Solution of stochastic partial differential equationsusing Galerkin finite element techniques
Manas K. Deb a.\ Iva M. Babuska b, J. Tinsley Oden b
a TIECD Software, 6400 Harrogate Drive, Austin. TX 78759, USAb Texas Institute for Computational and Applied Mathematics. The University of Texas. Austin TX, USA
To date, most computer simulations are based on deterministic mathematical models, where all inputdata are assumed to be perfectly known. In fact, this is never the case. All data contain a certain level ofuncertainty: material properties, loading scenarios, boundary conditions, domain geometry, etc., havesmaller or larger uncertainties which influence the quantities of interest. This resulting uncertainty in thesolution, can, of course, be larger or smaller than that in the input data.
The uncertainties of the input data can have different characteristics which must be taken into account,and must be related to the aims of the analysis. They can, for example, show stochastic probabilisticcharacter when the material properties are obtained experimentally. Unfortunately, probabilistic data areoften difficult to obtain and are generated by methods which themselves have additional uncertainties. Theinput data can be based on a "worst scenario" approach when bounds for the quantity of interest or in aprobabilistic framework are desired. If the uncertainties are small, perturbation theory is a very valuabletool for analyzing their effects. If they are larger, then perturbation theory is not applicable.
In this paper we address problems characterized by linear partial differential equations for which theinput data are stochastic; for example, the coefficients or the right-hand side (RHS) of the partial differ-ential equation (PDE) are the stochastic functions. The aim of the paper is to transform the stochastic PDEproblem into a deterministic problem where finite element methods can be used for obtaining useful nu-merical approximations. It is possible to use the theory of finite elements to obtain several useful results,including a posteriori error estimates, adaptive approaches, superconvergence computations of functionals,etc. (see [1,4]). The formulation of the stochastic boundary-value problem developed here also provides abasis for interpreting and analyzing numerous approaches suggested in the FEM literature from a unifiedpoint of view.
6360 M.K. Deb et al. I Comput. Methods Appl. Mech. Engrg. /90 (200/) 6359~372
This paper will concentrate on the stochastic input data in problems which are not white noise, i.e., thereare significant correlations. This is typical in many engineering applications. Differential equations withwhite noise are broadly studied in various contexts in physics, financial models, etc. For more on thissubject see, e.g. [10]. One obvious way to treat stochastic PDEs is the Monte Carlo Method. This method isexpensive especially when some higher-order accuracy for mean values, variation, etc. are sought. In ad-dition, errors in the numerical approximation of the exact solution must be characterized in a probabilisticway. It is worthwhile to mention our approach when adaptive finite element methods are used, has a clearrelation to an adapative Monte Carlo Method.
Various numerical methods to solve stochastic partial differential equations (SPDEs) have beenproposed in the literature. The work of Ghanem [9] and Ghanem and Spanos [8] advocate a hybridfinite element-spectral approach, while the monograph of Kleiber and Hien [I I] utilizes a perturbationapproach. Elishakoff and Ren [7] examine engineering finite element methods for structures with largestochastic variations and point to limitations of some approaches. A fuller description of work oncomputational methods for SPDEs used to model stochastic behavior in problems of mechanics can befound in the survey of Schueller and Pradwarter [14], in the books of Kleiber and Hien [1 I], Ghanemand Spanos [8], and in Deb [5,6] and the references therein. This paper is closely related to the Ph.D.thesis [6].
2. Model problems
We begin by considering two model problems of SPDEs of elliptic type.
Problem 1.
\7 . a(x)\7u(x) = f(x)u(x) = 0
on D,on aD. (2.1 )
Here a and u are stochastic functions while f is deterministic.
Problem 2.
\7 . a(x)\7u(x) = f(x)u(x) = 0
on D,on aD.
(2.2)
Here f and Ii are stochastic functions while a is deterministic. We assume that D E IR.d, d = 1,2, 3, is abounded Lipschitz domain. We formulated these two problems separately be~ause they have differentstructures. Of course, it is possible to analyze the problem when both a and f are stochastic, but con-sidering each case separately simplifies slightly the notation etc., and, in addition, allows us to exploitspecial properties of Problem 2. We also can treat analogously, the stochastic problem for other differentialequations; for example, those of linear elasticity, nonhomogeneous boundary conditions, stochasticboundaries aD etc. We refer to Problem 1 as a problem of the left-hand side (LHS) type of SPDE andanalogously Problem 2 will be referred to as the RHS type of SPDE.
