Stochastic model reduction for chaos representations

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Comput. Methods Appl. Mech. Engrg. 196 (2007) 3951–3966

Stochastic model reduction for chaos representations

Alireza Doostan a, Roger G. Ghanem b,*, John Red-Horse c

a Department of Civil Engineering, The Johns Hopkins University, Baltimore, MD 21218, USAb 210 KAP Hall, University of Southern California, Los Angeles, CA 90089, USA

c Sandia National Laboratories, Albuqurque, NM 87185, USA

Received 1 November 2005; accepted 30 October 2006Available online 27 March 2007

Abstract

This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galer-kin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for theefficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) inthe Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is devel-oped to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms athand, the solution of SPDE’s becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided forthe adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through itsapplication to an example problem from solid mechanics.� 2007 Elsevier B.V. All rights reserved.

Keywords: Uncertainty quantification; Stochastic finite elements; Stochastic model reduction

1. Introduction

Since the information available for predicting the behav-ior and evolution of many physical phenomena is incom-plete, limited, or known with a finite level of confidence,uncertainty quantification (UQ) is a critical componentfor the validation of associated predictive models. To thisend, various numerical approaches have been developedto characterize and propagate uncertainties from data tosystem response through partial differential equations withstochastic parameters.

The present paper adopts a probabilistic framework forpropagating uncertainty associated with the task of reduc-ing the computational costs. In a series of previous publica-tions, the authors have presented a mathematicalframework for the characterization and propagation of

0045-7825/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2006.10.047

* Corresponding author. Tel.: +1 213 740 9528; fax: +1 213 740 2037.E-mail addresses: [email protected] (A. Doostan), [email protected]

(R.G. Ghanem), [email protected] (J. Red-Horse).

uncertainty in physical systems [37,17,14,13,15,28,19,22,21,20,34,33,11,10,23,18]. This work is based on the adapta-tion of multiple Wiener integral representations [40,7] tofinite-dimensional spaces and their implementation into aweighted residual scheme for the stochastic characteriza-tion of the solution of SPDE’s. The restriction of therepresentations to finite-dimensional uncertainty (i.e. sto-chastic processes characterized by a finite-dimensional ran-dom vector) permits the generalization of the Wienerconstructions which had used polynomials orthogonal withrespect to Gaussian measure. Extensions of that work tothe Askey scheme were recently developed [42,41] togetherwith extensions using non-orthogonal representations [2,4]and representations in terms of wavelets [30,29]. While theabove extensions are limited to representations in terms ofindependent random variables, extensions using finite-dimensional dependent random vectors (such as appearing,for instance, in the finite-dimensional Karhunen–Loeverepresentation of an arbitrary stochastic process) have alsobeen completed [36]. These so-called polynomial chaos

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3952 A. Doostan et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3951–3966

(PC) expansions, coupled with stochastic projectionschemes, provide a general method for characterizing thesolution to problems of mathematical physics whoseparameters have been described as stochastic process. Ananalysis of the error associated with this method hasalready been developed [6,4]. The mathematical foundationof these developments, which permits their rigorous math-ematical analysis, lies in functional analysis and in theobservation that second-order random vectors (i.e. looselyspeaking, those with finite second order statistics) form aHilbert space.

While mathematically elegant, standard PC stochasticGalerkin schemes suffer from the curse of dimensionality:the number of basis functions increases exponentially withthe size of the random vector characterizing the system, ashigher order approximations become necessary to moreaccurately capture higher order response statistics. Thispaper concentrates on developing a simple procedure formitigating the computational cost associated with the sto-chastic finite element analysis of such systems. The benefitof the proposed algorithm lies in its ability to estimateresponse statistics with negligible loss of accuracy but sav-ing great amount of CPU time. This is achieved by identi-fying and characterizing a low-dimensional manifold onwhich the solution can be approximated within specifiedaccuracy.

The paper is organized as follows. Section 2 summarizesuncertainty propagation using Wiener Hermite polynomialchaos expansions together with some of challenges in theirapplication to complex systems. Following that, the pro-posed model reduction technique is introduced in Section3, which consists of two main components of the proposedalgorithm, namely a coarse scale analysis of the problemalong with a Hilbert–Karhunen–Loeve expansion of thecoarse scale response as the first component and the fine

scale analysis as the second constituent of the scheme.Finally, in Section 4, a numerical experiment is performedto demonstrate and delineate the performance of the pro-posed procedures.

2. Uncertainty propagation

Consider the stochastic linear elliptic boundary valueproblem, with stochastic operator and deterministic input,which consists of finding a stochastic functionuðx;xÞ : D� X! R, such that the following equationholds almost surely in X,

�r:ðaðx;xÞruðx;xÞÞ ¼ f ðxÞ x 2 D;

uðx;xÞ ¼ 0 x 2 oD;ð1Þ

where

0 < amin 6 aðx;xÞ as in X; 8x: ð2Þ

a : D� X! R is a stochastic function with continuous andsquare-integrable covariance function, X is the set of ele-mentary events, and x 2 X. It is assumed, without loss of

generality, that the input function f ðxÞ 2 L2ðDÞ is a deter-ministic function. For computational purposes, aðx;xÞ isusually assumed to be a function of a finite number of vari-ables. In particular in this work, it is assumed that aðx;xÞ isrepresented as, aðx;xÞ :¼ aðx; n1ðxÞ; . . . ; nqðxÞÞ, wherefnigq

i¼1 is a set of real mutually orthonormal Gaussian ran-dom variables with mean zero. Here, q is the number ofeffective coordinates in the stochastic characterization ofthe problem, and will henceforth be referred to as itsstochastic dimension. The solution of (1) is a mapping offx; n1ðxÞ; n2ðxÞ; . . . ; nqðxÞg,uðx;xÞ :¼ uðx; n1ðxÞ; . . . ; nqðxÞÞ; ð3Þ

which is in general a nonlinear mapping. For the sake ofcompleteness the fundamental concepts of the stochasticprojection approach are briefly reviewed next [21].

