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

Dinh Dũng

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.

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

Abstract

We establish a sparsity in terms of -summability and weighted -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.
Paper Structure (10 sections, 7 theorems, 143 equations)

This paper contains 10 sections, 7 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Laguerre GPWPC expansions
  3. Laguerre piecewise polynomials
  4. Laguerre GPWPC expansions
  5. Sparsity for Laguerre GPWPC expansion
  6. Semi-discrete parametric approximations
  7. Approximation by finite truncated GPWPC expansion
  8. Sparse-grid piecewise-polynomial interpolation
  9. Sparse-grid quadrature
  10. Extended least squares approximations

Key Result

Lemma 3.1

Given ${\boldsymbol{\delta}} \in {\mathbb D}$, assume $v \in L_2({\mathbb R}^\infty,V;{\boldsymbol{\lambda}}_a)$ and ${\mathcal{D}}^r_{\boldsymbol{\delta}} v \in L_2({\mathbb R}^\infty_{{\boldsymbol{\delta}}},V;{\boldsymbol{\lambda}}_a)$. Then it holds the equality

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.1
  • Theorem 4.2
  • ...and 6 more