Babuška - 2010 - A stochastic collocation method for elliptic partial differential equations with random input data.pdf

8/11/2019 Babuka - 2010 - A stochastic collocation method for elliptic partial differential equations with random input data.pdf

1/38


2/38

a posteriori error estimation [1, 3, 37], mesh adaptivity and the more recentmodeling error analysis [30, 31, 10]. All this has increased the accuracy ofnumerical predictions as well as our confidence in them.

Yet, many engineering applications are affected by a relatively large amountof uncertainty in the input data such as model coefficients, forcing terms, bound-ary conditions, geometry, etc. In this case, to obtain a reliable numerical pre-diction, one has to include uncertainty quantification due to the uncertainty inthe input data.

Uncertainty can be described in several ways, depending on the amountof information available; among others we mention: the worst case scenarioanalysis and fuzzy set theory, evidence theory, probabilistic setting, etc (see[6, 24] and the references therein).In this paper we focus on elliptic partialdifferential equations with a probabilistic description of the uncertainty in theinput data. The model problem has the form

L(a)u= f in D (1)

whereL is an elliptic operator in a domain D Rd, which depends on somecoefficients a(x, ), with x D, , and indicates the set of possibleoutcomes. Similarly, the forcing term f = f(x, ) can be assumed random aswell.

We will focus on the case where the probability space has a low dimension-

ality, that means, the stochastic problem depends only on a relatively smallnumber of random variables.This can be the case if, for instance, the mathematical model depends on few

parameters, which can be taken as random variables with a given joint proba-bility distribution. To make an example we might think at the deformation ofan elastic homogeneous material in which the Youngs modulus and the Pois-sons ratio (parameters that characterize the material properties), are randomvariables, either independent or not.

In other situations, the input data may vary randomly from one point of thephysical domainD to another and their uncertainty should rather be describedin terms of random fields with a given covariance structure (i.e. each point ofthe domain is a random variable and the correlation between two distinct pointsin the domain is known and non zero, in general; this case is sometimes referred

to as colored noise ).Examples of this situation are, for instance, the deformation of inhomoge-

neous materials such as wood, foams, or bio-materials such arteries, bones, etc.;groundwater flow problems where the permeability in each layer of sediments(rocks, sand, etc) should not be assumed constant; the action of wind (directionand point intensity) on structures; etc.

A possible way to describe such random fields consists in using a Karhunen-Loeve[27] or a Polynomial Chaos (PC) expansion [38,42]. The former representsthe random field as a linear combination of an infinite number of uncorrelatedrandom variables, while the latter uses polynomial expansions in terms of in-dependent random variables. Both expansions exist provided the random field

2


3/38

a: V, as a mapping from the probability space into a functional space V,has bounded second moment. Other non-linear expansions can be consideredas well (see e.g. [22]for a technique to express a stationary random field withgiven covariance structure and marginal distribution as a function of (infinite)independent random variables; non-linear transformations have been used alsoin [29, 39]). The use of non-polynomial expansions may be advantageous insome situations: for instance, in groundwater flow problems, the permeabilitycoefficient within each layer of sediments can feature huge variability, which isoften expressed in a logarithm scale. In this case, one might want to use aKarhunen-Loeve (or Polynomial Chaos) expansion for the logarithm of the per-meability, instead of the permeability field itself. This leads to an exponentialdependence of the permeability on the random variables and the resulting ran-dom field might even have unbounded second moments. An advantage of such anon-linear expansion is that it guarantees a positive permeability almost surely(a condition which is difficult to enforce, instead, with a standard truncatedKarhunen-Loeve or PC expansion).

Although such random fields are properly described only by means of aninfinite number of random variables, whenever the input data vary slowly inspace, with a correlation length comparable to the size of the domain, only fewterms in the above mentioned expansions are typically enough to describe therandom field with sufficient accuracy. Therefore, for this type of applications itis reasonable to limit the analysis to just a few number of random variables in

the expansion (see e.g. [2]).This argument is also strengthened by the fact that the amount of measured

data at ones disposal to identify the input data as random fields is in generalvery limited and barely sufficient to identify the first few random variables inthe expansion.

Conversely, situations in which the random fields are highly oscillatory witha short correlation length, as in the case of materials with a random microstruc-ture, do not fall in this category and will not be considered in the present work.The interested reader should refer, instead, to the wide literature in homoge-nization and multiscale analysis (see e.g. [16] and references therein).

To solve numerically the stochastic partial differential equation (1), a rela-tively new numerical technique, which has gained much attention in the last fewyears, is the so called Spectral Galerkin approximation (see e.g. [21]) . It em-ploys standard approximations in space (finite elements, finite volumes, spectralor h-p finite elements, etc. ) and polynomial approximation in the probabilitydomain, either by full polynomial spaces [41,29,20], tensor product polynomialspaces [4,18] or piecewise polynomial spaces [4,26].

The use of tensor product spaces is particularly attractive in the case of asmall number of random variables, since it allows naturally the use of anisotropicspaces where the polynomial degree is chosen differently in each direction inprobability. Moreover, whenever the random fields are expanded in a truncatedKarhunen-Loeve expansion and the underlying random variables are assumedindependent, a particular choice of the basis for the tensor product space (asproposed in [4, 5]), leads to the solution of uncoupled deterministic problems

3


4/38


5/38


6/38

Lemma 1 Under assumptionsA1andA2, problem(3)admits a unique solutionuVP,a, which satisfies the estimate

uP CPamin

D

E[f2] dx

12

. (4)

In the previous Lemma we have used the Poincare inequality

wL2(D)CPwL2(D) wH10 (D).

1.0.1 Weaker Assumptions on the random coefficients

It is possible to relax the assumptions A1 and A2substantially and still guaran-tee the existence and uniqueness of the solution u to problem (3). In particular,if the lower bound for the coefficient a is no longer a constant but a randomvariable, i.e. a(x, ) amin() > 0 a.s. a.e. on D, we have the followingestimate for the moments of the solution:

Lemma 2 (Moments estimates) Let p, q 0 with 1/p+ 1/q = 1. Take apositive integer k. Then if f LkpP (; L2(D)) and 1/amin LkqP() we havethatuLkP(; H10 (D)).Proof. Since

u(, )H10 (D)CPf(, )L2(D)amin() a.s.

the result is a direct application of Holders inequality:

u(, )kH10 (D)dP()CkP

f(, )L2(D)amin()

kdP()

CkP

f(, )kpL2(D)dP()1/p

1

amin()

qkdP()

1/q

Example 1 (Lognormal diffusion coefficient) As an application of the pre-vious lemma, we can conclude the well posedness of (3). with a lognormal dif-fusion coefficient. For instance, let

a(x, ) = exp

Nn=1

bn(x)Yn()

, YnN(0, 1) iid.

Use the lower bound

amin() = exp

Nn=1

bnL(D)|Yn()|

6


7/38

and then fork,q amin almost surely. On the other hand, the right hand side of (2) can berepresented as a truncated Karhunen-Loeve expansion

f(, x) = c0(x) +

1nN

ncn(x)Yn().

Remark 1 It is usual to have f and a to be independent, because the loadsand the material properties are seldom related. In such a situation we havea(Y(), x) = a(Ya(), x) and f(Y(), x) = f(Yf(), x), withY = [Ya, Yf] andthe vectorsYa, Yf independent.

7


8/38

After making Assumption 1, we have by DoobDynkins lemma (cf. [32]),that the solution u of the stochastic elliptic boundary value problem (3) can bedescribed by just a finite number of random variables, i.e. u(, x) =u(Y1(), . . . , Y N(), x).Thus, the goal is to approximate the function u = u(y, x), where y andx D. Observe that the stochastic variational formulation (3) has a deter-ministic equivalent which is the following: find uV,a such that

(au, v)L2(D)dy =

(f, v)L2(D)dy , vV,a (7)

noting that here and later in this work the gradient notation,, always meansdifferentiation with respect to x D only, unless otherwise stated. The spaceV,a is the analogue of VP,a with (, F, P) replaced with (, BN,dy). Thestochastic boundary value problem (2) now becomes a deterministic Dirich-let boundary value problem for an elliptic partial differential equation with anNdimensional parameter. For convenience, we consider the solution u as afunction u : H10 (D) and we use the notation u(y) whenever we want tohighlight the dependence on the parameter y . We use similar notations for thecoefficienta and the forcing termf. Then, it can be shown that problem ( 2) isequivalent to

D a(y)u(y) dx= D f(y)dx, H10 (D), -a.e. in . (8)

For our convenience, we will suppose that the coefficient a and the forcing termf admit a smooth extension on the -zero measure sets. Then, equation (8)can be extended a.e. in with respect to the Lebesgue measure (instead of themeasure).

Hence, making Assumption1is a crucial step, turning the original stochasticelliptic equation into a deterministic parametric elliptic one and allowing the useof finite element and finite difference techniques to approximate the solution ofthe resulting deterministic problem (cf. [25,13]).

Remark 2 Strictly speaking, equation (8) will hold only for those values ofy for which the coefficienta(y) is finite. In this paper we will assume thata(y) may go to infinity only at the boundary of the parameter domain.

2 Collocation method

We seek a numerical approximation to the exact solution of (7) in a finitedimensional subspace Vp,h based on a tensor product, Vp,h =Pp() Hh(D),where

Hh(D) H10 (D) is a standard finite element space of dimension Nh,which contains continuous piecewise polynomials defined on regular trian-gulationsTh that have a maximum mesh spacing parameter h >0.

8


9/38

Pp()L2() is the span of tensor product polynomials with degree atmostp = (p1, . . . , pN) i.e.Pp() =

Nn=1Ppn(n), with

Ppn(n) = span(ymn, m= 0, . . . , pn), n= 1, . . . , N .

Hence the dimension ofPp is Np =N

n=1(pn+ 1).

We first introduce the semi-discrete approximation uh : Hh(D), ob-tained by projecting equation (8) onto the subspace Hh(D), for each y, i.e.

D

a(y)uh(y) h dx=D

f(y)h dx, hHh(D), for a.e. y. (9)

The next step consists in collocating equation (9) on the zeros of orthogonalpolynomials and build the discrete solution uh,p Pp() Hh(D) by interpo-lating in y the collocated solutions.

To this end, we first introduce an auxiliary probability density function : R+ that can be seen as the joint probability of N independent randomvariables, i.e. it factorizes as

(y) =N

n=1n(yn), y, and is such that

L()


10/38


11/38

basis is then obtained by solving the following eigenvalue problems, for eachn= 1, . . . , N ,

n

zkn(z)v(z)n(z) dz = ckn

n

kn(z)v(z)n(z) dz, k= 1, . . . , pn+ 1.

The eigenvectors kn are normalized so as to satisfy the propertyn

kn(z)jn(z)n(z) dz = kj ,

n

zkn(z)jn(z)n(z) dz = cknkj .

See [4, 5]for further details on the double orthogonal basis.

We aim at analyzing, now, the analogies between the Collocation and theSpectral Galerkin methods. The Collocation method can be seen as a Pseudo-Spectral Galerkin method (see e.g. [33]) where the integrals over in (12) arereplaced by the quadrature formula (11): finduh,p Pp() Hh(D)such that

Ep

(auh,p, v)L2(D)

= Ep

(f, v)L2(D)

, v Pp()Hh(D). (13)

Indeed, by choosing in (13), the test functions of the form v(y, x) =lk(y)(x),where(x)

H

h(D) and l

k(y) is the Lagrange basis function associated to the

knot yk, k = 1, . . . , N p, one is led to solve a sequence of uncoupled problemsof the form (9) collocated in the points yk, which, ultimately, gives the samesolution as the Collocation method.

