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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion
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by Helmut Harbrecht, Michael Peters and Markus Siebenmorgen PDF
Math. Comp. 86 (2017), 771-797 Request permission

Abstract:

This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. In particular, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings.
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Additional Information
  • Helmut Harbrecht
  • Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland
  • Email: helmut.harbrecht@unibas.ch
  • Michael Peters
  • Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland
  • MR Author ID: 975028
  • Email: michael.peters@unibas.ch, markus@unibas.ch
  • Markus Siebenmorgen
  • Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland
  • Email: siebenmorgen@unibas.ch
  • Received by editor(s): November 6, 2013
  • Received by editor(s) in revised form: December 18, 2014, and July 6, 2015
  • Published electronically: March 24, 2016
  • Additional Notes: This research was supported by the Swiss National Science Foundation (SNSF) through the project “Rapid Solution of Boundary Value Problems on Stochastic Domains”.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 771-797
  • MSC (2010): Primary 65C05, 65C30, 60H25, 65N30
  • DOI: https://doi.org/10.1090/mcom/3107
  • MathSciNet review: 3584548