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Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

Dinh Dũng, Helmut Harbrecht, Van Kien Nguyen, Christoph Schwab

Abstract

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

Abstract

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.
Paper Structure (20 sections, 12 theorems, 156 equations, 5 figures)

This paper contains 20 sections, 12 theorems, 156 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Contents
  3. Notation
  4. Elliptic diffusion problems on variable domains
  5. A class of homothetic parametric domains
  6. Pullback of the solution to a reference domain
  7. Parametric analyticity of the pullback solution
  8. Random variations of the domain
  9. Log-Gaussian shape parametrization
  10. Holomorphy of the parametric pullback solution
  11. Derivative estimates
  12. Interpolation and Quadrature
  13. Sparse grid interpolation and quadrature
  14. Quasi-Monte Carlo integration
  15. Finite element discretization
  16. ...and 5 more sections

Key Result

Lemma 2.1

Assume that $a\in W^1_\infty({\mathbb S}^1;{\mathbb C})$ and let $h(\theta)$ be defined in h(theta). Assume also that Then there holds and

Figures (5)

  • Figure 1: Three realizations of the random domain with log-Gaussian random boundary in case of $\kappa = 0$ and for the sequence $\lambda_k = (|k|+1)^{-2}$ for all $k\in\mathbb{Z}$.
  • Figure 2: Three realizations of the random domain with log-Gaussian random boundary in case of $\kappa = 0$ and for the sequence $\lambda_k = (|k|+1)^{-3}$ for all $k\in\mathbb{Z}$.
  • Figure 3: Four realizations of the random domain under consideration with the associated solution of the underlying Poisson equation. Shape parametrization \ref{['a=trigonometric']} with $\kappa = 0$ and \ref{['eq:lkEple']}. The triangulation is the mapped version of the triangulation of the unit disk. The grading towards the origin is clearly visible.
  • Figure 4: Error of the approximation to $\|\mathbb{E}[\hat{u}]\|_{H^1(D_{{\operatorname*{ref}},0})}$ of the quasi-Monte Carlo method versus the number $N$ of sampling points.
  • Figure 5: The expectation of the random solution on the unit disk.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Example 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 17 more