ON QUASI-MONTE CARLO METHODS IN WEIGHTED ANOVA SPACES

P. Kritzer; F. Pillichshammer; G. W. Wasilkowski

doi:10.1090/mcom/3598

ON QUASI-MONTE CARLO METHODS IN WEIGHTED ANOVA SPACES

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

Computer Science

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Pages (from-to)	1381-1406
Number of pages	26
Journal	Mathematics of Computation
Volume	90
Issue number	329
DOIs	https://doi.org/10.1090/mcom/3598
State	Published - 2021

Bibliographical note

Funding 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”.

Publisher Copyright:
© 2021 American Mathematical Society. All Rights Reserved.

Keywords

ANOVA space
Quasi-Monte Carlo integration
weighted discrepancy
worst case error

ASJC Scopus subject areas

Algebra and Number Theory
Computational Mathematics
Applied Mathematics

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10.1090/mcom/3598

Cite this

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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 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.",

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author = "P. Kritzer and F. Pillichshammer and Wasilkowski, {G. W.}",

note = "Funding 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”. Publisher Copyright: {\textcopyright} 2021 American Mathematical Society. All Rights Reserved.",

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AU - Wasilkowski, G. W.

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PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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ER -

ON QUASI-MONTE CARLO METHODS IN WEIGHTED ANOVA SPACES

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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