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Truncation dimension for linear problems on multivariate function spaces

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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.

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Acknowledgments

The authors would like to thank two anonymous referees for their suggestions for improving the paper.

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Authors and Affiliations

  1. Institute for Analysis, Johannes Kepler University Linz, Altenbergerstr. 69, 4040, Linz, Austria

    Aicke Hinrichs

  2. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstr. 69, 4040, Linz, Austria

    Peter Kritzer

  3. Department of Financial Mathematics and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, 4040, Linz, Austria

    Friedrich Pillichshammer

  4. Computer Science Department, University of Kentucky, 301 David Marksbury Building, Lexington, KY, 40506, USA

    G. W. Wasilkowski

Authors
  1. Aicke Hinrichs
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  2. Peter Kritzer
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  3. Friedrich Pillichshammer
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  4. G. W. Wasilkowski
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Corresponding author

Correspondence to Peter Kritzer.

Additional information

A. Hinrichs, P. Kritzer, and F. Pillichshammer gratefully acknowledge the support of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna under the thematic program “Tractability of High Dimensional Problems and Discrepancy”.

A. Hinrichs is supported by the Austrian Science Fund (FWF), Project F5513-N26.

P. Kritzer is supported by the Austrian Science Fund (FWF), Project F5506-N26.

P. Kritzer and G.W. Wasilkowski acknowledge the support of the Statistical and Applied Mathematical Sciences Institute (SAMSI).

F. Pillichshammer is supported by the Austrian Science Fund (FWF) Project F5509-N26.

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Hinrichs, A., Kritzer, P., Pillichshammer, F. et al. Truncation dimension for linear problems on multivariate function spaces. Numer Algor 80, 661–685 (2019). https://doi.org/10.1007/s11075-018-0501-7

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  • DOI: https://doi.org/10.1007/s11075-018-0501-7

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