Piecewise-polynomial interpolations and quadratures for parametric PDEs with log-Laplace random inputs

TL;DR

This paper develops sparse polynomial chaos expansions and quadrature methods for solving parametric elliptic PDEs with log-Laplace random inputs, providing convergence rates based on coefficient sparsity.

Contribution

It establishes new sparsity results for Laguerre polynomial chaos coefficients and derives convergence rates for related approximation and quadrature methods.

Findings

Sparsity in polynomial chaos coefficients is characterized by $\,\ell_p$-summability.

Convergence rates are derived for semi-discrete parametric approximations.

Applicable to sparse-grid, piecewise-polynomial interpolations and quadratures.

Abstract

We establish a sparsity in terms of $\ell_p$-summability and weighted $\ell_2$-summability for the coefficients of the Laguerre generalized piecewise-polynomial chaos expansion of solutions to parametric elliptic PDEs with log-Laplace random inputs. From the sparsity, we derive convergence rates for semi-discrete approximations with respect to parametric variables. These rates are valid for sparse-grid, piecewise-polynomial interpolations and the generated quadratures, and to related extended least-squares approximations and generated quadratures.