpolating functions are used to construct the sparse interpolant, the error estimate is given by:

Where N is the number of sampling points,

If 1D polynomial interpolation functions are used, the error bound is:

assuming that f has k continuous derivatives

3535




Hierarchical integration So now, any function can now be approximated by the following reduced form

u

It is just a simple weighted sum of the value of the basis for all collocation points in the current sparse grid. Therefore, we can easily extract the

useful statistics of the solution from it.

For example, we can sample independently N times from the uniform distribution [0,1] to obtain one random vector Y, then we can place this vector into the above expression to obtain one realization of the solution.

In this way, it is easy to plot realizations of the solution as well as the PDF.

The mean of the random solution can be evaluated as follows:

where the probability density function is 1 since we have assumed uniform random variables on a unit hypercube .

Y

[0,1]N

3636




Hierarchical integration The 1D version of the integral in the equation above can be evaluated analytically:

Since the random variables are assumed independent of each other, the value of the multi-dimensional integral is simply the product of the 1D

integrals.

Denoting we rewrite the mean as

Thus, the mean is just an arithmetic sum of the hierarchical surpluses and the integral weights at each interpolation point.

3737




Hierarchical integration

To obtain the variance of the solution, we need to first obtain an approximate expression for 2u

Then the variance of the solution can be computed as

Therefore, it is easy to extract the mean and variance analytically, thus we only have the interpolation error. This is one of the advantages to use this interpolation based collocation method rather than quadrature based collocation method.

3838




Adaptive sparse grid collocation (ASGC)

Let us first revisit the 1D hierarchical interpolation For smooth functions, the hierarchical surpluses tend to zero as the interpolation level increases.

On the other hand, for non-smooth function, steep gradients/finite discontinuities are indicated by the magnitude of the hierarchical surplus.

The bigger the magnitude is, the stronger the underlying discontinuity is.

Therefore, the hierarchical surplus is a natural candidate for error control and implementation of adaptivity. If the hierarchical surplus is

larger than a pre-defined value (threshold), we simply add the 2N neighbor points of the current point.

3939




Adaptive sparse grid collocation

As we mentioned before, by using the equidistant nodes, it is easy to refine the grid locally around the non-smooth region.

It is easy to see this if we consider the 1D equidistant points of the sparse grid as a tree-like data structure

Then we can consider the interpolation level of a grid point Y as the depth of the tree D(Y). For example, the level of point 0.25 is 3.

Denote the father of a grid point as F(Y), where the father of the root 0.5 is itself, i.e. F(0.5) = 0.5.

Thus the conventional sparse grid in N-dimensional random space can be

reconsidered as

4040





Thus, we denote the sons of a grid point by 1, , NY Y Y

From this definition, it is noted that, in general, for each grid point there are two sons in each

dimension, therefore, for a grid point in a N-dimensional stochastic space, there are 2N sons.

The sons are also the neighbor points of the father, which are just the support nodes of the hierarchical basis functions in the next interpolation level.

By adding the neighbor points, we actually add the support nodes from the next interpolation level, i.e. we perform interpolation from level |i| to level |i|+1.

Therefore, in this way, we refine the grid locally while not violating the developments of the

Smolyak algorithm

4141





To construct the sparse grid, we can either work through the multi-index or the hierarchical basis. However, in order to adaptively refine the grid, we need to work through the hierarchical basis. Here we first show the equivalence of these two ways in a two dimensional domain.

We always start from the multi-index (1, 1), which means the interpolation level in first dimension is 1, in second dimension is also 1.

For level 1, the associated point is only 0.5.

So the tensor product of the coordinate is

0.5 0.5 (0.5,0.5) Thus, for interpolation level 0 of smolyak

construction, we have

(1,1)( , ) (1,1)(0.5,0.5)N NA w a

4242





Recalling, we define the interpolation level k as . ,N k NA

So, now we come to the next interpolation level, 1, 3k i

For this requirement, the following two multi-indices are satisfied:

(1,2), (2,1) Let us look at the index (1,2)

1 1i The associated set of points is 0.5 (j=1).

