Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

M. Gnewuch; M. Hefter; A. Hinrichs; K. Ritter; G. W. Wasilkowski

doi:10.1016/j.jco.2019.04.002

Embeddings for infinite-dimensional integration and L₂-approximation with increasing smoothness

M. Gnewuch, M. Hefter, A. Hinrichs, K. Ritter, G. W. Wasilkowski

Computer Science

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

6 Scopus citations

Abstract

We study integration and L₂-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

Original language	English
Article number	101406
Journal	Journal of Complexity
Volume	54
DOIs	https://doi.org/10.1016/j.jco.2019.04.002
State	Published - Oct 2019

Bibliographical note

Funding Information:
M. Hefter was supported by the Austrian Science Fund (FWF) , Project F5506-N26 . A. Hinrichs is supported by the Austrian Science Fund (FWF) , Project F5513-N26 . Both projects are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. K. Ritter was partially supported as a visiting professor at Kiel University, Germany .

Funding Information:
The authors would like to thank Michael Griebel and Henryk Woźniakowski for interesting discussions and valuable comments. The work on this paper was initiated during a special semester at the Institute for Computational and Experimental Research in Mathematics (ICERM) of Brown University. Part of the work was done during a visit of the University of New South Wales (UNSW) in Sydney and during a special semester at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. M. Hefter was supported by the Austrian Science Fund (FWF), Project F5506-N26. A. Hinrichs is supported by the Austrian Science Fund (FWF), Project F5513-N26. Both projects are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. K. Ritter was partially supported as a visiting professor at Kiel University, Germany.

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

Embedding theorems
High-dimensional integration
Infinite-dimensional integration
Reproducing kernel Hilbert spaces
Tractability

ASJC Scopus subject areas

Algebra and Number Theory
Statistics and Probability
Numerical Analysis
Mathematics (all)
Control and Optimization
Applied Mathematics

Access to Document

10.1016/j.jco.2019.04.002

Cite this

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abstract = "We study integration and L2-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.",

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note = "Funding Information: M. Hefter was supported by the Austrian Science Fund (FWF) , Project F5506-N26 . A. Hinrichs is supported by the Austrian Science Fund (FWF) , Project F5513-N26 . Both projects are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. K. Ritter was partially supported as a visiting professor at Kiel University, Germany . Funding Information: The authors would like to thank Michael Griebel and Henryk Wo{\'z}niakowski for interesting discussions and valuable comments. The work on this paper was initiated during a special semester at the Institute for Computational and Experimental Research in Mathematics (ICERM) of Brown University. Part of the work was done during a visit of the University of New South Wales (UNSW) in Sydney and during a special semester at the Erwin Schr{\"o}dinger International Institute for Mathematics and Physics (ESI) in Vienna. M. Hefter was supported by the Austrian Science Fund (FWF), Project F5506-N26. A. Hinrichs is supported by the Austrian Science Fund (FWF), Project F5513-N26. Both projects are part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. K. Ritter was partially supported as a visiting professor at Kiel University, Germany. Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

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