3. Mathematical formulation
Let (Q, F, P) be a probability space, where 0, F, P are the set of random events, the Ii-algebra of subsetsof Q and P the probability measure, respectively. If X is a real random variable in (Q, F, P) with X ELI (Q),we denote its expected value by
EIX] = ( X(w)dP(w) = (xdµ(x).l'i In
M.K. Deb el al. I Comput. Methods Appl. Mech. Engrg. 190 (200l) 6359~372
Here µ is the distribution probability measure for i, defined on the Borel set Band IRgiven by
µ(B) = P(X-I(B)).
6361
We will assume that µ(B) is absolutely continuous with respect to Lebesque measure; then there exists adensity function for i, P : IR-> IR+ such that
E[i] = 1p(x)dt.
Let us now define a random function. A function ~(x) = t/I(x, w) : D x Q -> IR will be called a randomfunction when it is jointly measurable (on the Borel sets (B(D) EB(Q))) and
E[1n t/l2(x, W)dX] < 00.
Remark 3.1. The last condition is not essential, but in this paper we will consider only such functions.
We assume that in (2.1) a(x) = a(x,w) is such that
0< (XI ~a(x,w) < (X2 < 00 a.e. on D x Q.
Let v(x, w) be defined on D x Q. Then we define
V = {lv(x,w)llIvll" < 00, v(x,w) = 0 on aD},where II . II v is the energy norm
The natural inner product in V is the bilinear form f!4 : V x V -> R
Obviously, V is a Hilbert space of random functions.
(3.1 )
(3.2)
(3.3)
(3.4)
Theorem 3.2. Let f(x) E L2(D). There exists unique weak solution uo(x,w) E V (i.e., a random function) ofthe problem (2.1) which satisfies
f!4(uo, v) = 10 (E[fv])dx = 2(v) Tlv E V.
Proof. The theorem follows immediately from the Lax-Milgram lemma. 0
(3.5)
Remark 3.3. We address here the LHS problem (2.1) when f is deterministic. The RHS problem is ad-dressed in Section 7.
Let us assume now that
M
a(x,w) = (E(a))(x) +L yT;ai(x)Ai(w},;=1
(3.6)
where A;(w), i= 1,2, ... ,M are real mutually independent random variables with mean value zero- . -2 . I .(E(Ai) = 0), vanance one (E(A;) = 1), and bounded Images ri of Q, r; = A;(Q) E IR , 1= 1, ,M.Further, we assume that each A; has a probability density function Pi : r; -> R+, i = 1, ,M, and
0< PI ~ Pi ~ P2 < 00 and a/(x) E LOC(D), A.j > 0, j = 1, ... ,M, and (3.1) holds.
6362 M.K. Deb et al. I Comput. Methods Appl. Mech. Engrg. /90 (2001) 6359~372
(3.7)
Remark 3.4. In (3.6), we can have M = 00 provided that the series converges in LX(D). Then, using (3.6) asthe truncated series, we can estimate the error caused by the truncation, but we will not address thisestimation here.
The expansion (3.6) is known as the Karhunen-Loeve (K-L) expansion [12]. Then A.; and a;(x) are theeigenvalues and eigenfunctions associated with the given covariance function C(x,x) of a(x) = a(x, w) anda;(x) have orthonormalities properties which can be utilized in the implementation. The probability den-sities Pi are arbitrary. For any of them, we get E[a(x, w)a(x, w)] = C(x,x).
2dxdy = IItlllw < 00, v(x,y) = 0 on oD for all y}.(3.9)
Then the space, W is the same as (equivalent to) V when a(x,w) is given in (3.6). Because of our as-sumptions on a and P, W is Hilbert space. Further, (3.4) becomes
.16'(u,v) =1p(y) Iv a(x,y)Y'xU(X,y)· Y'xv(x,y)dxdy
which will be used in the sequel. Further. we have
!t'(v) =1p(y) 1v(x,y)f(x)dxdy, v E W .