In the particular case where the diffusivity coefficient and the forcing termare multi-linear combinations of the random variables Yn(), and the randomvariables are independent, it turns out that the quadrature formula is exactif one chooses = . In this case, the solution obtained by the Collocationmethod actually coincides with the Spectral Galerkin one. This can be seeneasily observing that, with the above assumptions, the integrand in (12), i.e.(auh,p v) is a polynomial at most of degree 2pn+ 1 in the variable yn andthe Gauss quadrature formula is exact for polynomials up to degree 2pn + 1integrated against the weight .

The Collocation method is a natural generalization of the Spectral Galerkinapproach, and has the following advantages:

decouples the system of linear equations in Y also in the case where thediffusivity coefficienta and the forcing term fare non linear functions ofthe random variables Yn;

treats efficiently the case of non independent random variables with theintroduction of the auxiliary measure ;

can easily deal with random variables with unbounded support (see The-orem1 in Section4).

11


12/38

As it will be shown in Section4, the Collocation method preserves the sameaccuracy as the Spectral Galerkin approach and achieves exponential conver-gence if the coefficient a and forcing term fare infinitely differentiable withrespect to the random variablesYn, under very mild requirements on the growthof their derivatives in Y.

As a final remark, we show that the double orthogonal polynomials proposedin[4]coincide with the Lagrange basislk(y) and the eigenvaluesckn are nothingbut the Gauss knots of integration.

Lemma 3 Let R, : R a positive weight and{k}p+1k=1 the set ofdouble orthogonal polynomials of degreep satisfying

k(y)j(y)(y) dy = kj ,

yk(y)j(y)(y) dy = ckkj .

Then, the eigenvaluesck are the nodes of the Gaussian quadrature formula asso-ciated to the weight and the eigenfunctionsk are, up to multiplicative factors,the corresponding Lagrange polynomials build on the nodesck.

Proof. We have, for k = 1, . . . , p + 1,

(y ck)k(y)v(y)(y)dy = 0, vPp().Take v =

p+1j=1

j=k(y cj)Pp() in the above and let w =

p+1j=1(y cj). Then

w(y)k(y)(y)dy = 0, k= 1, . . . , p + 1.

Since{k}p+1k=1 defines a basis of the spacePp(), the previous relation impliesthat w is -orthogonal to Pp(). Besides, the functions (yck)k are alsoorthogonal to the same subspace: this yields, due to the one dimensional natureof the orthogonal complement ofPp() over Pp+1(),

(y ck)k =kw= kp+1

j=1(y cj), k= 1, . . . , p + 1

which gives

k =k

p+1j=1

j=k

(y cj), k= 1, . . . , p + 1

i.e. the double orthogonal polynomials k are collinear to Lagrange interpolantsat the nodes cj. Moreover, the eigenvalues cj are the roots of the polynomialw Pp+1(), which is -orthogonal to Pp() and therefore they coincide withthe nodes of the Gaussian quadrature formula associated with the weight .

12


13/38

3 Regularity results

Before going through the convergence analysis of the method, we need to statesome regularity assumptions on the data of the problem and consequent regu-larity results for the exact solution u and the semi-discrete solution uh.

In what follows we will need some restrictive assumptions on f and . Inparticular, we will assume fto be a continuous function in y , whose growth atinfinity, whenever the domain is unbounded, is at most exponential. At thesame time we will assume that behaves as a Gaussian weight at infinity, andso does the auxiliary density , in light of assumption (10).

Other types of growth offat infinity and corresponding decay of the prob-ability density , for instance of exponential type, could be considered as well.Yet, we will limit the analysis to the aforementioned case.

To make precise these assumptions, we introduce a weight (y) =N

n=1 n(yn)1 where

n(yn) =

1 if n is bounded

en|yn|, for some n > 0 if n is unbounded(14)

and the functional space

C0(; V) {v: V, v continuous in y, maxy

(y)v(y)V

0 andn strictly positive ifn is unbounded andzero otherwise.

The parameter n in (15) gives a scale for the decay of at infinity andprovides an estimate of the dispersion of the random variableYn. On the otherhand, the parameter n in (14) controls the growth of the forcing term f atinfinity.

Remark 3 (growth off) The convergence result given in Theorem1, in thenext section, extends to a wider class of functionsf. For instance we could take

f C0(; L2(D)) with = eP

Nn=1(nyn)

2/8. Yet, the class given in (14) isalready large enough for most practical applications (see Example2).

13


14/38

We can now choose any suitable auxiliary density (y) =N

n=1n(yn) thatsatisfies, for each n = 1, . . . , N

Cnmine(nyn)2 n(yn)< Cnmaxe(nyn)

2

, ynn (16)for some positive constants Cnmin and C

nmax that do not depend on yn.

Observe that this choice satisfies the requirement given in (10), i.e./L()C/Cmin with Cmin =

Nn=1 C

nmin.

Under the above assumptions, the following inclusions hold true

C0(; V)L

2(; V)L

2(; V)

with continuous embedding. Indeed, on one hand we have

vL2(;V)12

L()

vL2(;V)

CCmin

vL2(;V).

On the other hand,

v2L2(;V) =

(y)v(y)2Vdy v2C0(;V)

(y)

2(y)dy

Nn=1

Mnv2C0(;V)

withMn = n n/2n. Now, for n bounded, MnCnmax

|n

|, whereas if n is

unbounded

Mn =

n

e

(ny)2

2 +2n|y|

e(ny)

2

2 n(y) dyCnmax

2

ne2(n/n)

2

.

The first result we need is the following

Lemma 4 If f C0(; L2(D)) and a C0loc(; L(D)), uniformly boundedaway from zero, then the solution to problem (8) satisfiesuC0(; H10 (D)).The proof of this Lemma is immediate. The next result concerns the analyticityof the solutionu whenever the diffusivity coefficienta and the forcing termfareinfinitely differentiable w.r.t. y, under mild assumptions on the growth of their

derivatives in y. We will perform a one-dimensional analysis in each directionyn, n= 1, . . . , N . For this, we introduce the following notation: n =

Nj=1

j=nj ,

yn will denote an arbitrary element of n. Similarly, we set

n =

Nj=1

j=nj and

n =N

j=1

j=nj .