It is first noted at this point that it is desirable to havethe constraint given by Eq. (2) satisfied by any approxima-tion to the stochastic process aðx;xÞ. A number of proce-dures have been pursued to that end [22,2,36], in manycases conjuring up assumptions that are not physically sub-stantiated (e.g. requiring a particular functional form forthe process aðx;xÞ).

2.1. Spectral stochastic finite element method (SSFEM)

2.1.1. Variational formulationThe analysis of the numerical approximation of stochas-

tic functions can be greatly facilitated by using a tensorproduct framework. Consider two domains x 2 D;x 2 X,let H 1

0ðDÞ be the subspace of H 1ðDÞ consisting of functionswhich vanish on oD and are equipped with the normjjvjjH1

0ðDÞ ¼ f

RDjrvj2 dxg

12, also let L2ðXÞ be the space of

random variables with finite variance defined on X, andequipped with the inner product ðw; ~wÞL2ðXÞ ¼ hw~wi;8w; ~w 2 L2ðXÞ. Where h:i denotes the mathematical expec-tation operator. Then the tensor space H 1

0ðDÞ � L2ðXÞ isthe completion of the formal sums uðx;xÞ ¼P

i;j¼1;...;nviðxÞwjðxÞ, fvig � H 10ðDÞ; fwjg � L2ðXÞ, with

respect to the inner product

ðu; ~uÞH10ðDÞ�L2ðXÞ ¼

Xi;j

ðvi;~viÞH10ðDÞðwj; ~wjÞL2ðXÞ: ð4Þ

Consider the tensor product Hilbert space H ¼ H 10ðDÞ�

L2ðXÞ equipped with inner product ðv; uÞH ¼hRDrv � rudxi. Construct the bilinear form B : H�

H ! R by

Bðv;wÞ :¼ZD

arv � rwdx� �

; 8v;w 2 H ; ð5Þ

By the assumption of (2) on aðx;xÞ, the continuity andcoercivity of the bilinear form B are guaranteed [3], there-fore by the Lax–Milgram Lemma, [9], the following varia-tional formulation has a unique solution in H

Bðu; vÞ ¼LðvÞ; 8v 2 H ; ð6Þ

A. Doostan et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 3951–3966 3953

where

LðvÞ :¼ZD

fv dx� �

; 8v 2 H ð7Þ

is a bounded linear functional.

2.1.2. Polynomial chaos expansion

Any second-order functional of the Brownian motioncan be expressed as mean-square convergent series in termsof infinite-dimensional Hermite polynomials in Gaussianvariables [7], i.e.

vðxÞ ¼ x0H 0

þX1i1¼1

xi1 H 1ðni1ðxÞÞ

þX1i1¼1

Xi1

i2¼1

xi1i2 H 2ðni1ðxÞ; ni2ðxÞÞ

þX1i1¼1

Xi1

i2¼1

Xi2

i3¼1

xi1i2i3 H 3ðni1ðxÞ; ni2ðxÞ; ni3ðxÞÞ þ . . . ;

ð8Þ

where H nðni1 ; . . . ; ninÞ is a Hermite polynomial of order n invariables ðni1 ; . . . ; ninÞ. It will be notationally more conve-nient to rewrite the above equation in the form

vðxÞ ¼X1j¼0

xjwjðxÞ; ð9Þ

where wjðxÞ :¼ wjðnðxÞÞ and nðxÞ is the vector of indepen-dent Gaussian random variables ðni1 ; . . . ; ninÞ. Also there isa one-to-one correspondence between the functionals wjð�Þand H kð�Þ and also between the associated coefficients. Forcomputational purposes, the above series should be trun-cated with respect to both the dimension of vector n andalso the order of Hermite polynomials. An important prop-erty of the above polynomials which will be employed inthe following sections is their orthogonality with respectto the Gaussian probability measure, namely

wiwj

� �¼ w2

i

� �dij; ð10Þ

where dij is the Kronecker delta.It is noted at this point that the PC representation theo-

rem [7] was initially developed in an infinite-dimensionalsetting using multiple Wiener integrals. While this pointis crucial for representing parameters driven by unknownrandom sources, e.g. experimental measurements, forwhich the dimension of PC expansion cannot be fixed, itcan be readily mitigated when the source of random uncer-tainty has been identified with a finite number of randomvariables, as is the case with many problems of practicalinterest. It should moreover be noted that in the infinite-dimensional case, only expansions based on Hermite poly-nomials in Gaussian variables are known to converge inmean square.

2.1.3. Galerkin projection

The SSFEM seeks the solution of the above variationalproblem on the tensor product space consisting of the finitedimensional space of piecewise continuous polynomialscorresponding to a partition Th of D as the spatial discret-ization and the space of random variables spanned by poly-nomial chaos [21] of order up to p as the discretization ofthe random dimension of uðx;xÞ. More specifically, con-sider a family of finite element approximation spaces,X :¼ X h � H 1

0ðDÞ, consisting of piecewise linear continu-ous functions on the corresponding discretization of D,and Y :¼ Y p ¼ �p

i¼1Hi � L2ðXÞ, where Hi represents theith Homogeneous chaos, the space spanned by all ith orderHermite polynomial chaos, constructed from the setfnigq

i¼1, [21]. Then the finite dimensional Galerkin approx-imation of the exact solution uðx;xÞ on the tensor productspace of X and Y is obtained by solving

Bðuh;p; vÞ ¼LðvÞ; 8v 2 X � Y : ð11Þ

More clearly, one can write the approximation solution

uh;pðx;xÞ ¼X

i;j

uijN iðxÞwjðxÞ; ð12Þ

and use the test function vðx;xÞ ¼ NkðxÞwlðxÞ to find thecoefficients uij. Then (11) gives the following system of lin-ear equations for uijX

i;j

hwlwjðað�; �ÞrN i;rN kÞL2ðDÞiuij

¼ hwlðf ð�Þ;NkÞL2ðDÞi8k; l: ð13Þ

In the above equations, fNigNsi¼1 and fwig

Pi¼0 are bases for X

and Y, respectively.

