Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients

TL;DR

This paper analyzes the convergence of polynomial expansions for solutions of parametric elliptic PDEs with affine coefficients, providing improved bounds and demonstrating their sharpness through numerical experiments.

Contribution

It introduces enhanced summability estimates for polynomial expansions of PDE solutions with affine parametric coefficients, advancing previous theoretical bounds.

Findings

Improved bounds on polynomial expansion convergence for affine parametric PDEs

Results extend to Jacobi polynomial expansions

Numerical experiments confirm the sharpness of the bounds

Abstract

We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments.