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Quasi-Monte Carlo and discontinuous Galerkin

Vesa Kaarnioja, Andreas Rupp

TL;DR

It is proved that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen.

Abstract

In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.

Quasi-Monte Carlo and discontinuous Galerkin

TL;DR

It is proved that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen.

Abstract

In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.
Paper Structure (21 sections, 14 theorems, 112 equations, 2 figures)

This paper contains 21 sections, 14 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations, preliminaries, and assumptions
  3. Uniform and affine setting
  4. Lognormal setting
  5. Quasi-Monte Carlo cubature
  6. Uniform and affine model
  7. Lognormal model
  8. Conforming FE methods
  9. Uniform and affine setting
  10. Lognormal setting
  11. Discontinuous Galerkin FE methods
  12. Preliminaries
  13. Bounds for derivatives with respect to the random variable
  14. A brief note on error estimates
  15. Parametric regularity analysis
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $G$ belong to the weighted unanchored Sobolev space over $[0,1]^s$ with weights $\boldsymbol{\gamma}=(\gamma_{\mathrm{\mathfrak{u}}})_{\mathrm{\mathfrak{u}}\subseteq\{1:s\}}$. A randomly shifted lattice rule with $n=2^m$ points in $s$ dimensions can be constructed by a CBC algorithm such that fo where Here, $\zeta(x):=\sum_{k=1}^\infty k^{-x}$ is the Riemann zeta function for $x>1$ and $\math

Figures (2)

  • Figure 7.1: Root mean squared error for the affine case for first order polynomial DG methods. The NIPG method with $\eta = 10$ is depited blue, while the SIPG method with $\eta = 100$ is depicted red and the SIPG method with $\eta = 10$ is depicted green, and the conforming finite element solution is black. The green graph is omitted in the right picture.
  • Figure 7.2: Root mean squared error for the lognormal case. The NIPG method with $\eta = 10$ is depicted in red, the SIPG method with $\eta = 10$ is depicted in blue, and the conforming finite element solution is black. The left picture illustrates DG and conforming finite elements for locally linear polynomial approximation, while the second picture shows only DG for a locally quadratic approximation.

Theorems & Definitions (21)

  • Lemma 3.1: cf. kuonuyenssurvey
  • Lemma 3.2: cf. nicholskuo and log
  • Lemma 5.1
  • Lemma 5.2: PietroE12
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • Theorem 5.5
  • Corollary 5.6
  • ...and 11 more