2.1.4. Modeling stochastic coefficient aðx;xÞIn the context of SSFEM, as described extensively in

[16,15,34], one has to represent the stochastic function a

of (13) in its polynomial chaos decomposition, i.e.

aðx;xÞ :¼ aðx; n1ðxÞ; . . . ; nqðxÞÞ ¼X

i

aiðxÞwiðxÞ: ð14Þ

The above representation is viewed as a generalized Fou-rier expansion with generalized coefficients aiðxÞ being

aiðxÞ :¼ haðx; �Þwiihw2

i i: ð15Þ

In the examples shown later in this paper, the coefficientaðx;xÞ is modeled as a truncated lognormal process. Inparticular, let gðx;xÞ ¼ g0ðxÞ þ

PLi¼1nigiðxÞ be a truncated

Karhunen–Loeve expansion of a Gaussian process definedon D with given mean g0ðxÞ and covariance functionRggðx; yÞ. Therefore, the fnigL

i¼1 are uncorrelated standardGaussian random variables and giðxÞ denotes the ithweighted eigenfunction of the covariance kernel Rggðx; yÞ,i.e.ZD

Rggðx; yÞ/iðyÞdy ¼ ki/iðxÞ; x 2 D; ð16Þ


and,

giðxÞ :¼ffiffiffiffiki

p/iðxÞ: ð17Þ

Consider next the process aðx;xÞ :¼ egðx;xÞ ¼ exp½g0ðxÞþPniðxÞgiðxÞ� obtained from gðx;xÞ by exponentiation,

thus leading aðx;xÞ having a lognormal marginal probabil-ity distribution. Since the sequence fgigiP1 is uniformlybounded in L1ðDÞ, the variational problem (6) is well-posed in-spite the fact that aðx;xÞ violates the conditionof (2) for a deterministic lower bound amin [3]. Following[15], a more detailed polynomial chaos representation oflognormal process aðx;xÞ is available by simplifying Eq.(15) and is given by

aðx;xÞ ¼ a0ðxÞ 1þXL

i¼1

nigi þXL

i¼1

XL

j¼1

ðninj � dijÞhðninj � dijÞ2i

gigj þ . . .

!;

ð18Þ

where a0ðxÞ ¼ exp½g0ðxÞ þRggðx;xÞ

2�, and the consecutive sum-

mations of the above equation correspond to differentpolynomial chaos orders. Clearly, for computational pur-poses, the above representation should be truncated atsome finite order which depends on the choice of prescribedstatistical model for aðx;xÞ as well as the necessary require-ments for stability of the variational problem (6). Accord-ing to [31], in order for the discrete Galerkin solution of (6)to be well-posed in the sense of Hadamard, the order ofpolynomial chaos expansion of a in (18) should be at leasttwice as that of the expansion of response u in (12).

2.2. Computational cost of SSFEM

The computational cost of the SSFEM depends on thespatial discretization size, h, the dimension of PC expan-sion of the coefficient a, i.e. q, and the order of PC expan-sion of solution u, i.e. p. In particular, the total number ofterms P + 1 with order less than or equal to p in q stochas-tic dimensions is given by

P þ 1 ¼ ðp þ qÞ!p!q!

: ð19Þ

Therefore, the size of the algebraic system of equations tobe solved for the response coefficients uij is NsðP þ 1Þ�NsðP þ 1Þ, which can be extremely large for the problemssuffering certain complexities in their spatial or stochasticdimension. An example of such complexities in the spatialdomain is the simulation of crack evolution in a solid body.In the stochastic dimension, this issue is highlighted inproblems exhibiting highly variable input parameters[21,11], or long transient behavior as in some stochasticadvection-diffusion problems, [42,39]. The former problemrequires high resolution of spatial discretization around thecrack tip and the latter necessitates high-order approxima-tion in the random dimension. Efficient model reduction inthe product space is thus essential for the accurate resolu-tion of many such physical problems.

Usually, and in order to maintain completeness up to aspecified order p in Y, a chaos expansion of a full polyno-mial order in Y is carried out, resulting in an exponentialgrowth in the cardinality of the approximating space asthe stochastic dimension is increased. Following this tradi-tional approach, the resulting complexity of this problem isgoverned by the stochastic complexity of the data, and notthat of the solution. In a number of problems of interest, itcan be observed that the physical system acts as a filter,smoothing out to some extent, fluctuations in the data.In such cases, developing computational tools that areadapted to the complexity in the solution, rather than inthe data, carry the promise of extending the realm of sto-chastic projection methods to a much wider class of com-putationally intensive problems.

3. Stochastic model reduction

An intrinsic alternative optimal basis to fwigPi¼0 can be

obtained by an adapted proper orthogonal decomposition(POD) or a modified version of the Karhunen–Loeveexpansion, which is the subject of the present section. Ithas been shown, in numerous recent work [27,8,1,24,35,32,25], that by considering the subspace characterized byonly a few of these basis functions, a reduced order modelof the problem can be developed which is capable of wellapproximating the quantities of interest.

