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 quasiMonte 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.
|Number of pages||26|
|Journal||Mathematics of Computation|
|State||Published - 2021|
Bibliographical noteFunding Information:
Received by the editor January 16, 2020, and, in revised form, August 4, 2020, and September 1, 2020. 2020 Mathematics Subject Classification. Primary 65D30, 65C05, 11K38. Key words and phrases. Quasi-Monte Carlo integration, ANOVA space, worst case error, weighted discrepancy. 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”.
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- ANOVA space
- Quasi-Monte Carlo integration
- weighted discrepancy
- worst case error
ASJC Scopus subject areas
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics