Truncation dimension for linear problems on multivariate function spaces

Aicke Hinrichs; Peter Kritzer; Friedrich Pillichshammer; G. W. Wasilkowski

doi:10.1007/s11075-018-0501-7

Truncation dimension for linear problems on multivariate function spaces

Aicke Hinrichs, Peter Kritzer, Friedrich Pillichshammer, G. W. Wasilkowski

Computer Science

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

7 Scopus citations

Abstract

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are the following: when is it possible to approximate the solution for the original function of very many variables by the solution for the same function, however with all but the first k variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number k = k(ε) such that the corresponding error is bounded by a given error demand ε? Surprisingly, k(ε) could be very small even for weights with a modest speed of convergence to zero.

Original language	English
Pages (from-to)	661-685
Number of pages	25
Journal	Numerical Algorithms
Volume	80
Issue number	2
DOIs	https://doi.org/10.1007/s11075-018-0501-7
State	Published - Feb 6 2019

Bibliographical note

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Multivariate problems
Truncation algorithms
Truncation dimension
Weighted function spaces

ASJC Scopus subject areas

Applied Mathematics

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10.1007/s11075-018-0501-7

Cite this

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AU - Kritzer, Peter

AU - Pillichshammer, Friedrich

AU - Wasilkowski, G. W.

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N2 - The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are the following: when is it possible to approximate the solution for the original function of very many variables by the solution for the same function, however with all but the first k variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number k = k(ε) such that the corresponding error is bounded by a given error demand ε? Surprisingly, k(ε) could be very small even for weights with a modest speed of convergence to zero.

AB - The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are the following: when is it possible to approximate the solution for the original function of very many variables by the solution for the same function, however with all but the first k variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number k = k(ε) such that the corresponding error is bounded by a given error demand ε? Surprisingly, k(ε) could be very small even for weights with a modest speed of convergence to zero.

KW - Multivariate problems

KW - Truncation algorithms

KW - Truncation dimension

KW - Weighted function spaces

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Truncation dimension for linear problems on multivariate function spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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