Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

被引:110
作者
Chkifa, Abdellah [1 ,2 ]
Cohen, Albert [1 ,2 ]
Schwab, Christoph [3 ]
机构
[1] Univ Paris 06, Lab Jacques Louis Lions, UMR 7598, F-75005 Paris, France
[2] CNRS, Lab Jacques Louis Lions, UMR 7598, F-75005 Paris, France
[3] ETH, Seminar Appl Math, CH-8092 Zurich, Switzerland
来源
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | 2015年 / 103卷 / 02期
关键词
Parametric PDEs; Curse of dimensionality; Sparse Legendre series; Sparse high dimensional interpolation; PARTIAL-DIFFERENTIAL-EQUATIONS; STOCHASTIC COLLOCATION METHOD; ANALYTIC REGULARITY; CONVERGENCE;
D O I
10.1016/j.matpur.2014.04.009
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The numerical approximation of parametric partial differential equations D(u, y) = 0 is a computational challenge when the dimension d of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown in [1,2] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map y -> u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis in [1,2] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for classes of parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z -> u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying semi-axes lengths of the ellipses in each coordinate z(j) reflect the anisotropy of the solution map with respect to the corresponding parametric variables y(j). This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case d = infinity. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDEs that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDEs, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allow us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions. (C) 2014 Elsevier Masson SAS. All rights reserved.
引用
收藏
页码:400 / 428
页数:29
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