Lemma 5 Under the assumption that, for everyy = (yn, yn), there existsn < + such that

kyna(y)

a(y)

L(D)

knk! andkynf(y)L2(D)1 + f(y)L2(D)

knk!, (17)

14


15/38

the solutionu(yn, yn, x) as a function ofyn, u: n C0n(n; H10 (D)) admits

an analytic extensionu(z, yn, x), z C, in the region of the complex plane(n; n) {z C, dist(z, n)n}. (18)

with0< n < 1/(2n). Moreover,z(n; n),

n(Re z) u(z)C0n

(n;H10(D))

CPenn

amin(1 2nn) (2fC0(;H10(D))

+ 1) (19)

with the constantCp as in (4).

Proof. In every point y , the k-th derivative of u w.r.t yn satisfies theproblem

B(y; kynu, v) =k

l=1

kl

lynB(y;

klyn

u, v) + (kynf, v), vH10 (D),

whereB is the parametric bilinear form B (y; u, v) =D

a(y)u v dx. Hence

a(y)kynuL2(D)k

l=1

kl

lyna(y)

a(y)

L(D)

a(y)klyn uL2(D)+ Cp

aminkynfL2(D).

SettingRk =a(y)kynuL2(D)/k! and using the bounds on the derivativesofa and f, we obtain the recursive inequalityRk

kl=1

lnRkl+ Cp

aminkn(1 + fL2(D)).

The generic term Rk admits the bound

Rk(2n)kR0+ Cpamin

(1 + fL2(D))knk1l=0

2l.

Observing, now that R0=

a(y)u(y)L2(D) Cpaminf(y)L2(D) and

kynuL2(D)k!

Rkamin

,

we get the final estimate on the growth of the derivatives ofu

kynu(y)L2(D)k!

Cpamin

(2f(y)L2(D)+ 1)(2n)k.

We now define for every ynn the power series u : C C0n(n, H10 (D)) as

u(z, yn, x) =

k=0

(z yn)kk!

kynu(yn, yn, x).

15


16/38

Hence,

n(yn)u(z)C0n

(n,H10 (D))

k=0

|z yn|kk!

n(yn)kynu(yn)C0n(n;H10 (D))

CPamin

maxynn

n(yn)

2f(yn)C0

n(n;L

2(D))+ 1

k=0

(|z yn|2n)k

CPamin

(2fC0(;L2(D))+ 1)

k=0(|z yn|2n)k

where we have exploited the fact that n(yn) 1 for all yn n; the seriesconverges for all z Csuch that|z yn| n < 1/(2n). Moreover, in the ball|z yn| n, we have, by virtue of (14),n(Re z)ennn(yn) and then

n(Re z)u(z)C0n

(n,H10(D))

CPenn

amin(1 2nn) (2fC0(;L

2(D))+ 1)

The power series converges for every yn n, hence, by a continuation argu-ment, the functionu can be extended analytically on the whole region (n; n)and estimate (19) follows.

Example 3 Let us consider the case where the diffusivity coefficient a is ex-panded in a linear truncated Karhunen-Loeve expansion

a(, x) = b0(x) +N

n=1

nbn(x)Yn(),

provided that such expansion guaranteesa(, x)amin for almost everyandxD [34]. In this case we have

kyna

a

L(D)

nbnL(D)/amin fork= 10 fork >1

and we can safely taken = nbnL(D)/amin in (17).If we consider, instead, a truncated exponential expansion

a(, x) = amin+ eb0(x)+

PNn=1

nbn(x)Yn()

we have kyn

a

a

L(D)

nbnL(D)k

and we can taken =

nbnL(D). Hence, both choices fulfill the assumptionin Lemma5.

16


17/38

Example 4 Similarly to the previous case, let us consider a forcing termf ofthe form

f(, x) =c0(x) +N

n=1

cn(x)Yn()

where the random variablesYnare Gaussian (either independent or not), and thefunctionscn(x)are square integrable for anyn = 1, . . . , N . Then, the functionfbelongs to the spaceC0(; L

2(D)), with weight defined in (14), for any choiceof the exponent coefficientsn > 0.

Moreover,

kynf(y)L2(D)1 + f(y)L2(D)

cnL2(D) fork= 1

0 fork >1

and we can safely take n =cnL2(D) in (17). Hence, such a forcing termsatisfies the assumptions of Lemma 5. In this case, though, the solution u islinear with respect to the random variablesYn (hence, clearly analytic) and ourtheory is not needed.

Observe that the regularity results are valid also for the semidiscrete solutionuh, as well.

4 Convergence analysis

Our aim is to give a priori estimates for the total error = uuh,pin the naturalnormL2() H10 (D). The next Theorem states the sought convergence result,and the rest of the section will be devoted to its proof. In particular, we willprove that the error decays (sub)exponentially fast w.r.t. p under the regularityassumptions made in Section 3. The convergence w.r.t. h will be dictated bystandard approximability properties of the finite element space Hh(D) and theregularity in space of the solution u (see e.g. [12,11]).

Theorem 1 Under the assumptions of Lemmas 4 and 5, there exist positiveconstantsrn, n= 1, . . . , N , andC, independent ofh andp, such that

u uh,pL2H10 1amin

infv

L2

HhD a|(u v)|2

12

+ CN

n=1

n(pn)exp{rnpnn}(20)

where

ifn is bounded

n = n = 1

rn = log2n|n|

1 +

1 + |n|

2

42n

ifn is unbounded

n = 1/2, n = O(

pn)

rn = nn

17


18/38

n is the distance betweenn and the nearest singularity in the complex plane,as defined in Lemma5, andn is defined as in (15).

