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On quasi-Monte Carlo methods in weighted ANOVA spaces


Authors: P. Kritzer, F. Pillichshammer and G. W. Wasilkowski
Journal: Math. Comp. 90 (2021), 1381-1406
MSC (2020): Primary 65D30, 65C05, 11K38
DOI: https://doi.org/10.1090/mcom/3598
Published electronically: January 11, 2021
MathSciNet review: 4232228
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Abstract:

In the present paper we study quasi-Monte Carlo rules for approximating integrals over the $d$-dimensional unit cube for functions from weighted Sobolev spaces of regularity one. While the properties of these rules are well understood for anchored Sobolev spaces, this is not the case for the ANOVA spaces, which are another very important type of reference spaces for quasi-Monte Carlo rules.

Using a direct approach we provide a formula for the worst case error of quasi-Monte Carlo rules for functions from weighted ANOVA spaces. As a consequence we bound the worst case error from above in terms of weighted discrepancy of the employed integration nodes. On the other hand we also obtain a general lower bound in terms of the number $n$ of used integration nodes.

For the one-dimensional case our results lead to the optimal integration rule and also in the two-dimensional case we provide rules yielding optimal convergence rates.


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Additional Information

P. Kritzer
Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria
MR Author ID: 773334
ORCID: 0000-0002-7919-7672
Email: peter.kritzer@oeaw.ac.at

F. Pillichshammer
Affiliation: Institut für Finanzmathematik und Angewandte Zahlentheorie, Johannes Kepler Universität Linz, Altenbergerstr. 69, 4040 Linz, Austria
MR Author ID: 661956
ORCID: 0000-0001-6952-9218
Email: friedrich.pillichshammer@jku.at

G. W. Wasilkowski
Affiliation: Computer Science Department, University of Kentucky, 301 David Marksbury Building, 329 Rose Street, Lexington, Kentucky 40506
MR Author ID: 189251
ORCID: 0000-0003-4727-7368
Email: greg@cs.uky.edu

Keywords: Quasi-Monte Carlo integration, ANOVA space, worst case error, weighted discrepancy
Received by editor(s): January 16, 2020
Received by editor(s) in revised form: August 4, 2020, and September 1, 2020
Published electronically: January 11, 2021
Additional Notes: The first author was supported by the Austrian Science Fund (FWF): Project F5506-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
The second author was supported by the Austrian Science Fund (FWF): Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
Article copyright: © Copyright 2021 American Mathematical Society