The standard form of the Karhunen–Loeve expansion[26] has been constructed as an optimal representation inthe L2 sense. However, an important class of stochasticfunctions, such as the solution of the SPDE of (1), havecertain regularity and degree of smoothness that justifiestheir representation in a proper subset of L2 (X), namelysome Sobolev spaces. These ideas have been previouslydeveloped and implemented in various contexts, such asminimal dynamical systems from PDEs [24], functionaldata analysis [35,32], and image analysis [25]. In the presentwork, and following common nomenclature [25], this rep-resentation is referred to as the Hilbert–Karhunen–Loeveexpansion.

3.1. Hilbert–Karhunen–Loeve expansion of uðx;xÞ

Let u 2 H 10ðDÞ � L2ðXÞ be the solution to (1). Consider

the following orthogonal representation of u,

uðx;xÞ ¼ huðxÞi þX1i¼1

ðuð�;xÞ � hui;uiÞH10ðDÞuiðxÞ; ð20Þ

where ð�; �ÞH10ðDÞ denotes the inner product in H 1

0ðDÞ andfuig is an orthonormal basis in H 1

0ðDÞ, i.e.

ðw; vÞH10ðDÞ ¼

ZD

rw � rvdx 8w; v 2 H 10ðDÞ; ð21Þ

and,

ðui;ujÞH10ðDÞ ¼ dij 8ui;uj 2 fuig: ð22Þ


Clearly, ðuð�;xÞ � hui;uiÞH10ðDÞ is a random variable, and

can thus be expressed asffiffiffiffiki

pgiðxÞ :¼ ðuð�;xÞ � hui;uiÞH1

0ðDÞ; ð23Þ

where gi(x) is a zero mean random variable with unit var-iance, through the normalizing constant ki, whose proba-bility measure is defined from that of u through themapping (23).

Particular choice of {ui} in (20), as discussed in thesequel, leads to an optimal representation of u in the sensethat the remainder error of M-term truncation of (20) isminimum in H 1

0ðDÞ � L2ðXÞ compared to any other ortho-normal basis {ui} of H 1

0ðDÞ.Let Rðx; yÞ ¼ hðuðx; �Þ � huiÞðuðy; �Þ � huiÞi be the covari-

ance kernel of u, and let T : H 10ðDÞ ! H 1

0ðDÞ be the follow-ing compact self-adjoint operator of the Hilbert–Schmidttype,

Tvð�Þ :¼ ðRð�; yÞ; vÞH10ðDÞ 8v 2 H 1

0ðDÞ: ð24Þ

The operator T has a nondecreasing sequence of non-neg-ative eigenvalues {mi} which corresponds to a sequence oforthonormal eigenfunctions {ui}, i.e.

ðRð�; yÞ;uiÞH10ðDÞ ¼ miuiðyÞ 8i: ð25Þ

Using orthonormal eigenfunctions {ui} of (25) in (20) andthe sequence of eigenvalues {mi} as normalizing constants{ki} of (23), it is straightforward to show that the randomvariables {gi} are also mutually orthonormal, i.e.hgigji ¼ dij8i; j. Having the representation

uðx;xÞ ¼ huðxÞi þX1i¼1

ffiffiffiffimip

giðxÞuiðxÞ ð26Þ

for u, using eigenfunctions {ui} of (25), one can alterna-tively perform the stochastic Galerkin scheme describedin Section 2.1.3 with the adapted orthogonal basis {gi} ofEq. (26). However, For the case of SFEM, since the covari-ance structure of u in not a priori available, the aboveexpansion of u is not defined. More importantly, even ifthe covariance structure of u is a priori known, the exactjoint probability density function associated with randomvariables of (26) requires the complete statistical represen-tation of u. Thus the orthonormal basis {gi} of (26) cannotbe directly adopted in the stochastic Galerkin projection ofSection 2.1.3. In the sequel, a simple but effective algorithmis proposed to overcome the above difficulties. Particularly,an SSFEM analysis of the problem is performed on acoarse discretization, Thc , of D with original PC basisfunctions of order less or equal p. The solution of this anal-ysis is then processed to extract the representation of the set{gi} in terms of fwig

Pi¼1. The reduced-order model of the

original problem then consists of discretization Thfof D,

referred to here as fine discretization, and the set {gi} asthe basis in random dimension. It should be noted that inthis work, a fine discretization does not necessarily implya small error in the spatial dimension of the problem,rather it denotes any desirable discretization level. A coarse

mesh can be defined in terms of a discretization on whichthe dominant {gi} can be accurately characterized inL2ðXÞ. The premise in the foregoing is that such a mesh willtypically be much coarser than required for resolving thephysics of the problem.

3.2. Coarse discretization analysis

Consider a coarse mesh discretization Thc of D. A p-order PC expansion of solution of (1) on Thc approximatesu as

uhc;pðx;xÞ ¼X

i;j

uijN iðxÞwjðxÞ: ð27Þ

Having the above approximation of u, the covariancekernel of u is estimated as (starting summation over j andl at 1),

RCðx; yÞ ¼X

i;j

uijN iðxÞwj

! Xk;l

uklN kðyÞwl

!* +

¼XP

j¼1

Xi;k

uijukjN iðxÞNkðyÞ !

hw2j i: ð28Þ

Now the Hilbert–Karhunen–Loeve expansion of uhc;pðx;xÞcan be obtained by solving Eq. (25) using a Galerkin pro-jection scheme on the coarse mesh, which leads to a dis-crete generalized eigenvalue problem whose size isdetermined by the number of degrees of freedom of thecoarse discretization Thc . More specifically, let

uhcðxÞ ¼

Xi

fiN iðxÞ ð29Þ

be the approximation of the eigenfunction us in H 10ðDÞ.