The first term on the right hand side of (20) concerns the space approxima-bility of u in the subspace Hh(D) and is controlled by the mesh size h. Theactual rate of convergence will depend on the regularity in space of a(y) andf(y) for each y as well as on the smoothness on the domain D. Observethat anh orh padaptive strategy to reduce the error in space is not precludedby this approach.

The exponential rate of convergence in the Ydirection depends on the con-

stants rn, which on their turn, are related to distances n to the nearest sin-gularity in the complex plane. In Examples 3 and 4 we have estimated theseconstants in the case where the random fields a and fare represented by ei-ther a linear or exponential truncated Karhunen-Loeve expansion. Hence, a fullcharacterization of the convergence rate is available in these cases.

Observe that in Theorem 1 it is not necessary to assume the finiteness ofthe second moment of the coefficient a. Before proving the theorem, we recall

some known results of approximation theory for a function f defined on a onedimensional domain (bounded or unbounded) with values in a Banach spaceV, f : R V. As in Section 2, let : R+ be a positive weightwhich satisfies, for all y, (y)CMe(y)2 for some CM >0 and strictlypositive is is unbounded and zero otherwise; yk , k = 1, . . . , p+ 1 theset of zeros of the polynomial of degree p orthogonal to the spacePp1 withrespect to the weight; an extra positive weight such that(y)Cme(y)2/4for some Cm > 0. With this choice, the embedding C

0(; V) L2(; V) is

continuous. Observe that the condition on is satisfied both by a Gaussianweight = e(y)

2

with /2 and by an exponential weight = e|y|for any 0. Finally, we denote byIp the Lagrange interpolant operator: Ipv(y) =p+1

k=1 v(yk)lk(y), for every continuous function v and by k =

l2k(y)(y) dythe weights of the Gaussian quadrature formula built uponIp.

The following two lemmas are a slight generalization of a classical result byErdos and Turan[17].

Lemma 6 The operatorI

p : C0

(; V)

L2

(; V) is continuous.

Proof. We have, indeed, that for anyvC0(; V)

Ipv2L2(;V) =

p+1k=1

v(yk)lk(y)2V(y) dy

p+1k=1

v(yk)Vlk(y)2

(y) dy.

18


19/38

Thanks to the orthogonality property

lj(y)lk(y)(y) dy= jk , we have

Ipv2L2(;V)

p+1k=1

v(yk)2Vl2k(y)(y) dy

maxk=1,...,p+1

v(yk)2V2(yk)p+1k=1

l2k(y)(y)

2(yk) dy

v

2C0(;V)

p+1

k=1k

2(yk

).

In the case of bounded, we have Cm andp+1

k=1 k = 1 for any p and

the result follows immediately. For unbounded, since (y) CMe(y)2 ,all the even moments c2m =

y2m(y) dy are bounded, up to a constant, by

the moments of the Gaussian density e(y)2

. Therefore, using a result fromUspensky 28[36], it follows

p+1k=1

k2(yk)

p

(y)

2(y)dy CM

C2m

2

and we conclude that

IpvL2(;V)C1vC0(;V)

Lemma 7 For every functionvC0(; V) the interpolation error satisfiesv IpvL2(;V)C2 infwPp()V v wC0(;V).

with a constantC2 independent ofp.

Proof. Let us observe thatw Pp() V, it holdsIpw= w. Then,

v IpvL2(;V) v wL2(;V)+ Ip(w v)L2(;V)C2v wC0(;V).

Sincew is arbitrary in the right hand side, the result follows.

The previous Lemma relates the approximation error (v Ipv) in the L2-norm with thebest approximationerror in the weighted C0-norm, for any weight

(y)Cme(y)2/4. We analyze now the best approximation error for a func-tionv : V which admits an analytic extension in the complex plane, in theregion (; ) ={z C, dist(z, )< } for some >0. We will still denotethe extension by v; in this case, represents the distance between R andthe nearest singularity ofv (z) in the complex plane.

19


20/38

We study separately the two cases of bounded and unbounded. We startwith the bounded case, in which the extra weight is set equal to 1. Thefollowing result is an immediate extension of the result given in [14,Chapter 7,Section 8]

Lemma 8 Given a functionv C0(; V) which admits an analytic extensionin the region of the complex plane(; ) ={z C, dist(z, )} for some >0, it holds:

minwPpV

v

w

C0(;V)

2

1ep log() max

z(;) v(z)

V

where1< = 2

|| +

1 + 42

||2 .

Proof. We sketch the proof for completeness. We first make a change of vari-

ables, y(t) = y0+||2

t, where y0 is the midpoint of . Hence, y([1, 1]) = .We set v(t) =v(y(t)). Clearly, v can be extended analytically in the region ofthe complex plane ([1, 1];2 /||) {z C, dist(z, [1, 1])2 /||}.

We then introduce the Chebyshev polynomials Ck(t) on [1, 1] and the ex-pansion of v: [1, 1]V as

v(t) =a0

2 +

k=1

akCk(t), (21)

where the Fourier coefficients akV, k = 0, 1, . . ., are defined as

ak = 1

v(cos(t)) cos(kt) dt.

It is well known (see e.g. [14,9]) that the series (21) converges in any ellipticdisc D C, with >1, delimited by the ellipse

E ={z = t + isC, t= + 1

2 cos , s=

12

sin(), [0, 2)}

in which the function v is analytic. Moreover (see[14] for details) we have

akV2k maxzD

v(z)V

.

If we denote by pv Pp() Vthe truncated Chebyshev expansion up to thepolynomial degreep and we observe that|Ck(t)| 1, for allt[1, 1], we have

minwPpV

v wC0(;V) v pvC0([1,1];V)

k=p+1

akV 2

1 p max

zDv(z)

V.

20


21/38


22/38

Lemma 10 Letv be a function inC0(R; V). We suppose thatv admits an ana-lytic extension in the strip of the complex plane(R; ) ={z C, dist(z, R)} for some >0, and

z = (y+ iw)(R; ), (y)v(z)VCv().

