594 RWTH Publication No: 956771        2017       
TITLE Fully discrete approximation of parametric and stochastic elliptic PDEs
AUTHORS Markus Bachmayr, Albert Cohen, Dinh Dũng, Christoph Schwab
ABSTRACT It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in ℓp summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for non-Hilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.
KEYWORDS affine coefficients, lognormal coefficients, sparse tensor product polynomials, n-term approximation, wavelets, finite elements
DOI 10.1137/17M111626X
PUBLICATION SIAM Journal on Numerical AnalysisVol. 55, Iss. 5 (2017)