2 2i The associated set of points is 0 (j=1) and 1 (j=2).

So the tensor product is

(0.5) (0,1) (0.5,0), (0.5,1) Then the associated sum is

(1,2) 1,2(1,1) 1,2(0.5,0) (0.5,1)w a w a

4343





In the same way, for multi-index (2,1), we have the corresponding collocation points

(0,1) (0.5) (0,0.5), (1,0.5) So the associated sum is

(2,1) 2,1(1,1) 2,1(0,0.5) (1,0.5)w a w a

Recall 1, ( ) ( )( )Niiq N

q

A f f

i

So for Smolyak interpolation level 1, we have

1 (1,1) (1,2)2 1,2 (1,1) (1,1)

3

1,2 (2,1) 2,11,2 (1,1) 2,1

( ) (0.5,0.5) (0.5,0)

(0.5,1) (0,0.5) (1,0.5)

NiiA f w a w a

w a w a w a

i

4444





So, from this equation, it is noted that we can store the points with the surplus sequentially. When calculating interpolating value, we only need to perform a simple sum of each term. We don’t need to search for the point. Thus, it is very efficient to generate the realizations.

Here, we give the first three level of multi-index and corresponding nodes.

(1,1) (0.5,0.5)

(1,2) (0.5,0), (0.5,1)

(2,1) (0,0.5), (1,0.5)

(2,2) (0,0), (1,0), (0,1), (1,1)

(1,3) (0.5,0.25), (0.5,0.75)

(3,1) (0.25,0.5), (0.75,0.5)

4545





In fact, the most important formula is the following

From this equation, in order to calculate the surplus, we need the interpolation value at just previous level. That is why it requires that we construct the sparse grid level by level.

Now, let us revisit the algorithm in a different view. we start from the hierarchical basis. We still use the two-dimension grid as an example.

Recalling

4646





As previous given, the first level in a sparse grid is always a point (0.5,0.5). There is a possibility that the function value at this point is 0 and

thus the refinement terminates immediately. In order to avoid the early stop for the refinement process, we always refine the first level and keep a provision on the first few hierarchical surpluses.

It is noted here, the associated multi-index with this point (0.5, 0.5) is (1,1),i.e. and it corresponds to level of interpolation k = 0. (0.5,0.5) (1,1)

So the associated sum is still

(1,1)( , ) (1,1)(0.5,0.5)N NA w a

Now if the surplus is larger than the threshold, we add sons (neighbors) of this point to the grid.

4747





In one-dimension, the sons of the point 0.5 are 0,1.

According to the definition, the sons of the point (0.5,0.5) are

(0,0.5), (1,0.5), (0.5,0), (0.5,1)which are just the points in the level 1 of sparse grid. Indeed, the associated multi-index is

(0,0.5), (0,1) (2,1), (0.5,0), (0.5,1) (1, 2)

Recalling when we start from multi-index(1, 2) (0.5,0), (0.5,1), (2,1) (0,0.5), (1, 0.5)

So actually, we can see they are equivalent. Now, let us summarize what we have

1: (0,0.5), (1,0.5), (0.5,0), (0.5,1)k So the associated sum is

(1,2) 1,2 (2,1) 2,1(1,1) 1,2 (1,1) 2,1(0.5,0) (0.5,1) (0,0.5) (1,0.5)w a w a w a w a

4848





After calculating the hierarchical surplus, now we need to refine. Here we assume we refine all of the points, i.e. threshold is 0.

Notice here, all of the son points are actually from next interpolation level, so we still construct the sparse grid level by level.

First, in one dimension, the sons of point 0 is 0.25, 1 is 0.75.

So the sons of each point in this level are:

(0,0.5) (0.25,0.5), (0,1), (0,0)

(1,0.5) (0.75,0.5), (1,1), (1,0)

(0.5,0) (0.5,0.25), (1,0), (0,0)

(0.5,1) (0.5,0.75), (1,1), (0,1)

Notice here, we have some redundant points. Therefore, it is important to keep track of the uniqueness of the newly generated (active) points.