Then, for any >0, there exists a constantC, independent ofp, and a function(p) =O(

p), such that

minwPp

VmaxyR v(y) w(y)Ve

(y)24 C(p)ep.

Proof. We introduce the change of variablet = y/

2 and we denote by v(t) =

v(y(t)). Observe that vC0(R; V) with weight = e2|t|. We consider the

expansion of v in Hermite polynomials

v(t) =

k=0

vkHk(t), where vkV , vk =R

v(t)Hk(t)et2 dt. (25)

We set, now, f(z) = v(z)ez2

2 . Observe that the Hermite expansion of f asdefined in (22) has the same Fourier coefficients as the expansion of v definedin (25). Indeed

fk =R

f(t)hk(t) dt=R

v(t)Hk(t)et2 dt= vk.

Clearly, f(z) is analytic in the strip (R; 2

), being the product of analytic

functions. Moreover,

f(y+ iw)V

=|e (y+iw)2

2 |v(z)Ve y

2w22 e

2|y|Cv().

Setting

C() = max


23/38

It is well known (see e.g. [8]) that the Hermite functionshk(t) satisfy |hk(t)|< 1for all tR and all k = 0, 1, . . .. Hence, the previous series can be bound as

Ep(v)

k=p+1

vkVC

k=p+1

e

2

2k+1.

Lemma 14in Appendix provides a bound for such a series and this concludesthe proof.

We are now ready to prove Theorem 1.

Proof. [of Theorem1] The error naturally splits into = (u uh) + (uh uh,p).The first term depends on the space discretization only and can be estimatedeasily; indeed the functionuhis the orthogonal projection ofuonto the subspaceL2() H10 (D) with respect to the inner product

Da| |2. Hence

u uhL2()H10(D) 1

amin

D

a|(u uh)|2 1

2

1amin

infvL2()Hh(D)

D

a|(u v)|2 1

2

The second term uhuh,p is an interpolation error. We recall, indeed, thatuh,p =Ipuh. To lighten the notation, we will drop the subscript h, beingunderstood that we work on the semidiscrete solution. We recall, moreover,thatuh has the same regularity as the exact solution u w.r.t y .

To analyze this term we employ a one-dimensional argument. We first passfrom the norm L2 toL

2:

u IpuL2H10 12

L()

u IpuL2H10

Here we adopt the same notation as in Section 3,namely we indicate withn aquantity relative to the direction yn andn the analogous quantity relative toall other directions yj , j=n. We focus on the first directiony1 and define aninterpolation operator

I1: C

01(1; L

21

H10 )

L21(1; L

21

H10 ),

Ip1v(y1, y1 , x) =p1+1k=1

v(y1,k, y1 , x)l1,k(y1).

Then, the global interpolantIp can be written as the composition of two in-terpolation operatorsIp =I1 I(1)p whereI(1)p is the interpolation operator inall directions y2, y3, . . . , yN except y1: I(1)p : C01 (1; H10 ) L21 (1; H10 ). Wehave, then

u IpuL2H10 u I1uL2H10

I

+ I1(u I(1)p u)L2H10 II

23


24/38

Let us bound the first term. We think of u as a function of y1 with valuesin a Banach space V, u L21(1; V), where V = L21 (1)H10 (D). UnderAssumption 2, in Section 3, and the choice of given in (16), the followinginclusions hold true

C01(1; V)C0G1(1; V)L21(1; V)

with 1 = G1 = 1 if 1 is bounded and 1 = e1|y1|, G1 = e

(1y1)2

4 if 1 isunbounded. We know also from Lemma6that the interpolation operatorI1 iscontinuous both as an operator from C01(1; V) with values in L

21

(1; V) and

from C0G1(1; V) in L21(1; V). In particular, we can estimate

I =u I1uL21

(1;V)C2 infwPp1Vu wC0

G1(;V).

To bound the best approximation error in C0G1(; V), in the case 1 boundedwe use Lemma 8 whereas if 1 is unbounded we employ Lemma 10 and thefact thatuC01(1; V) (see Lemma4). In both cases, we need the analyticityresult, for the solution u, stated in Lemma5. Putting everything together, wecan say that

I

Cer1p1 , 1 boundedC(p1)e

r1p1 , 1 unbounded

the value ofr1 being specified in Lemmas 8and10. To bound the term II, weuse Lemma6:

IIC1u I(1)p uC01(1;V).The term on the right hand side is again an interpolation error. So we have tobound the interpolation error in all the other N 1 directions, uniformly withrespect to y1 (in the weighted norm C

01). We can proceed iteratively, defining

an interpolationI2, bounding the resulting error in the direction y2 and so on.

4.1 Convergence of moments

In some cases one might be interested only in computing the first few moments

of the solution, namely E[um

], m= 1, 2, . . .. We show in the next two lemmasthat the error in the first two moments, measured in a suitable spatial norm, isbounded by themean square erroru uh,pL2H10 , which, upon Theorem1, isexponentially convergent with respect to the polynomial degree p employed inthe probability directions. In particular, without extra regularity assumptionson the solution u of the problem, we have optimal convergence for the error inthe mean value (first moment) measured in L2(D) or H1(D) and for the errorin the second moment measures in L1(D).

Lemma 11 (approximation of mean value)

E[u uh,p]V(D) u uh,pL2()V(D), withV(D) = L2(D) orH1(D).

24


25/38

The proof is immediate and omitted. Although the previous estimate impliesexponential convergence with respect to p, under the assumptions of Theorem1, the above estimate is suboptimal and can be improved by a duality argument(see[4] and Remark 5.2 from[5]).

Lemma 12 (approximation of the second moment)

E[u2 u2h,p]L1(D)Cu uh,pL2()L2(D) uC0(;H1(D))withCindependent of the discretization parametersh andp.