Then uo(x,y) E W defined in (3.5) can be written in the form (3.8) and satisfies
.9l(uo, v) = !t'(v) "tv E W.
(3.10)
(3.11)
(3.12)
(3.13)
Because of our assumptions, uo(x,y) E W is uniquely determined from (3. ]2); we transformed the stochasticproblem (2.1) (resp. (3.5)) into the deterministic problem (3.12).
Because uo(x, w) E V (resp. uo(x,y) E W), we have
E[uo(x, w)] = 1p(y)uo(x,y)dy E HI (D),
E[u~(x,w)] = lp(y)u~(x,Y)dYEL'(D),
[ouo ] 1 ouo 2E OX; (x,w) = rP(y) OX; (x,y)dy E L (D),
E [ (~:~ (x, w) r] = 1p(y) (~:~ (x,y) Y dy E LI (D),
C(x,x) = E[uo(x,w)uo(x,w)] E L2(D x D).
Equations (3.] 3) shows that the mean value, variance, standard deviation and covariance are well-definedfunctions. Their smoothness depends on a;(x) in (3.6), I(x), and smoothness of oD.
M.K Deb et al. I Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359~372
4. The finite element solution of the LHS type of SPDE
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We have seen in Section 3 that the LHS type of SPDE can be cast in the form (3.10). This form is verysuitable for finite element approximation, and allows us to use the theories of FEM, the a posteriori errorestimation, adaptivity, etc.
Let ~(D) be a mesh on D which satisfies all of the usual assumptions, common in the finite elementtheory and t(~(D)) be the elements with h(D) = max(diamt(~(n). Then we denote by~(n = ~(rl) x··· x ~(rM) the rectangular mesh on r and let t(~(n) = t(~(rd) x··· x t(~(rM)) theassociated elements. We denote by ~(D,r) the mesh on D x r, and t(~(D,n) = t(~(D)) x t(~(n) theelements on D x r. Let
Sp.q(D, n = {v(x,y) E Wlv(x,y) I t(Dr) is polynomial of degree p in x, 'Vy E r and of degree q
in Yl,Y2, .. ·Ym, 'Vx ED}. (4.1)
If t(D) are quadrilaterals, then v(x,y) are polynomials of degree p in every variable Xi> i = 1, ... , d. If themesh on D is curvilinear, then we use the standard pull-back polynomials of degree p.
Remark 4.1. The finite element approximation then reads: Find usp.q(D,fl E Sp,q(D, n such that
(4.2)
where fJ4 and II! are defined in (3.10) and (3.11). From standard finite element theory, we get the basicresult:
Theorem 4.2. The finite element solution UsP.q(D.r)E Sp.q(D, n exists, is unique, and
Hence
Iluo - USM(D.TlIIW(D,r) -+ 0 as h(~(D») and h(~(n) -+ 0
and
Iluo - USp.q(D.nIIW(D.f) -+ 0 as p, q -+ 00.
(4.3)
(4.4 )
(4.5)
Remark 4.3. Condition (4.4) expresses convergence of the h version while (4.5) that of the p version of thefinite element method.
Utilizing the smoothness of the data and of the solution, we can, in standard ways, prove a priori es-timates for the error in the energy norm, L2, norm and the error in the data at interest for the mean value,standard derivation, covariance etc. When data are smooth, the following result follows from standardFEM procedures.
Theorem 4.4. Let the input data a;(x),f(x) and aD be sufficiently smooth. Then we have
6364 M. K. Deb et al. I Comput. Methods Appl. Mech. Engrg. /9() (200/) 6359~372
5. A simple model problem: algebraic stocbastic equations
We next consider a very simple problem which has many of the same essential properties as the LHS typeof SPDE. To show the major ideas, we consider the stochastic problem
au = 1,
where a = a(w) is a stochastic function (in fact, a random variable) with
0< amin:::;; a(w):::;; amax < 00
:md Ii= u(w) is the stochastic solution of (5.1).We assume that
M
a(w) = au + I>iAi(w),i~1
(5.1)
(5.2)
(5.3)
where Ai (w), i = 1, .. , , M are independent random variables wi th the same assumptions as in Section 2,ai E R Conditions (5.2) are the expressions (3.6) when a(x, w) is x independent. Let Ti = Ai(Q) =(-IXil IXi) = Ii) Ct.i> 0, Ai(W) = Yi E Ii, i = 1, ... ,M, T = T; x ' , , x T M, Pi0'i) be the probability density ofAi and PlY) = PI 0'1)P2lv2) ... P(YM), Y = (YI, .. , ,YM).
