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Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

Content Provider	Springer Nature Link
Author	DeVore, Ronald Cohen, Albert Schwab, Christoph
Copyright Year	2010
Abstract	Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝd are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as random parameters y=y(ω)=(y i(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x,y) is a function of both the space variable x∈D and the in general countably many parameters y.We establish new regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to $V=H^{1}_{0}(D)$ . These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a so-called “generalized polynomial chaos” (gpc) expansion of u.Convergence estimates of approximations of u by best N-term truncated V valued polynomials in the variable y∈U are established. These estimates are of the form N −r , where the rate of convergence r depends only on the decay of the random input expansion. It is shown that r exceeds the benchmark rate 1/2 afforded by Monte Carlo simulations with N “samples” (i.e., deterministic solves) under mild smoothness conditions on the random diffusion coefficients.A class of fully discrete approximations is obtained by Galerkin approximation from a hierarchic family $\{V_{l}\}_{l=0}^{\infty}\subset V$ of finite element spaces in D of the coefficients in the N-term truncated gpc expansions of u(x,y). In contrast to previous works, the level l of spatial resolution is adapted to the gpc coefficient. New regularity theorems describing the smoothness properties of the solution u as a map from y∈U=(−1,1)∞ to a smoothness space W⊂V are established leading to analytic estimates on the W norms of the gpc coefficients and on their space discretization error. The space W coincides with $H^{2}(D)\cap H^{1}_{0}(D)$ in the case where D is a smooth or convex domain.Our analysis shows that in realistic settings a convergence rate $N_{\mathrm{dof}}^{-s}$ in terms of the total number of degrees of freedom N dof can be obtained. Here the rate s is determined by both the best N-term approximation rate r and the approximation order of the space discretization in D.
Ending Page	646
Page Count	32
Starting Page	615
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	6
Volume Number	10
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-07-31
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Wiener polynomial chaos Economics general Stochastic and parametric elliptic equations Sparsity Numerical Analysis Computer Science Linear and Multilinear Algebras, Matrix Theory Stochastic differential and integral equations Math Applications in Computer Science Applications of Mathematics Approximation rates Nonlinear approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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