Substituting this approximation into Eq. (25) yields the fol-lowing expression for the error

� ¼X

i

fi

ZD

ryRCðx; yÞ � ryN iðyÞdy � msN iðxÞ� �

: ð30Þ

Requiring the error to be orthogonal to the approximatingspace yields equations of the following form,

ð�;N jÞH10ðDÞ ¼ 0 8j: ð31Þ

Equivalently,

Xi

fi rx

ZD

ryRCðx; yÞ � ryN iðyÞdy� ��

� rxN jðxÞdx�ms

ZD

rxN iðxÞ � rxNjðxÞdx�¼ 0 8j:

ð32Þ

Substituting the first expression of Eq. (28) into Eq. (32)yields the following generalized eigenvalue problem,

QTCQf ¼ mQf; ð33Þ


where,

Qij ¼ZD

rN iðxÞ � rNjðxÞdx; ð34Þ

is the ijth entry of the stiffness matrix corresponding to thecase where the coefficient a of (1) is unit constant over D.Clearly, for the cases where the mean of a is constant overD, the matrix Q is readily available as a factor of the stiff-ness matrix corresponding to the mean of a which is al-ready obtained when the problem is solved on the coarsemesh.

Moreover, C is the covariance matrix of the nodal solu-tion of (1) on Thc , and is given as

Cij ¼XP

k¼1

uikujkhw2ki: ð35Þ

Having {mi} and ff ig, as solutions to the eigenproblem (33),the Hilbert–Karhunen–Loeve and reduced-order represen-tations of uhc;pðx;xÞ are given by

uhc;pðx;xÞ ¼ huhc;pðxÞi þX

i

ffiffiffiffimip

giðxÞuiðxÞ ð36Þ

and,

uhc;pðx;xÞ huhc;pðxÞi þXl

i¼1

ffiffiffiffimip

giðxÞuiðxÞ; ð37Þ

respectively. The upper limit l in the above equation is thesize of the dominant eigenspace of (33) such thatPl

i¼1mi=P

imi is sufficiently close to one.It is important to note that one only needs to solve (37)

for its modes associated with the l largest eigenvalues, thusrequiring an a priori estimate of

Pimi.

Having the above setting, the set of random variablesfgig

li¼1 are re-written as a linear transformation of the set

of polynomial chaos basis fwigPi¼1. In particular, the argu-

ments of the inner product of Eq. (23) are simply replacedby their corresponding discrete approximation on thecoarse mesh, i.e.

ðuhc;pð�;xÞ � huhc;pi;ulÞH10ðDÞ

¼Xi¼1

Xp

j¼1

uijN iwjðxÞ;X

i

fiN i

!H1

0ðDÞ

¼ ffiffiffiffimlp

glðxÞ 8l: ð38Þ

Equivalently,

glðxÞ ¼XP

j¼1

aljwjðxÞ 8l; ð39Þ

where the coefficient alj; l ¼ 1; . . . ; l and j ¼ 1; . . . ; P , is de-fined as

alj :¼ 1ffiffiffiffimlp

Xm;n

ujmQmnfðlÞn : ð40Þ

Here fðlÞn denotes the nth component of the lth eigenvector.It is worth mentioning that each element of the set fgig

li¼1

generally contains up to p-order polynomials in fnigqi¼1.

3.2.1. Adaptive selection of expansion order p

Clearly, one important factor in the above representa-tion is the proper choice of expansion order p whichdirectly bears upon the computational costs of the SFEManalysis. Recently, general guidelines have been proposedfor the selection of a suitable p in the context of a posteriorierror analysis of stochastic Galerkin projections [4,5,12].The aforementioned guidelines apply only to very specificconstructions of stochastic coefficients a and cannot bedirectly applied to the present problem, as defined in Sec-tion 2.1.4. Based on the results of the above coarse analy-sis, a relatively simple, but effective, error analysis is nextintroduced to determine the minimum degree of expansionp to achieve a target accuracy of the approximations on thefine mesh. The algorithm is based on performing successiveapproximations in random dimension starting from loworder approximation and increasing p until the contribu-tion of the pth order approximation falls below a thresholdvalue. In particular, let uðjÞhc ;pðx;xÞ ¼

Pli¼1

ffiffiffiffimip ðaijwjðxÞÞ

uiðxÞ be the contribution of the jth polynomial chaosbasis function in the representation of (37). A measure ofthe magnitude of uðjÞhc;p in H 1

0ðDÞ � L2ðXÞ is defined asfollows:

hj :¼ ðuðjÞhc ;p; uðjÞhc;pÞH1

0ðDÞ�L2ðXÞ ¼

ZDjruðjÞhc;pðx; �Þj

2 dx� �

¼Xl

i¼1

mia2ijhw

2j i: ð41Þ

Where the last equality follows from Eq. (22). Now the rel-ative contribution of each polynomial chaos basis functionis obtained by

Hj ¼hjPPk¼1hk

8j: ð42Þ

The adaptivity procedure is continued until the contribu-tion of bases associated with ðp þ 1Þth order expansionfalls below a target acceptance level. It is important to no-tice that the above adaptive approximations are performedon the coarse mesh rather than the original fine mesh, withthe results being valid for the fine analysis. This is mainlydue to the decoupling of the two sources of error, namelythe spatial discretization error and random discretization

error in the stochastic Galerkin scheme. While the formercomponent of error is related to the partition Th of Dand/or the degree of shape functions fNig, the later hasto do with the order of spectral decomposition of u in Y.In particular,

u� uhc;p ¼ ðu� upÞ þ ðup � uhc ;pÞ; ð43Þ

shows the contribution of these two sources of error in theoverall error of the coarse analysis. The first and second

w=

1.0

X

Y

Thickness, t=1.0Plane strain conditionsPoisson’s ratio, ν=0.15Elastic modules, E=Random field

Edge fixed in YY direction

Edg

e fix

ed in

XX

dire

ctio

n

[email protected]=2.0

2@1.

0=2.

