## On quasi-Monte Carlo methods in weighted ANOVA spaces

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P. Kritzer, F. Pillichshammer and G. W. Wasilkowski
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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
- 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”. - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1381-1406 - MSC (2020): Primary 65D30, 65C05, 11K38
- DOI: https://doi.org/10.1090/mcom/3598
- MathSciNet review: 4232228