4949




Adaptive sparse grid collocation After delete the same points, and arrange ,we can have

(0,0), (1,0), (0,1), (1,1) (2,2)

(0.5,0.25), (0.5,0.75) (1,3)

(0.25,0.5), (0.75,0.5) (3,1)

which are just all of the support nodes in k=2. Therefore, we can see by setting the threshold equal zero, conventional

sparse grid is exactly recovered.

We actually work on point by point instead of the multi-index.

In this way, we can control the local refinement of the point.

Here, we equally treat each dimension locally, i.e., we still add the two neighbor points in all N dimensions. So there are generally 2N son (neighbor) points.

If the discontinuity is along one dimension, this method can still detect it. So in this way, we say this method can also detect the important dimension.

5050




Adaptive sparse grid collocation: implementation

As shown before, we need to keep track of the uniqueness of the newly active points.

To this end, we use the data structure <set> from the standard template library in C++ to store all the active points and we refer to this as the

active index set.

Due to the sorted nature of the <set>, the searching and inserting is always very efficient. Another advantage of using this data structure is that it is easy for a parallel code implementation. Since we store all of the active points from the next level in the <set>, we can evaluate the surplus for each point in parallel, which increases the performance significantly.

In addition, when the discontinuity is very strong, the hierarchical surpluses may decrease very slowly and the algorithm may not stop until a sufficiently high interpolation level. Therefore, a maximum interpolation level is always specified as another stopping criterion.

5151




Adaptive sparse grid collocation: Algorithms Let be the parameter for the adaptive refinement threshold. We propose the

following iterative algorithm beginning with a coarsest possible sparse grid , i.e.

with the N-dimensional multi-index i = (1,…,1), which is just the point (0.5,…,0.5).

0

1) Set level of Smolyak construction k = 0.

2) Construct the first level adaptive sparse grid .

• Calculate the function value at the point (0.5, …, 0.5).

• Generate the 2N neighbor points and add them to the active index set.

• Set k = k + 1;

3) While and the active index set is not emptymaxk k• Copy the points in the active index set to an old index set and clear the active index set.

• Calculate in parallel the hierarchical surplus of each point in the old index set according to

Here, we use all of the existing collocation points in the current adaptive sparse grid

This allows us to evaluate the surplus for each point from the old index set in parallel.

5252




Adaptive sparse grid collocation: Algorithms

• For each point in the old index set, if

Generate 2N neighbor points of the current active point according to the above equation;

Add them to the active index set.

• Add the points in the old index set to the existing adaptive sparse grid Now the adaptive sparse grid becomes

• Set k = k + 1;

4) Calculate the mean and variance, the PDF and if needed realizations of the solution.

In practice, instead of using surplus of the function value, it is sometimes preferable to use surplus of the square of the function value as the error indicator. This is because

the hierarchical surplus is related to the calculation of the variance. In principle, we can consider it like a local variance. Thus, is more sensitive to the local variation of the

stochastic function than . Error estimate of the adaptive interpolant

5353




NUMERICAL EXAMPLES

5454




Adaptive sparse grid interpolationAbility to detect and reconstruct steep gradients

5555




A dynamic system with discontinuityA dynamic system with discontinuity

We consider the following governing equation for a particle moving under the influence of a potential field and friction force:

2

2

d X dX dhfdt dxdt

0 00 , / 0X t x dX dt t v

If we set the potential field as , then the differential equation has two stable fixed points and an unstable fixed point at .

4 235 / 8 15 / 4h x x x 15 / 35x

The stochastic version of this problem with random initial position in the interval can be expressed as

with stochastic initial condition

where denotes the response of the stochastic system, ,

and is uniformly distributed random variable over .

23

2

35 15,

2 2

d X dXf X Xdtdt

0 00, , | 0t

dXX t Y X X Y

dt

,X t Y 0 1 2 / 2X x x

Y 1,1 .