We introduce the space W. Let v0'),y E r = Ii x h x ... X 1M. Then
(5.4)
where a0') = aD + 2:::1 aiYi,aj E Rj = 0, ... ,M. Further, for u E W, v E W we define the bilinear form
PJ(u,v) = !.p(Y)a(Y)u0')v(Y)dY
and the functional
.Y"(v) = 1p0')v(y)dy.
The problem (5.1) can be formulated as follows: Find uu0') E W such that
PJ(ua, v) = .2"(v), Vv E W.
(5.5a)
(5.5b)
(5.6)
L I h h A(/) . - a I Il(i) - hi _.j j-I h - 0 - 1 (.) Thet on ;, t e mes U i . -IXi - Yi < Y; < ... < Y; - lXi, i - Yi - Yi , i-max} ITi- 1"" n I. enby /).(1) we denote the rectangular mesh /).(1) = Li(rl) x /).(r2) x Li(r M) and -r(Li(1)) = -reLiCTi)) xr(Li(T2)) x ... x -r(Li(r M)) be the elements on r. Then we let
S'l = {v(y) E Wlvlt(li(T) is polynomial of degree q separately in y" ... 'YM}'
and the finite element sq(1) E S'I satisfies
PJ(U,S<I,v) = .2"(v) Vv ESq.
Obviously, the exact solution of (5.1) is
1ua(w) = uo0') = a0') .
Hence, because of (5.2) and (5.3), it is easy to see that u(y) is smooth on r and hence we have
(
M ) 1/2lIua - us~llw:::;; C ~h7(q+ll
(5.7)
(5.8)
(5.9)
(5,10)
M.K. Deb et al. 1 Comput, Methods Appl. Mech. Engrg. /90 (2001) 6359~372
Further, as an example, consider
6365
(5.11)
which is the error in the data of interest, namely E[uo]. Proceeding analogously as in the classical finiteelement method, we define the influence function G E W by
~(u, G) = ./. updy Vu E W.
Because of the symmetry of ~(u, v) and G(y) = Ija(y), G(y) is smooth,
Next, considering implementational aspects, we assume that q = 0, i.e., we use constant shape functions onthe mesh A(r), and we have
where
(5.17)
for yrl < Yj < /j,elsewhere, (5.18)
and
Obviously,
~(tf;kl .... ,kM' tf;ll .....t.J = ° .
6366 M.K. Deb et a/. I Comput. Met/zod5 Appl. Mech. Engrg, 190 (2001) 6359~372
o
-2
~ -4
~• -6t( -8~.E!-10
-12
-14o
ErrorJn_Mean,SIope=1.9859
2 3 4 5
Fig, 1. Convergence rate for M = I.all = al = 1.,:x",."la",;n = 1.5. The observed rate is very close to the theoretical one also for crudemesh,
for (k.,."kM)=I-(f, ... fM) and the system to solve is diagonal. As usual, we call the number of basisfunctions in (5.15) the number of degrees of freedom and denote it by N. We can of course use q > 0 andorthogonalize the shape functions on 5'(LI(r): then we solve systems of equations with a diagonalmatrix.
Let us consider a numerical example. Assume M = I,ao = a) = I and that 0:) is selected so thatamaxlamin = 1.5 and the density PI is uniform, Further, consider a uniform mesh Ll(Td. Because the exactsolution is known, i.e., its probability distribution: we now can compute the error
as well the errors in the mean standard derivation and variation.In Fig. I, we show in the en - fn scale the errors and their observed rate obtained by a least-square fit of
the data. The theoretical rate for the L2 norm is I and the other data is 2. We see that high accuracy wasachieved with small numbers of degrees of freedom M and the rates match perfectly the theoretical esti-mates.
