random fields - efficient analysis and simulation...karhunen-loeve expansion is very useful for...

Random FieldsEfficient Analysis and Simulation

Christian Bucher & Sebastian Wolff

Vienna University of Technology

& DYNARDO Austria GmbH, Vienna

Overview

Introduction

Elementary properties

Conditional random fields

Computational aspects

Example

Concluding remarks

2/25 c⃝ Christian Bucher 2010-2014

Random field

Real valued function H(x) defined in an n-dimensional space

H ∈ R; x = [x1, x2, . . . xn]T ∈ D ⊂ Rn

Ensemble of all possible realisations

Describe statistics in terms of mean and variance

Need to consider the correlation structure between values of

H at different locations x and y

H(x, ω)

x, y

L

ω

xy


Second order statistics of random field

Mean value function

H(x) = E[H(x)]

Autocovariance function

CHH(x, y) = E[{H(x)− H(x)}{H(y)− H(y)}]

A random field H(x) is called weakly homogeneous if

H(x) = const. ∀x ∈ D; CHH(x, x+ξ) = CHH(ξ) ∀x ∈ D

A homogeneous random field H(x) is called isotropic if

CHH(ξ) = CHH(||ξ||) ∀ξ

Correlation distance (characteristic length Lc)


Example: Random field in a square plate

Simulated random samples of isotropic field


Conditional Random Fields 1

Assume that the values of the random field H(x) are known

at the locations xk , k = 1 . . . m

Stochastic interpolation for the conditional random field:

H(xi) = a(x) +

m∑k=1

bk(x)H(xk)

a(x) and bk(x) are random interpolating functions.

Make the mean value of the difference between the random

field and the conditional field zero

E[H(x)−H(x)] = 0

Minimize the variance of the difference

E[(H(x)−H(x))2]→ Min.


Conditional Random Fields 2Mean value of the conditional random field.

¯H(x) =[CHH(x, x1) . . . CHH(x, xm)

]C−1HH

H(x1)...H(xm)

CHH denotes the covariance matrix of the random field H(x)

at the locations of the measurements.

Covariance matrix of the conditional random field

C(x, y) = C(x, y)−

−[CHH(x, x1) . . . CHH(x, xm)

]C−1HH

CHH(y, x1)...

CHH(y, xm)

Zero at the measurement points.


Spectral decomposition

Perform a Fourier-type series expansion using deterministic

basis functions ϕk and random coefficients ck

H(x) =

∞∑k=1

ckϕk(x), ck ∈ R, ϕk ∈ R, x ∈ D

Optimal choice of the basis functions is given by an

eigenvalue (”spectral”) decomposition of the

auto-covariance function (Karhunen-Loeve expansion)

CHH =

∞∑k=1

λkϕk(x)ϕk(y),

∫DCHH(x, y)ϕx(x)dx = λkϕk(y)

The basis functions ϕk are orthogonal and the coefficients ckare uncorrelated.


Discrete version

Discrete random field

Hi = H(xi), i = 1 . . . N (1)

Spectral decomposition is given by

Hi − E(Hi) =N∑k=1

ϕk(xi)ck =

N∑k=1

ϕikck (1)

In matrix-vector notation

H = Φc+ H

Computation of basis vectors by solving for the eigenvalues

λk of the covariance matrix CHH

CHHϕk = λkϕk ; λk ≥ 0; k = 1 . . . N


Example: Random field in a square plate

Basis vectors


Modeling random fields

Most important: Correlation structure, most significant

parameter is the correlation length Lc .

Estimate for the correlation length can be obtained by

applying statistical methods to observed data

Type of probability distribution of the material/geometrical

parameters. Statistical methods can be applied to infer

distribution information from observed measurements.

Helpful to identify the exact type of correlation (or

covariance) function, and to check for homogeneity. This will

be feasible only if a fairly large set of experimental data is

available.


Computational aspects

Need to set up the covariance matrix from covariance

function of field

Storage requirements of O(M2)Covariance matrix is full

Karhunen-Loeve expansion is realised using numerical

methods from linear algebra (eigenvalue analysis)

Numerical complexity of O(M3)


Simulation for small correlation length

Assemble sparse covariance matrix (e.g. based on piecewise

polynomial covariance functions)

Cl ,p(d) = (1− d/l)p+ , p > 1

Perform a decomposition of the covariance matrix, possible

C = LLT , eg. by a sparse Cholesky factorization.

Simulate N field vectors uk of statistically independent

standard-normal random variables, one number for each

node.

Apply the correlation in standard normal space for each

sample k : zk = Luk .

Transform the correlated field samples into the space of the

desired random field: xk,i = F(−1) (N(zk,i)).

Does not reduce the number of variables


Simulation for large correlation length 1

Typical covariance function

Cl(d) = exp

(−d2

2l2

)A spectral decomposition is used to factorize the covariance

matrix by Cov = Φdiag(λi)ΦT with eigenvalues λi and

orthogonal eigenvectors Φ = [ϕi ].

This decomposition is used to reduce the number of random

variables. Given a moderately large correlation length, only a

few (eg. 3-5) eigenvectors are required to represent more

than 90% of the total variability.

Perform a decomposition of the covariance matrix

CHH = Φdiag(λi i)ΦT and choose m basis vectors ϕi being

associated with the largest eigenvalues.



Simulate N vectors uk of statistically independent

standard-normal random variables, each vector is of

dimension m.

Apply the (decomposed) covariance in standard normal space

for each sample k

zk =

m∑i

√λiϕiuk,i


desired random field: xk,j = F(−1) (N(zk,j)).



A global error measure ϵ may be based on the total variability

being explained by the selected eigenvalues, i.e.

ϵ = 1−∑i=1 cλi∑i=1 nλi

= 1−1

n

∑i=1

cλi

wherein n is the number of discrete points.

This procedure allows the generation of random field samples

with relatively large correlation length parameters

It is based on a model order reduction, i.e. only a portion of

the desired variability can be retained.

Covariance matrix is stored as a dense matrix. Hence, the

size of the FEM mesh is effectively limited to ≈ 30.000nodes (covariance matrix has 9x108 entries, i.e. > 7GB).


Efficient simulation strategy

Randomly select M support points from the finite element

mesh.

Assemble the covariance matrix for the selected sub-space.

Perform a decomposition of the covariance matrix

C = Φdiag(λi)ΦT and choose m basis vectors ϕi .

Create basis vectors ψi by interpolating the values of ϕi on

the FEM mesh.

Simulate N vectors uk of statistically independent

standard-normal random variables, each vector is of

dimension m.

Apply the (decomposed) covariance in standard normal space

for each sample k

zk =

m∑i

ψiuk,i


desired random field: xk,j = F(−1) (N(zk,j)).


Expansion Optimal Linear Estimator 1

Expansion Optimal Linear Estimation (EOLE) is an

extension of Kriging

Kriging interpolates a random field based on samples being

measured at a sub-set of mesh points.

Assume that the sub-space is described by the field values

yk = {zk,1, . . . , zk,M} = {m∑i=1

√λiϕi ,1uk,i , . . . ,

m∑i=1

√λiϕi ,Muk,i}


Expansion Optimal Linear Estimator 2

Minimization of the variance between the target random field

and its approximation under the constraint of equal mean

values of both results in:

ψi = CTzyC

−1yy

ϕi√λi

with Cyy denoting the correlation matrix between the

sub-space points and Czy denoting the (rectangular)

covariance matrix between the sub-space points and the

nodes in full space.


Example

Sheet metal forming application

Modelled by 4-node shell elements using 8786 finite element

nodes

Homogeneoues field, exponential correlation function

Maximum dimension is 540 mm, correlation length

parameter is chosen to be 100 mm.

Truncated Gaussian distribution with mean value 5, standard

deviation 15, lower bound −20 and upper bound 30Sub-space dimension is chosen to be small (between 50 and

1000 points)


Example

M = 50 M = 200

M = 500


Example - Basis vectors (M = 50)


Example - Basis vectors (full)


Errors

MAC values of various shapes (reference of comparison: full

model) for different numbers of support points n.

n MAC ψ1 MAC ψ2 MAC ψ5 MAC ψ1050 0.999 0.999 0.949 0.393

100 0.999 0.999 0.999 0.986

200 0.999 0.999 0.999 0.999

400 0.999 0.999 0.999 0.999

800 1 1 0.999 0.999

8786 1 1 1 1


Concluding Remarks

Karhunen-Loeve expansion is very useful for reduction of

number of variables

Solution of eigenvalue problem may run into computational

problems (storage, time)

Suitable reduction methods reduce storage and time

requirements drastically

→ Software Statistics on Structures - SoS by


random fields - efficient analysis and simulation...karhunen-loeve expansion is very useful for...

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