On efficient weighted integration via a change of variables

P. Kritzer, F. Pillichshammer, L. Plaskota, G. W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

In this paper, we study the approximation of d-dimensional ϱ-weighted integrals over unbounded domains R+d or Rd using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded Lp norm of mixed partial derivatives of first order, where p∈ [1 , + ∞]. The main results give sufficient conditions on the change of variables ν which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on ϱ and p. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

Original languageEnglish
Pages (from-to)545-570
Number of pages26
JournalNumerische Mathematik
Volume146
Issue number3
DOIs
StatePublished - Nov 1 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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