1 2| | / 2X x x

0x

[ 1,1]

5656




A dynamic system with discontinuityA dynamic system with discontinuity

If we choose and a relatively large friction coefficient , a steady –state solution is achieved in a short time. The analytical steady-state solution is given by

This results in the following statistical moments

In the following computations, the time integration of the ODE is performed using a fourth-order Runge-Kutta scheme and a time step was used.

0 0.05,X 0.2X 2f

, 15 / 35, 0.25

, 15 / 35, 0.25

X t Y Y

X t Y Y

1, 15 / 35

4X t Y

45Var ,

112X t Y

0.001t

The analytic PDF is two dirac mass with unequal strength.

5757




Failure of gPC expansionFailure of gPC expansion

-2 -1.5 -1 -0.5 0 0.50

0.5

1

1.5

2

2.5

3

3.5

4

Y

PD

F

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

Y

X

-2.5 -2 -1.5 -1 -0.5 0 0.50

0.5

1

1.5

2

2.5

3

3.5

Y

PD

F

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

Y

X

Evolution of ( , ) for 0 10X t Y t PDF at 10t

10p

20p

Solution at 10t

5858





Order

Mea

n(X

)

5 10 15 20-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

gPCExact

Order

Var

(X)

5 10 15 200.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

gPCExact

Mean and variances exhibit oscillations with the increase of expansion order

5959





Y

X

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

p = 5p = 10p = 15p = 20Exact

Steady-state solution at 25t

Therefore, the gPC expansion fails to converge the exact solutions with increasing expansion orders. It fails to capture the discontinuity in the

stochastic space due to its global support, which is the well known “Gibbs phenomenon” when applying global spectral decomposition to problems with discontinuity in the random space.

6060




Failure of Lagrange polynomial interpolationFailure of Lagrange polynomial interpolation

-1 -0.5 0 0.5 10

1

2

3

4

5

6

7

8

9

10

Y

PD

F

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Y

X

-1 -0.5 0 0.5 10

5

10

15

20

25

30

35

Y

PD

F

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Y

X

Evolution of ( , ) for 0 10X t Y t Solution at 10t PDF at 10t

Much better than gPC, but still exhibit oscillation near discontinuity point.

5k

10k

6161





Level

Mea

n(X

)

2 4 6 8 10 120.1

0.2

0.3

0.4

0.5

Lagrange Polynomial interpolationExact

Level

Mea

n(X

)

10 11 12 130.16

0.162

0.164

0.166

0.168

0.17


Level

Var

(X)

10 11 12 130.4

0.401

0.402

0.403


Level

Var

(X)

2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5


6262





Y

X

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

k = 3k = 5k = 7k = 9Exact

Steady-state solution at 25t

Similarly, the CSGC method with Lagrange polynomials also fails to capture the discontinuity. However, better predictions of the low-order moments are obtained than the gPC method. Though the results indeed converge, the converged values are not accurate.

Near the discontinuity point of the solution, there are still several wriggles, which will not disappear even for higher interpolation levels.

6363




Sparse grid collocation with linear basis function

Evolution of ( , ) for 0 10X t Y t

5k 10k

6464




Sparse grid collocation with linear basis function

Y

X

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1-1

-0.5

0

0.5

1

k = 3k = 5k = 7k = 9Exact

Y

PD

F

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80

10

20

30

40

50

60

Solution at 25t PDF at 25t

Thus, it is correctly predicting the discontinuity behavior using linear basis function. And it is also noted that there are no oscillations in the solutions shown here.

6565




Adaptivity

#Points

Abs

olut

eer

ror

ofva

rian

ce

101 102 103 104 10510-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

CSGCASGC: = 10-2

Y

Lev

el

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10

5

10

15

20

For CSGC method, a maximum interpolation level of 15 is used and 32769 points are need to reach accuracy of . For the ASGC method, a maximum interpolation level of 30 is used. Only 115 number of points are required to achieve accuracy of .

51 10

103 10

6666




Highly discontinuous solution

Now we consider a more difficult problem. We set , and . All other conditions remain as before. The reduction of the friction coefficient,

together with higher initial energy will result in the oscillation of the particle for several cycles between each of the two stable equilibrium points.

