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

Dinh Dũng, Van Kien Nguyen, Viet Ha Hoang

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.

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

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 . The established sparsity is quantified by -summability and weighted -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 -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.
Paper Structure (9 sections, 14 theorems, 122 equations)

This paper contains 9 sections, 14 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Sparsity for parametric PDEs
  3. The Laguerre GPC expansion
  4. Sparsity for the Laguerre GPC expansion
  5. Semi-discrete approximations
  6. Sparse-grid polynomial interpolation
  7. Sparse-grid quadrature for numerical integration
  8. Extended least squares approximation
  9. Parametric PDEs with Gaussian log-normal inputs

Key Result

Lemma 2.1

Assume that there exist a positive sequence ${\boldsymbol{\rho}}=(\rho_j)_{j\in {\mathbb N}}$ and a positive number $\kappa$ satisfying Then we have the following. Let ${\boldsymbol{y}} \in {\mathbb R}^\infty$ with $b({\boldsymbol{y}})\in L_\infty(D)$ and ${\boldsymbol{s}}\in {\mathbb F}$ such that $\operatorname{supp}({\boldsymbol{s}})\subseteq \operatorname{supp}({\boldsymbol{\rho}})$. Then we

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 16 more