0

a b

Fig. 1. 2D solid mechanics problem: (a) geometry of the problem; (b) a sample realization of the modules of elasticity E.

w w

w w

a b

c d

Fig. 2. Different discretizations of D: (a) Mesh 1 (32 DOF, refers to the coarse discretization); (b) Mesh 2 (82 DOF); (c) Mesh 3 (158 DOF); (d) Mesh 4(474 DOF, refers to the fine discretization).

Table 1Required parameters to define each case study

Case study Mesh lg rg q pa p lx ¼ ly

Case 1 1–4 0.0 0.85 1 8 3, 4 1.0Case 2 1–4 0.0 0.25 8 8 1–4 1.0Case 3 1–4 0.0 0.85 8 8 1–4 1.0


term in the right hand side of the above representation arereferred as random discretization error and spatial dis-

cretization error, respectively. Clearly, following the adap-tive algorithm of selecting p, one is reducing the firstcomponent of error to zero. Accordingly, the significantsource of error in the fine scale analysis is the spatial


discretization error, i.e. the second term in the right handside of

u� uhf ;p ¼ ðu� upÞ þ ðup � uhf ;pÞ: ð44Þ

Also, at any stage of the approximation process, bases withnegligible contribution, can be detected and eliminated ashigher order approximations are sought. This, however,is only valid if certain degree of regularity of solution uexists.

3.3. Fine discretization analysis

The fine discretization analysis aims at solving the varia-tional problem of Section 2.1.3 on the tensor product spaceof X and Y g. Here X :¼ X hf

� H 10ðDÞ is the finite dimen-

sional space of piecewise continuous polynomials, corre-sponding to a fine partition Thf

of D. The randomcomponent of the solution is represented by Y g, the spaceof random variables spanned by the set f1; fgig

li¼1g which

was characterized using the coarse discretization analysis.Therefore, the finite dimensional Galerkin approximationof the exact solution uðx;xÞ on the tensor product spaceX � Y g is obtained by solving

Bðuhf ;p; vÞ ¼LðvÞ; 8v 2 X � Y g: ð45Þ

1 20

0.5

1

Mode

Nor

mal

ized

eig

enva

lue

XY

a

b

Fig. 3. Coarse discretization analysis with different discretization sizes, all whorizontal displacement, Mesh 1; (c) first eigenfunction for vertical displacem

Specifically, let the approximate solution of (1) on Thfbe

written as

uhf ;pðx;xÞ ¼X

i;j

uijN iðxÞgjðxÞ; ð46Þ

with g0ðxÞ ¼ 1, and use the test function vklðx;xÞ ¼NkðxÞglðxÞ to find the coefficients uij. Then (46) gives thefollowing system of linear equations for uij on ThfX

i;j

hglgjðað�; �ÞrN i;rNkÞL2ðDÞiuij

¼ hglðf ð�Þ;N kÞL2ðDÞi 8k; l: ð47Þ

Clearly, in the process of calculating the coefficients of theabove algebraic system of equations, one needs to calculatethe expectations of the typical triple product of the formdijk :¼ hwigjgki in which the first component is due toexpansion of the stochastic function aðx;xÞ described inSection 2.1.4. For computational purposes, dijk can be sim-ply recast as

dijk ¼ wi

XP

m¼1

ajmwm

! XP

n¼1

aknwn

!* +

¼XP

m¼1

XP

n¼1

ajmaknhwiwmwni ¼XP

m¼1

XP

n¼1

ajmakncimn; ð48Þ

3 4 number

Mesh 4 Mesh 3 Mesh 2 Mesh 1

XY

c

ith p = 4, q = 1: (a) normalized eigenvalues; (b) first eigenfunction forent, Mesh 1.


where the coefficients cimn :¼ wiwmwn can be readily com-puted [21]. Recall that coefficients ajm and akn in the aboverepresentation were calculated in Eq. (40) based on thecoarse mesh analysis of Section 3.2.

Remark 1. In order for random variables {gi}, associatedwith dominant eigenmodes of the Hilbert–Karhunen–Loeve expansion of (26), to be well approximated in thecoarse scale analysis, the corresponding projection opera-tor on the eigenspace must be well approximated, withoutnecessarily having the eigenvectors themselves well approx-imated. Such coarse discretization should be identifiedthrough adaptivity in the spatial domain of the problemand, as will be shown in the numerical analysis, is far frombeing fine enough to approximate u.

4. Numerical examples

In this section, a two-dimensional example from solidmechanics is chosen to illustrate different aspects of theproposed algorithm. Fig. 1 depicts the L-shaped geometryof the problem subjected to a unit pressure applied on theleft edge. The Dirichlet boundary conditions are imposedby setting the y-displacement of the upper edge as well asthe x-displacements of the far right edge to zero. The only

1 2 30

2

4

6

8

10

12

14

PC Index

Rel

ativ

e er

ror

(%)

Mesh 4Mesh 3Mesh 2Mesh 1

1 2 3 40

5

10

15

20

25

PC Index

Rel

ativ

e er

ror

(%)


a

c

Fig. 4. Accuracy of the algorithm in estimating optimal basis with respect to acorresponding to meshes 1–4 for the case of p = 3; (b) aij corresponding tcorresponding to meshes 1–4 for the case of p = 4; (d) aij corresponding to m

source of randomness is the elastic modulus of the platewhich is modeled as a truncated lognormal random fieldas described in Section 2.1.4.