0.05f 0 1X 0.1X

YX

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Y

X

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

Failure of gPC (left) and polynomial interpolation (right) at t = 250s. The failure of Lagrange polynomial interpolation is much more obvious.

6767




Highly discontinuous solution

Y

X

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8k = 4

Y

X

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8k = 6

Y

X

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8k = 8

Y

X

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8k = 10

Results using the CSGC

method with linear basis functions.

6868




The Kraichnan-Orszag three-mode problem

The governing equations are (X.Wan, G. Karniadakis, 2005),

12 3

21 3

31 22

dxx x

dtdx

x xdtdx

x xdt

subject to stochastic initial conditions:

1 1 2 2 3 3(0) (0; ), (0) (0; ), (0) (0; )x x x x x x

There is a discontinuity when the random initial condition approaches

(x1=x2) . So let us first look at the discontinuity in the

deterministic solution.

1 21 and 1x x

6969





Singularity at plane

Time

x2

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

x1(0) = 1, x2(0) = 0.95, x3(0) = 1x1(0) = 1, x2(0) = 0.97, x3(0) = 1x1(0) = 1, x2(0) = 0.99, x3(0) = 1x1(0) = 1, x2(0) = 1.00, x3(0) = 1

2 1x

7070




The Kraichnan-Orszag three-mode problem For computational convenience and clarity we first perform the following

transformation

The discontinuity then occurs at the planes and .

Thus, we first study the stochastic response subject to the following random 1D initial input:

where . Since the random initial data can cross the plane we know that gPC will fail for this case.

(X.Wan, G. Karniadakis, 2005)

The resulting governing equations become:

2 231 21 3 2 3 1 2, ,

dydy dyy y y y y y

dt dt dt

1 2 3(0; ) 1, (0; ) 0.1 (0; ), (0; ) 0y y Y y ( 1,1)Y U 2 (0; )y

2 0y

1 1

2 2

3 3

2 20

2 2

2 20

2 20 0 1

y x

y x

y x

1 0y 2 0y

7171





Time

y1

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2y1(0) = 0.1, y2(0) = 1.0, y3(0) = 1.0y1(0) = 0.05, y2(0) = 1.0, y3(0) = 1.0y1(0) = 0.02, y2(0) = 1.0, y3(0) = 1.0y1(0) = 0.0, y2(0) = 1.0, y3(0) = 1.0

Singularity at plane 1 0y

7272




One-dimensional random input: variance

Time

Var

ianc

eof

y 3

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8 MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 30

Time

Var

ianc

eof

y 2

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 30

Time

Var

ianc

eof

y 1

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 30

gPC fails after t = 6s while the ASGC method converges even with a large threshold. The solid line is the result from MC-SOBOL sequence with 106

iterations.

1Var( )y 2Var( )y

3Var( )y

7373




One-dimensional random input: adaptive sparse grid

Y

Lev

el

-1 -0.5 0 0.5 10

5

10

15

From the adaptive sparse grid, it is seen that most of the refinement after level 8 occurs around the discontinuity point Y = 0.0, which is consistent with previous discussion.

The refinement stops at level 16, which corresponds to 425 number of points, while the conventional sparse grid requires 65537 points.

Adaptive sparse grid

with

210

7474




One-dimensional random input

The ‘exact’ solution is taken

as the results given by MC-

SOBOL

The error level is defined as

, 301,2,3max Var( ) Var( )i i MC ti

y y

7575




One-dimensional random input: long term behavior

Time

Mea

nof

y 1

0 20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MC-SOBOL: 1,000,000ASGC: = 10-2

ASGC: = 10-3

Time

Var

ianc

eof

y 1

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

MC-SOBOL: 1,000,000ASGC: = 10-2

ASGC: = 10-3

Time

Var

ianc

eof

y 2

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

MC-SOBOL: 1,000,000ASGC: = 10-2

ASGC: = 10-3

Time

Var

ianc

eof

y 3

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7 MC-SOBOL: 1,000,000ASGC: = 10-2

ASGC: = 10-3

7676




One-dimensional random input: Realizations

These realizations are reconstructed using hierarchical surpluses. It is seen that at earlier time, the discontinuity has not yet been developed, which explains the reason that the gPC is accurate at earlier times.

7777




Two-dimensional random input

Next, we study the K-O problem with 2D input:

1 2 1 3 2(0; ) 1, (0; ) 0.1 (0; ), (0; ) (0; )y y Y y Y

Now, instead of a point, the discontinuity region becomes a line.

Time

Var

ianc

eof

y 1

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 10

Time

Var

ianc

eof

y 2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 10

7878





Time

Var

ianc

eof

y 3

0 2 4 6 8 100

0.2

0.4

0.6

0.8MC-SOBOL: 1,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

gPC: p = 10

Y1

Y2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

It can be seen that even though the gPC fails at a larger time, the adaptive collocation method converges to the reference solution given by MC-SOBOL with 106 iterations.

7979





The error level is defined as

, 101,2,3max Var( ) Var( )i i MC ti

y y

The number of points of the conventional

grid with interpolation 20

is 12582913.

8080




Two-dimensional random input: Long term behavior

0 10 20 30 40 50-2

-1

0

1

2

3

4

5 x 10-3

Time

Err

or i

n m

ean

ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

0 10 20 30 40 50-2

-1

0

1

2

3x 10

-3

Time

Err

or i

n va

rian

ce

ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

Error in the mean (left) and the variance (right) of y1 with different thresholds for the 2D K-O problem in [0, 50].

8181




Three-dimensional random input Next, we study the K-O problem with 3D input:

1 1 2 2 3 3(0; ) (0; ), (0; ) (0; ), (0; ) (0; )y Y y Y y Y This problem is much more difficult than any of the other problems examined previously. This is due to the strong discontinuity ( the discontinuity region now consists of the planes Y1 = 0 and Y2 = 0) and the

moderately-high dimension.

It can be verified from comparison with the result obtained by MC-SOBOL that unlike the previous results, here 2 x 106 iterations are needed to

correctly resolve the discontinuity.

Due to the symmetry of y1 and y2 and the corresponding random input, the variances of y1 and y2 are the same.

Finally, in order to show the accuracy of the current implementation of the h-adaptive ME-gPC, we provide the results for this problem.

8282




Three-dimensional random input:ASGC

Time

Var

ianc

eof

y 1

0 1 2 3 4 5 6 7 8 9 100.33

0.34

0.35

0.36

0.37

0.38

0.39

MC-SOBOL: 2,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

Time

Var

ianc

eof

y 3

0 1 2 3 4 5 6 7 8 9 100.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

MC-SOBOL: 2,000,000ASGC: = 10-1

ASGC: = 10-2

ASGC: = 10-3

Evolution of the variance of y1=y2 (left) and y3 (right) for the 3D K-O problem using ASGC

8383




Three-dimensional random input: ME-gPC

Time

Var

ianc

eof

y 1

0 1 2 3 4 5 6 7 8 9 100.33

0.34

0.35

0.36

0.37

0.38

0.39

MC-SOBOL: 2,000,000ME-gPC: p = 3ME-gPC: p = 4ME-gPC: p = 5ME-gPC: p = 6

Time

Var

ianc

eof

y 3

0 1 2 3 4 5 6 7 8 9 100.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

MC-SOBOL: 2,000,000ME-gPC: p = 3ME-gPC: p = 4ME-gPC: p = 5ME-gPC: p = 6

Evolution of the variance of y1=y2 (left) and y3 (right) for the 3D K-O problem using ME-gPC with and 4

1 10 32 10

8484




Three-dimensional random input

Due to the strong discontinuity in this

problem, it took much longer time for both methods

to arrive at the same accuracy.

8585




Stochastic elliptic problem In this example, we compare the convergence rate of the CSGC and ASGC methods through a stochastic elliptic problem in two spatial dimensions.

