Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains

TL;DR

This paper investigates how localized function representations improve the efficiency of polynomial surrogates for parametric elliptic PDEs, especially on parametric domains, through theoretical analysis and numerical experiments.

Contribution

It introduces a localized support approach for polynomial surrogates in parametric PDEs, demonstrating improved convergence and efficiency over traditional methods.

Findings

Localized representations lead to higher convergence rates.

Numerical experiments confirm efficiency gains.

Method extends to other elliptic and parabolic PDEs.

Abstract

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a \textsl{highly nonlinear} way on a parameter $y\in [-1,1]^{\mathbb{N}}$. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove $\ell^p$-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.