Sparsity for parametric PDEs with log-gamma random inputs and applications

TL;DR

This paper introduces a new approach to demonstrate the sparsity of polynomial chaos expansion coefficients for solutions to parametric elliptic PDEs with log-gamma inputs, enabling improved convergence analysis and applications.

Contribution

It establishes novel sparsity results for Laguerre polynomial chaos expansions in PDEs with log-gamma inputs and extends these results to log-normal inputs, improving prior conditions.

Findings

Sparsity quantified by $\, ext{ell}_p$-summability and weighted $\, ext{ell}_2$-summability.

Derived convergence rates for sparse polynomial approximations and quadrature rules.

Improved sufficient conditions for $\, ext{ell}_p$-summability in log-normal input cases.

Abstract

We propose a novel method for establishing the sparsity of the coefficients of the Laguerre generalized polynomial chaos expansion of solutions to parametric elliptic PDEs with log-gamma inputs on $\mathbb{R}_+^\infty$. The established sparsity is quantified by $\ell_p$-summability and weighted $\ell_2$-summability of the coefficients. Building on these sparsity results, we derive convergence rates for semi-discrete approximations in the parametric variables. These rates apply to sparse-grid polynomial interpolations, extended least-squares approximations and the associated semi-discrete quadrature rules. Moreover, a counterpart of our method for parametric elliptic PDEs with log-normal inputs yields a significant improvement in the sufficient condition for $\ell_p$-summability when the component functions in the log-normal representation of the parametric diffusion coefficients have global support, compared with results obtained in prior works.