As shown in previous examples, the ASGC method can accurately capture the non-smooth region in the stochastic space. Therefore, when the non-smooth region is along a particular dimension (i.e. one dimension is more important than others), the ASGC method is expected to identify it and resolve it.

In this example, we demonstrate the ability of the ASGC method to detect important dimensions when each dimension weigh unequally.

This is similar to the dimension-adaptive method, especially in high stochastic dimension problems.

8686




Stochastic elliptic problem Here, we adopt the model problem

, , , in

, 0 on

N Na u f D

u D

with the physical domain . To avoid introducing errors of any significance from the domain discretization, we take a deterministic smooth load with homogeneous boundary conditions.

2, 0,1D x x y

, , cos sinNf x y x y

The deterministic problem is solved using the finite element method with 900 bilinear quadrilateral elements. Furthermore, in order to eliminate the errors associated with a numerical K-L expansion solver and to keep the random diffusivity strictly positive, we construct the random diffusion

coefficient with 1D spatial dependence as ,Na x

2 2 21/ 2 1

8

1

log , 0.5 1 cos 2 12

n LN

N nn

La x e x n Y

where are independent uniformly distributed random variables in the interval

, 1,nY n N 3, 3

8787




Stochastic elliptic problem This expansion is similar to a K-L expansion of a 1D random field with stationary covariance

2

1 21 2 2

log , 0.5 , expN

x xa x x x

L

Small values of the correlation L correspond to a slow decay, i.e. each stochastic dimension weighs almost equally. On the other hand, large

values of L results in fast decay rates, i.e., the first several stochastic dimensions corresponds to large eigenvalues weigh relatively more. By using this expansion, it is assumed that we are given an analytic stochastic input. Thus, there is no truncation error. This is different from the discretization of a random filed using the K-L expansion, where for different correlation lengths we keep different terms accordingly.

In this example, we fix N and change L to adjust the importance of each stochastic dimension. In this way, we investigate the effects of L on the ability of the ASGC method to detect the important dimensions.

8888




Stochastic elliptic problem: N = 5

8989




Stochastic elliptic problem: N = 5 We estimate the approximation error for the mean and variance.

Specifically, to estimate the computation error in the q-th level, we fix N and compare the results at two consecutive levels, e.g. the error for the mean is

2L D

, 1,q N N q N NE A u A u

The previous figures shows results for different correlation lengths at N=5.

For small L, the convergence rates for the CSGC and ASGC are nearly the same. On the other hand, for large L, the ASGC method requires

much less number of collocation points than the CSGC for the same accuracy.

This is because more points are placed only along the important dimensions which are associated with large eigenvalues.

Next, we study some higher-dimensional cases. Due to the rapid increase in the number of collocation points, we focus on moderate to higher

correlation lengths so that the ASGC is effective.

9090




Stochastic elliptic problem: higher-dimensions

#points:4193

Comparison:

CSGC: 3.5991x1011

9191




Stochastic elliptic problem: higher-dimensions In order to further verify our results, we compare the mean and the variance when N = 75 using the AGSC method with with the ‘exact’

solution obtained by MC-SOBOL simulation with 106 iterations.

x

0

0.2

0.4

0.6

0.8

1

y

0

0.2

0.4

0.6

0.8

1

0

0.0001

0.0002

0.0003

0.0004

Relative error of mean

x

0

0.2

0.4

0.6

0.8

1

y

0

0.2

0.4

0.6

0.8

1

0

0.0005

0.001

0.0015

Relative error of variance

810

,q N N MC

MC

E A u E u

E u

Computational time: ASGC : 0.5 hour MC: 2 hour

9292




Application to Rayleigh-Bénard instability

Cold wall = -0.5

hot wall X

Y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

Insu

late

d

Insu

late

d

Filled with air (Pr = 0.7). Ra = 2500, which is larger than the critical Rayleigh number, so that convection can be initialized by varying the hot wall temperature. Thus, the hot wall temperature is assumed to be a random variable.

Prior stochastic simulation, several deterministic computations were performed in order to find out the range where the critical point lies in.

9393




Rayleigh-Bénard instability: deterministic case

Time

[K]1/

2

20 40 60 80 100

10-3

10-2

10-1

100

h = 0.30h = 0.40h = 0.50h = 0.55h = 0.60h = 0.70

Evolution of the average kinetic

energy for different hot wall temperatures

As we know, below the critical temperature, the flow vanishes, which corresponds to a decaying curve. On the other hand, the growing curve represents the existence of heat convection. Therefore, the critical temperature lies in the range [0.5,0.55].

9494




Rayleigh-Bénard instability: deterministic formulation

Steady state Nusselt number which denotes the rate of heat transfer:

1

00

1y

h c

Nu dxy

1 Conduction

1 Convection

Nu

Nu

This again verifies the critical hot wall temperature lies in the range [0.5,0.55]. However, the exact critical value is not known to us. So, now we try to capture this unstable equilibrium using the ASGC method.

9595




Rayleigh-Bénard instability: stochastic formulation

We assume the following stochastic boundary condition for the hot wall temperature:

0.4 0.3h Y where Y is a uniform random variable in the interval [0,1]. Following the discussion before, both conductive and convective modes occur in this range.

h

Lev

el

0.4 0.45 0.5 0.55 0.6 0.65 0.7

2

4

6

8

10

12

h

Nu

0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

0.05

0.1

0.15

1Nu Nu 0.541

9696





h

0.4 0.45 0.5 0.55 0.6 0.65 0.7-0.05

0

0.05

0.1

0.15

0.2

0.25

h

u

0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

0.02

0.04

0.06

0.08

h

v

0.4 0.45 0.5 0.55 0.6 0.65 0.7

0

1

2

3

Solution of the variables versus hot-wall temperature at

point (0.1,0.5)

u v

Linear: conduction

nonlinear: convection

0.541

9797





0.350.250.150.05

-0.05-0.15-0.25-0.35-0.45

Max = 0.436984

0.350.250.150.05

-0.05-0.15-0.25-0.35-0.45

Max = 0.436984

Prediction of the temperature when using ASGC (left) and the solution of the deterministic problem using the same (right).

Fluid flow vanishes, and the contour distribution of the temperature is characterized by parallel horizontal lines which is a typical distribution for

heat conduction.

0.436984h

h

9898





0.60.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.667891

43210

-1-2-3-4

Max = 4.33161

43210

-1-2-3-4

Max = 4.31858

43210

-1-2-3-4

Max = 4.3208

43210

-1-2-3-4

Max = 4.33384

0.60.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.667891

Prediction of the temperature when using ASGC (top) and the solution of the deterministic problem using the same (bottom).

0.66789h

h

9999




Rayleigh-Bénard instability: Mean

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73709

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73123

0.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.55

0.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.549918

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73715

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73128

ASGC (top) MC-SOBOL (bottom)

100100




Rayleigh-Bénard instability: Variance

0.0120.0110.010.0090.0080.0070.0060.0050.0040.0030.0020.001

Max = 0.0129218

32.62.21.81.410.60.2

Max = 3.30617

32.62.21.81.410.60.2

Max = 3.28535

32.62.21.81.410.60.2

Max = 3.27903

32.62.21.81.410.60.2

Max = 3.29984

0.0120.0110.010.0090.0080.0070.0060.0050.0040.0030.0020.001

Max = 0.0128783

ASGC (top) MC-SOBOL (bottom)

101101




Conclusion

An adaptive hierarchical sparse grid collocation method is developed based on the error control of local hierarchical surpluses.

Besides the detection of important and unimportant dimensions as dimension-adaptive methods can do, additional singularity and local behavior of the solution can also be revealed.

By employing adaptivity, great computational saving is also achieved than the conventional sparse grid method and traditional gPC related method.

Solution statistics is easily extracted with a higher accuracy due to the analyticity of the linear basis function used.

The current computational code is a highly efficient parallel program and has a wider application in various uncertainty quantification area.

nicholas zabaras and xiang ma

Documents