More specifically, the underlying Gaussian process g isfirst represented by a q-term Karhunen–Loeve expansionof a Gaussian process with constant mean lg and covari-

ance kernel Rggðx; yÞ ¼ r2g expf�jx1�y1j

lxþ �jx2�y2j

lyg defined on

Ds ¼ ½�1; 1� � ½�1; 1�, where rg, lx, and ly are the standarddeviation, correlation length in x-direction, and correlationlength in y-direction of the process g, respectively. Havinga closed-form representation of the eigenvalues and eigen-functions associated with this particular kernel [38], theGaussian process g is obtained on the L-shape domain Dsimply by taking the required values of g on Ds. Thereforethe spatial functions fgig of Eq. (18) defined on D areextracted from the corresponding functions fgig on Ds.Alternatively one may start by representing g directly onD and solving Eq. (16) numerically [21]. Clearly, for thepresent example, the choice of the domain of definitionof a is a modeling decision and does not affect the general-ity of the algorithm. In particular the objective of thisexample is to investigate the accuracy of the proposedreduced order analysis compare to the original model. Itcan be expected that the results from the proposed analysiswill converge in some suitable sense to those of the original

1 2 3–0.2

0

0.2

0.4

0.6

0.8

1

1.2

PC Index

α ij

Mesh 5Mesh 4Mesh 3Mesh 2Mesh 1

1 2 3 4–0.2

0

0.2

0.4

0.6

0.8

1

1.2

PC Index

α ij


b

d

relatively fine mesh (Mesh 5, 930 DOF): (a) relative errors in estimating aij

o meshes 1–4 for the case of p = 3; (c) relative errors in estimating aij

eshes 1–4 for the case of p = 4.


one as the dominant subspace characterizing the solution isbetter approximated. This can be achieved by considering asequence of meshes with different discretization levels, asshown in Fig. 2. The coarse discretization analysis of Sec-tion 3.2 is performed on Meshes 1–4 and the fine discretiza-

tion analysis of (3.3) is done on Mesh 4.In what follows, three different case studies are exam-

ined in order to clarify different aspects of the algorithm.The parameters of each case are defined in Table 1.

–1000 0 1000–2

–1.5

–1

–0.5

0

0.5

1

1.5

2x 10

7

ξ

η 1


–1000 0 1000–20

–15

–10

–5

0

5x 10

8

ξ

η 1


a b

c d

Fig. 5. Optimal basis for q = 1: (a) shape of g1ðnÞ, p = 3; (b) pd

1000 0 1000–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

ξ

PC

bas

is

ψ1/103

ψ2/106

ψ3/109

ψ4/1012

a

Fig. 6. PC basis for q = 1: (a) shape

4.1. Case 1

The objective of this case is to compare the robustness ofthe algorithm in characterizing the dominant subspace ofthe solution on different discretization levels. Taking thenumber of effective coordinates of the probability spaceq = 1, enables one to plot the optimal basis g and comparethem with the PC bases. While the shapes of the optimalbasis functions are expected to be different when the coarse

–6 –5 –4 –3 –2 –1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

η1

pdf η


–6 –5 –4 –3 –2 –1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

η1

pdf η


f of g1, p = 3; (c) shape of g1ðnÞ, p = 4; (d) pdf of g1, p = 4.

5 0 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ψi

pdf ψ

i

ψ1

ψ2

ψ3

ψ4

b

of PC basis; (b) pdf of PC basis.


analyses are performed on different orders of PC expan-sion, the analysis of this section shows that in fact the sta-tistics, and more specifically the probability distributionfunctions, of the basis are invariant with respect to theorder of PC expansion. This is partially due to the fact that,armed with a certain regularity of the solution, with respectto stochastic parameters, the dominant subspace on whichthe solution is defined can be characterized by low orderhomogeneous chaos. In order to test this hypothesis, theanalyses are repeated for p = 3 and p = 4. The coarse dis-cretization analysis of Fig. 3 shows that there is only onerandom variable, g1, that needs to be characterized.

As seen from plots (b) and (d) of Fig. 4, the first and sec-ond order polynomial chaos provide the main contributionin representing the optimal basis g1 and, as depicted inplots (a) and (c), the associated coefficients aij are estimatedwith a good accuracy on the coarsest mesh (Mesh 1). Moreclearly, the absolute relative error in estimating aij coeffi-cients decreases to zero as the coarse mesh gets refinedtoward the original fine mesh. Fig. 5 shows the shapeand the probability distribution of the optimal basis g1ðnÞobtained from four different meshes. Again it is observedthat the space on which principal components of thesolution exist is well approximated by a relatively coarsediscretization of the spatial domain. For the sake of com-

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

a

c

Fig. 7. Adaptive selection of expansion order p, rg ¼ 0:

parison, the PC basis along with their probability distribu-tion functions are plotted in Fig. 6.

4.2. Cases 2 and 3

In this part, the accuracy of the reduced order model inapproximating the response of the system is explored. Therandom dimension of the problem is taken to be 8 (q = 8).Case 2 and Case 3 refer to situations where the standarddeviation of the Gaussian process g is rg ¼ 0:85 andrg ¼ 0:25, respectively, thus requiring different order ofPC approximation for solution u. The required order ofapproximation p is detected by the adaptive algorithmdescribed in Section 3.2.1 with Ha ¼ 10�4 as the acceptancelimit for Hj of Eq. (42). Moreover, the basis functions ofthe pth order homogeneous chaos whose relative contribu-tion Hj is less than Ha are omitted from the fine discretiza-tion part of the algorithm. In order to investigate theaccuracy of the reduced order analysis, jjujjL2ðDÞ�L2ðXÞ aswell as the probability distribution of the horizontal dis-placement of the node located at ð0:0; 1:0Þ are taken asthe quantities of interest to be compared with those ofthe full order analysis. It is worth noting that the coeffi-cients of variation of the solution process over the spatialdomain vary between 16% �18% for Case 2 and between

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

b

d

25: (a) Mesh 1; (b) Mesh 2; (c) Mesh 3; (d) Mesh 4.


60% and 65% for Case 3. Also the variation of error asso-ciated with approximating each quantity of interest withrespect to different discretization levels is probed.

