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

by A. Cohen and A. Chkifa and Ch. Schwab

(Report number 2013-25)

Abstract
The numerical approximation of parametric partial differential equations \(\cal D(u,y)=0\) is a computational challenge when the dimension \(d\) of of the parameter vector \(y\) is large, due to the so-called {\it curse of dimensionality}. It was recently shown in [5, 6] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there \cs{exist} polynomial approximations to the solution map \(y\mapsto u(y)\) with an algebraic convergence rate that is \cs{independent of the parametric dimension \(d\)}. The analysis in [5, 6] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy \cs{for establishing similar results for some classes parametric PDEs that do not necessarily fall in this category.} Our approach is based on building an analytic extension \(z\mapsto u(z)\) 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 radii 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 \(d=\infty\). 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 PDE's 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 \cs{parametric} elliptic PDE's, and (iv) elliptic equations set on a domain that is parametrized by the vector \(y\). We give general strategies that allows us to derive the analytic extension in a \cs{unified} abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces, generalizing recent results in [13, 15]. 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.

Keywords:

@Techreport{CCS13_522,
  author = {A. Cohen and A. Chkifa and Ch. Schwab},
  title = {Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-25.pdf },
  year = {2013}
}

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