In Fig. 2, we show analogous results for M = 2,ao = a, = a2, Pi and !Xl = !X2 uniform and arnaxlamin =199. We observe that also for crude mesh, convergence is in the asymptotic range.
6. The general case of random LHS of SPD, an example
Let us consider the two-dimensional problem (2.1), respectively (3.10) and (3.11), and its approximatesolution based on (4.1).
Let D = (-0.5,0.5) x (-0.5,0.5), <Ta = 0.1
3
a(XI,X2,W) = aO(XI ,X2) + <Ta L ~aj(XI ,x2)Ai(w);=1
M.K. Deb et al. I Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359~372 6367
6.56
ErroU n_Mean, Slope= 1,9703
Error_ln_L2_nonn. Slope=O.9812
Error_ln_ Variance, Slopec1.9025
Errocln_SlandanLDevlatlon,
5.55
0
-2
T
.' -4t•I -6
!.Ii -8
-10
-124.5
Fig, 2. Convergence for M = 2. ao = at = a2 = 1. amax/amin = 199. The observed rate is very close to the theoretical one.
with r = (-.;3, .;3)3,p(y) = constant, and A; and a;(x) are obtained by taking products of K-L expansionfor single dimension with (one-dimensional) covariance function C = cr2e-1-X\-X21, cr = 0.1, andf(XJ,X2) = 2(0.5 - xi - xD· Because the problem is symmetric, we need address only a quarter of thedomain D. We consider on D uniform mesh of squares; particularly we use an 8 x 8 mesh on the quarter ofthe domain D. Also on r a uniform square mesh is considered; particularly we use 8 x 8 x 8 mesh on r. OnD we use bilinear elements and on r the constant elements. Denoted by uo(x,y) and u(x,y;h(D),h(r» theexact and the finite element solution. We have
In Figs. 3(a) and (b) we show the mean and the variance computed numerically. In Figs. 4(a) and (b) weshow the mean and variance using 65,536 realizations using a uniform mesh in D.
7. Solution of random RSH type SPDE
In the previous sections, we addressed the problem of random LHS type in a SPDE, The analysis ofRHS type is very similar. Nevertheless, in this case we can, in addition, directly compute the mean valueand the covariance. As before we will assume that
M
f(x, OJ) = fo(x) +L: Jf;.fi(x)A;(OJ);=1
(7.1)
and assume about the random variables A](OJ). the same as before. Then we can set
f(x,OJ) = f(x,y), xED and y E r (7.2)
368 M.K. Deb et al. I Comput, Methodr Appl. Mech. Engrg. /90 (2001) 6359~372
19. 3. Computation based on (4.1) and (4.2). (a) The mean value of the solution. max value is 0.06323. (b) the variance of the solution.; max value 0.1876£ - 04.
'ig. 4. (a) Computation based on Monte Carlo method: the mean value of the solution, max value is 0.06324. (b) The variance of theolution is max value is 0.1881E - 0.4.
As a a numerical example, consider the one-dimensional problem,
d2-U -- dx2 = f(x) on D = (-0.5,0.5)
u(±0.5) = O.
Assume that E[f] = 0 and the covariance off to be
with (J = 0, I.Fig. 5 shows the covariance of the solution computed from Theorem 7.2.Theorem 7.2 shows that the covariance of the random RHS type SPDF can be computed as a deter-
ministic problem on D x D. The strong form is
L,LxC(x,:X) = F(x, x) on D x D,
where LXl Lx are the operators of the LHS of the equation and C(x,x) = 0 on aD x D and D x aD.
5E-05 joU.s;o
If)
Fig. 5. The covariance of the solution computed by Theorem 7.2.
(7.11 )
M.K. Deb et a/. 1 Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359~372 6371
Remark 7.3. The space dJI is the typical space of functions with dominant mixed derivatives. For more,see [2].
Remark 7.4. The proof that the covariance C(x,x) satisfies (7.11) was proven in [3].