Proof. We have

E[u2 u2h,p]L1(D) E[(u uh,p)(u + uh,p)]L1(D) u uh,pL2()L2(D)u + uh,pL2()L2(D) u uh,pL2()L2(D)

uL2()L2(D)+ uh,pL2()L2(D)

.

The termuh,pL2L2 can be bounded asuh,pL2()L2(D) =IpuhL2()L2(D)C1uhC0(;L2(D))CuC0(;H1(D))where we have used the boundedness of the interpolation operatorIp stated inLemma6. The last inequality follows from the fact that the semidiscrete solutionuh is the orthogonal projection of the exact solution u onto the subspace Hhwith respect to the energy inner product, hence

a(y)uh(y)L2(D)

a(y)u(y)L2(D), yand the energy norm is equivalent to the H1 norm.

Similarly, it is possible to estimate the approximation error in the covariancefunction of the solution u.

On the other hand, to estimate the convergence rate of the error in higherorder moments, or of the second moment in higher norms, we need extra reg-ularity assumptions on the solution to ensure proper integrability and then beable to use analyticity.

5 Numerical Examples

This section illustrates the convergence of the collocation method for a stochasticelliptic problem in two dimensions. The computational results are in accordancewith the convergence rate predicted by the theory.

The problem to solve is

(au) =0, on D,u=0, on DD,

anu=1, on DNnu=0, on (D (DD DN)),

25


26/38

withD={(x, z)R2 :1.5x0,0.4z0.8},DD ={(x, z)R2 :1x 0.5, z= 0.8},DN ={(x, z) R2 :1.5x0, z =0.4},

cf. Figure1.

(au) = 0nu= 0 nu= 0

nu= 0 nu= 0u= 0

a nu= 1

Figure 1: Geometry and boundary conditions for the numerical example.

The random diffusivity coefficient is a nonlinear function of the randomvector Y , namely

a(, x) =amin+

exp

[Y1() cos(z) + Y3()sin(z)] e 18 +

[Y2()cos(x) + Y4()sin(x)] e 18

.

(26)

Here amin = 1/100 and the real random variables Yn, n = 1, . . . , 4 are inde-pendent and identically distributed with mean value zero and unit variance. Toillustrate on the behavior of the collocation method with either unbounded orbounded random variables Yn, this section presents two different cases, corre-sponding to either Gaussian or Uniform densities. The corresponding collocationpoints are then cartesian products determined by the roots of either Hermite orLegendre polynomials.

Observe that the collocation method only requires the solution of uncoupleddeterministic problems in the collocation points, also in presence of a diffusivitycoefficient which depends non-linearly on the random variables as in (26). This isa great advantage with respect to the classical Stochastic-Galerkin finite element

26


27/38

method as considered in [4]or [29] (see also the considerations given in Section2.1). Observe, moreover, how easily the Collocation method can deal withrandom variables with unbounded support.

Figure2shows some realizations of the logarithm of the diffusivity coefficientwhile Figures3and4show the mean and variance of the corresponding solutions.

The finite element space for spatial discretization is the span of continuousfunctions that are piecewise polynomials with degree five over a triangulationwith 1178 triangles and 642 vertices, see Figure 5. This triangulation has beenadaptively graded to control the singularities at the boundary points (1, 0.8)and (

0.5, 0.8). These singularities occur where the Dirichlet and Neumann

boundaries meet and they essentially behave liker, with r being the distanceto the closest singularity point.

To study the convergence of the tensor product collocation method we in-crease the order p for the approximating polynomial spaces,Pp(), followingthe adaptive algorithm described on page 1287 of the work [5]. This adaptivealgorithm increases the tensor polynomial degree with an anisotropic strategy:it increases the order of approximation in one direction as much as possiblebefore considering the next direction.

The computational results for the H10 (D) approximation error in the ex-pected value, E[u], are shown on Figure 6 while those corresponding to theapproximation of the second moment, E[u2], are shown on Figure 7. To esti-mate the computational error in the i-th direction, corresponding to a multi

index p = (p1, . . . , pi, . . . , pN), we approximate it by E[e] E[uh,p uh,p],with p = (p1, . . . , pi + 1, . . . , pN). We proceed similarly for the error in theapproximation of the second moment.

As expected, the estimated approximation error decreases exponentially fastas the polynomial order increases, for both the computation ofE[u] and E[u2],with either Gaussian or Uniform probability densities.

6 Conclusions

In this work we have proposed a Collocation method for the solution of ellipticpartial differential equations with random coefficients and forcing terms. Thismethod has the advantages of: leading to uncoupled deterministic problems

also in case of input data which depend non-linearly on the random variables;treating efficiently the case of non independent random variables with the in-troduction of an auxiliary density ; dealing easily with random variables withunbounded support, such as Gaussian or exponential ones; dealing with no dif-ficulty with a diffusivity coefficient a with unbounded second moment.

We have provided a full convergence analysis and proved exponential con-vergence in probability for a broad range of situations. The theoretical resultis given in Theorem1 and confirmed numerically by the tests presented in Sec-tion5.

The method is very versatile and very accurate for the class of problemsconsidered (as accurate as the Stochastic Galerkin approach). It leads to the

27


28/38

Figure 2: Some realizations of log(a).

28


29/38

Figure 3: Results for the computation of the expected value for the solution,E[u].

Figure 4: Results for the computation of the variance of the solution, V ar[u].

29


30/38


31/38

1 1.5 2 2.5 3 3.5 410

8

107

106

105

104

103

102

101

p1

H1 0

Errorin

E[u]

Convergence with respect to the polynomial order p1, random variable Y1

Gaussian density

Uniform density

1 1.5 2 2.5 3 3.5 4 4.5 510

8

107

106

105

104

103

102

p2

H1 0

Errorin

E[u]


Gaussian density

Uniform density

1 1.5 2 2.5 3 3.5 410

8

107

106

105

104

103

102

101

p3

H1 0

Errorin

E[u]


Gaussian densityUniform density

1 1.5 2 2.5 3 3.5 4 4.5 510

8

107

106

105

104

103

102

p4

H1 0

Errorin

E[u]



Figure 6: Convergence results for the approximation of the expected value, E[u].