Figs. 7 and 8 show the adaptive selection of p in cases 2and 3. Based on the chosen value of Ha, indicated by thedotted horizontal line, the plots suggest that a secondand third order representation is sufficient to approximatethe response u for cases 2 and 3, respectively. From Figs. 7and 8 it is noted that the order selection procedure isinsensitive to the discretization size, thus confirming the

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

a

c

Fig. 8. Adaptive selection of expansion order p, rg ¼ 0:

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mode number

Nor

mal

ized

eig

enva

lue


a b

Fig. 9. Decay of the normalized eigenvalues mi for different disc

decoupling argument of Section 3.2.1. The basis functionswhose corresponding H falls above Ha are then selectedfor the reduced order representation on the fine mesh.The number of dominant modes in the Hilbert–Karh-unen–Loeve expansion of uhc;p is dictated by l for whichPl

i¼1mi=P

mi P 0:99 and is found to be l = 4 for bothcases as depicted in Fig. 9. Therefore, only 5 basis func-tions, including the zeroth order, are needed for the finescale analysis, which is significantly smaller than the stan-dard choice of PC basis functions with P = 45, for Case

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

100

101

102

10–8

10–6

10–4

10–2

100

PC Index

Θ

p=1 (P=8)p=2 (P=44)p=3 (P=164)

b

d

85: (a) Mesh 1; (b) Mesh 2; (c) Mesh 3; (d) Mesh 4.

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mode number

Nor

mal

ized

eig

enva

lue


retization sizes: (a) Case 2, rg ¼ 0:25; (b) Case 3, rg ¼ 0:85.

101

102

103

102

Number of nodes (log scale)

RE

(lo

g sc

ale)

Case 2Case 3

1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Mode number (i)

⟨(η i(f

) –η i(c

) )2 ⟩

Mesh 3Mesh 2Mesh 1

1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Mode number (i)

⟨(η i(f

) –ηi(c

) )2 ⟩

Mesh 3Mesh 2Mesh 1

a b

c

Fig. 10. Variation of the relative error, RE, REg with different discretization sizes. (a) RE; (b) variance of the error in estimating gi;REg, Case 2; (c)variance of the error in estimating gi;REg, Case 3; Superscripts (c) and (f) refer to the analysis on Mesh 1 (2 or 3) and Mesh 4, respectively.

Fig. 11. Relative error in mean of horizontal displacement: (a) Case 2, rg ¼ 0:25; (b) Case 3, rg ¼ 0:85.


2, and P = 165, for Case 3, respectively (Figs. 10–13). Asshown in Fig. 10, for each coarse discretization level, theaccuracy of estimating the set fgig

4i¼1 corresponding to

the fine mesh from the analysis of coarse mesh can beexamined by computing the variance of the error, REg,between these two cases.

Having the response of the system for both the reducedorder and the full order analyses, it is possible to compare

the norms of the error between these two cases. The follow-ing relative global error, RE, is selected to investigate thevariation of such error with respect to different discretiza-tion sizes, Fig. 10

RE :¼

RDjurm

hf ;pðx; �Þ � ufm

hf ;pðx; �Þj2 dx

D ERDjufm

hf ;pðx; �Þj2dx

D E : ð49Þ

Fig. 12. Relative error in variance of horizontal displacement: (a) Case 2, rg ¼ 0:25; (b) Case 3, rg ¼ 0:85.

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

Horizontal displacement u(0.0,1.0)

pdf o

f u(0

.0,1

.0)

Full model analysisReduced model, Mesh 1Reduced model, Mesh 2Reduced model, Mesh 3Reduced model, Mesh 4

15 16 17 18 19 200

1

2

3x 10

–3

Horizontal displacement, u(0.0,1.0)

pdf o

f u(0

.0,1

.0),

rig

ht ta

il


10 0 10 20 30 40 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1


pdf o

f u(0

.0,1

.0)


30 35 40 450

0.5

1

1.5

2x 10

–3


pdf o

f u(0

.0,1

.0),

rig

ht ta

il


a b

c d

Fig. 13. Probability distribution function (pdf) of horizontal displacement of point ðx1; x2Þ ¼ ð0:0; 1:0Þ: (a) Case 2, rg ¼ 0:25; (b) Case 2, rg ¼ 0:25, righttail of pdf; (c) Case 3, rg ¼ 0:85; (d) Case 3, rg ¼ 0:85, right tail of pdf.


where urmhf ;p

and ufmhf ;p

refer to the solution of the reducedorder analysis, as mentioned in Section 3.3, and the fullorder analysis, respectively.

5. Conclusion

A model reduction technique for chaos representationsof solution of SPDE’s is proposed which enables one totackle the curse of dimensionality through adaptation.The algorithm is based on a dual scale analysis of the prob-

lem. In the coarse scale analysis, based on a Hilbert–Karh-unen–Loeve decomposition, the low dimensional subspaceon which a good approximation of the response of theSPDE exists is identified. This lower dimensional manifoldis then used in the fine scale analysis of the algorithm. Thenumber of basis functions characterizing the randomdimension of the fine scale analysis is considerably less thanthat of the full model analysis (5 compare to 45 and 165 forCases 2 and 3, respectively). The proposed efficient algo-rithm requires a full order analysis as well as the solution


of a generalized eigenvalue problem, both on a coarsediscretization whose computational costs are considerablyless than the solution of the full order model on the finediscretization. An important feature of the algorithm thatcan be used to further reduce the computational cost, isthe fact that all unnecessary bases in the PC expansioncan be detected and eliminated from further analysis. Theaccuracy and convergence of the proposed algorithm isshown through an example from solid mechanics. In addi-tion to its value in enhancing the computational efficiencyof stochastic projection method, the proposed algorithmcan also be readily used in an adaptive approximationcontext.

Acknowledgements

This work was supported by the ONR, NSF and SandiaNational Laboratories.

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Stochastic model reduction for chaos representations