Remark 7.5. Computation of the covariance function directly from (7.6a) and (7.6b) is much more ex-pensive than computation from Theorem 7.2. Using Theorem 7.2 C(XI ,X2) exists for all FE q/ and hencethe convergence of the series (7.1) can be very weak. Further, in practice, only the covariance functionF(x] ,X2) is available and K-L expansion utilizes it. Hence we should utilize F(XJ,X2) as much as possible,i.e., to compute directly the covariance by Theorem 7.2,
Remark 7.6. The perturbation approach for dealing with the LHS problem essentially transforms it to theRHS problem.
8. Comments and summary
The aim of the present work was to cast the problem of stochastic PDEs into a framework similar tothose familiar in deterministic problems which are suitable for approximation by the finite element method.The FEM is very well-developed, theoretically and practically. A posteriori error estimation is available aswell, as are adaptive procedures for h,p and h-p versions of the FEM. The major difficulty with the methoddescribed here is the high dimensionality of the problem. This aspect has to be taken into account in theimplementation. Obviously. adaptive procedures are necessary for successful solutions. The approachpresented in this paper gives a basis for utilizing the large body of known results and methods pertaining tothe FEM. Implementational aspects are not addressed here.
It is worthwhile to mention that the use of the Wiener chaos polynomials used e.g., in [8]is essentially thep-version in the framework in this paper. We mention only models problems. Analogously, it is possible toanalyze stochastic boundary conditions and the problem of stochastic domains. Also the approach pre-sented here can be generalized for other types of differential equations. There are many problems whichremain to be addressed. One essential problem is the case in which the K-L series converges very slowly andmany terms are needed. It is necessary to weaken the sense of convergence so that the problem with "al-most" white noise will also be solvable. We assumed that the data in the stochastic formulation are per-fectly known. Of course this is not generally the case and uncertainties are present here too. The influence ofthese uncertainties has to be analyzed as well. This will be the subject of forthcoming papers.
Acknowledgements
The authors gratefully acknowledge the support of ONR under contract NOOO14-95-1-0401, NSF GrantDMS 9802367, and support of related work by Sandia National Laboratories under grant BF-2070.
References
[I] M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley. New York, 2000.[2] T.1. Amanov, Spaces of Differentiable Functions with Dominant Mixed Derivatives. Nauka, Moscow. 1976 (in Russian),[3] J. Babuska, On Randomized Solution of Laplace's Equation. Casopis Pest Mat" 1961.[4] l. Babuska, T. Strouboulis, The Finite Element Method and its Reliability, Oxford University Press. Oxford, 2001.[51 M. Deb.!. Babuska, J.T. Odell. Stochastic Finite Eiement using Galcrkin Approximation, Fifth US National Congress on
Computational Mechanics. Boulder. 1999.[6] M, Deb, Solution of stochastic partial differential equations (SPDEs) using Galerkin method: theory and applications. PhD.
Dissertation. The University of Texas. Austin, 2000.[7] l. ElishakofT. Y. Ren, The bird's eye view on finite element method for structures with large stochastic variations. Comput.
Methods Appl. Mech. Engrg. 168 (1-4) (1999) 51-61.[8] R. Ghanem. P. Spanos. Stochastic Finite Elements: A Spectral Approach, Springer. Berlin, 1991.
6372 M.K. Deb et al. I Comput. Methods Appl. Mech. Engrg. 190 (2001) 6359~372
[9] R. Ghanem, Ingredients for a general purpose stochastic finite elements implementation, Comput. Methods Appl. Mech. Engrg.168 (1-4) (\999) 19-33.
[101 H, Holden, B. Oksendal. J. Uboe. T.S. Zhang. Stochastic Partial Differential Equations - A Modeling White Noise FunctionalApproach. Birkhauser, Basel, 1966.
[Ill M. Kleiber. T.D, Hien. The Stochastic Finite Element Method. Wiley. New York. 1992.[12] M. Loeve. Probability Theory. fourth ed .• Springer, l\ew York. 1977.[13] B. Oksendal. Stochastic Differential Equations - An Introduction with Application. Springer, Berlin. 1998.[14J G,l. Schueller and H.I. Pradwarter, Computational stochastic mechanics - current developments and prospects, in: S. Idelsohn.
E. Onate. E. Dvorkin (Eds.). Computational Mechanics: New Trends and Applications. CIMNE. Barcelona, 1998.