31


32/38

1 1.5 2 2.5 3 3.5 410

8

107

106

105

104

103

102

10

1

100

p1

H1 0

Errorin

E[u2]



1 1.5 2 2.5 3 3.5 4 4.5 510

7

106

105

104

103

102

101

p2

H1 0

Errorin

E[u2]



1 1.5 2 2.5 3 3.5 410

7

106

105

104

103

102

101

100

p3

H1 0

Errorin

E[u2]



1 1.5 2 2.5 3 3.5 4 4.5 510

7

106

105

104

103

102

101

100

p4

H1 0

Errorin

E[u2]



Figure 7: Convergence results for the approximation of the second moment,E[u2].

32


33/38

solution of uncoupled deterministic problems and, as such, is fully parallelizablelike a Monte Carlo method. The extension of the analysis to other classes oflinear and non-linear problems is an ongoing research.

The use of tensor product polynomials suffers from the curse of dimensional-ity. Hence, this method is efficient only for a small number of random variables.For a moderate or large dimensionality of the probability space one should ratherturn tosparse tensor product spaces. This aspect will be investigated in a futurework.

AcknowledgmentsThe first author was partially supported by the Sandia National Lab (Con-tract Number 268687) and the Office of Naval Research (Grant N00014-99-1-0724). The second author was partially supported by M.U.R.S.T. Cofin 2004Grant Metodi Numerici Avanzati per Equazioni alle Derivate Parziali di In-teresse Applicativo and a J.T. Oden Visiting Faculty Fellowship. The thirdauthor was partially supported by the the ICES Postdoctoral Fellowship Pro-gram, UdelaR in Uruguay and the European Network HYKE (HYperbolic andKinetic Equations: Aymptotics, Numerics, Analysis), funded by the EC (Con-tract HPRN-CT-2002-00282).

Appendix

Lemma 13 Letr R+, r


34/38

withfk = (2k+ 1), gk =rk and Gk = (1 rk+1)/(1 r). Then

nk=0

(2k+ 1)rk = (2n + 1)1 rn+1

1 r n1k=0

21 rk+1

1 r

= (2n + 1)1 rn+1

1 r 2

1 r

n r 1 rn

1 r

= 1

1 r

(2n + 1) (2n + 1)rn+1 2n + 2r 1 rn

1 r

= 11 r

1 + 2r1 r rn+1

(2n + 1) + 21 r

which gives the first result. Clearly,

k=0

(2k+ 1)rk = 1 + r

(1 r)2 .

Then, computing the tail series as

k=n+1

(2k+ 1)rk =

k=0

(2k+ 1)rk n

k=0

(2k+ 1)rk

we obtain easily the second result as well.

Lemma 14 Letr R+, r


35/38


36/38

[12] P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, New York, 1978.

[13] Manas K. Deb, Ivo M. Babuska, and J. Tinsley Oden. Solution of stochas-tic partial differential equations using Galerkin finite element techniques.Comput. Methods Appl. Mech. Engrg., 190(48):63596372, 2001.

[14] Ronald A. DeVore and George G. Lorentz. Constructive approximation,volume 303 of Grundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1993.

[15] Howard C. Elman, Oliver G. Ernst, Dianne P. OLeary, and Michael Stew-art. Efficient iterative algorithms for the stochastic finite element methodwith application to acoustic scattering. Comput. Methods Appl. Mech. En-grg., 194(9-11):10371055, 2005.

[16] B. Engquist, P. Lostedt, and O. Runborg, editors. Multiscale Methods inScience and Engineering, volume 44 ofLecture Notes in ComputetionalScience and Engineering. Springer, 2005.

[17] P. Erdos and P. Turan. On interpolation. I. Quadrature- and mean-convergence in the Lagrange-interpolation. Ann. of Math. (2), 38(1):142155, 1937.

[18] Philipp Frauenfelder, Christoph Schwab, and Radu Alexandru Todor. Fi-nite elements for elliptic problems with stochastic coefficients. ComputerMethods in Applied Mechanics and Engineering, 194(2-5):205228, 2005.

[19] Daniele Funaro and Otared Kavian. Approximation of some diffusion evolu-tion equations in unbounded domains by Hermite functions. Math. Comp.,57(196):597619, 1991.

[20] Roger Ghanem. Ingredients for a general purpose stochastic finite elementsimplementation. Comput. Methods Appl. Mech. Engrg., 168(1-4):1934,1999.

[21] Roger G. Ghanem and Pol D. Spanos.Stochastic finite elements: a spectralapproach. Springer-Verlag, New York, 1991.

[22] Mircea Grigoriu.Stochastic calculus. Birkhauser Boston Inc., Boston, MA,2002. Applications in science and engineering.

[23] Einar Hille. Contributions to the theory of Hermitian series. II. The rep-resentation problem. Trans. Amer. Math. Soc., 47:8094, 1940.

[24] J. Hlavacek, I. Chleboun, and I. Babuska. Uncertain input data problemsand the worst scenario method. Elsevier, Amsterdam, 2004.

[25] S. Larsen. Numerical analysis of elliptic partial differential equations withstochastic input data. PhD thesis, University of Maryland, 1986.

36


37/38


38/38

[41] Dongbin Xiu and George Em Karniadakis. Modeling uncertainty in steadystate diffusion problems via generalized polynomial chaos. Comput. Meth-ods Appl. Mech. Engrg., 191(43):49274948, 2002.

[42] Dongbin Xiu and George Em Karniadakis. The Wiener-Askey polyno-mial chaos for stochastic differential equations. SIAM J. Sci. Comput.,24(2):619644 (electronic), 2002.

Documents

Babuška - 2010 - A stochastic collocation method for elliptic partial differential equations with random input data.pdf