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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Integration in reproducing kernel Hilbert spaces of Gaussian kernels
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by Toni Karvonen, Chris J. Oates and Mark Girolami HTML | PDF
Math. Comp. 90 (2021), 2209-2233 Request permission

Abstract:

The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm for efficient numerical integration of functions reproduced by Gaussian kernels has not been fully solved. In this article we construct two classes of algorithms that use $N$ evaluations to integrate $d$-variate functions reproduced by Gaussian kernels and prove the exponential or super-algebraic decay of their worst-case errors. In contrast to earlier work, no constraints are placed on the length-scale parameter of the Gaussian kernel. The first class of algorithms is obtained via an appropriate scaling of the classical Gauss–Hermite rules. For these algorithms we derive lower and upper bounds on the worst-case error of the forms $\exp (-c_1 N^{1/d}) N^{1/(4d)}$ and $\exp (-c_2 N^{1/d}) N^{-1/(4d)}$, respectively, for positive constants $c_1 > c_2$. The second class of algorithms we construct is more flexible and uses worst-case optimal weights for points that may be taken as a nested sequence. For these algorithms we derive upper bounds of the form $\exp (-c_3 N^{1/(2d)})$ for a positive constant $c_3$.
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Additional Information
  • Toni Karvonen
  • Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom
  • MR Author ID: 1256975
  • ORCID: 0000-0002-5984-7295
  • Chris J. Oates
  • Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom; and School of Mathematics, Statistics & Physics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
  • MR Author ID: 998570
  • Mark Girolami
  • Affiliation: The Alan Turing Institute, London NW1 2DB, United Kingdom; and Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
  • MR Author ID: 665653
  • ORCID: 0000-0003-3008-253X
  • Received by editor(s): July 1, 2020
  • Received by editor(s) in revised form: December 16, 2020
  • Published electronically: June 18, 2021
  • Additional Notes: The authors were supported by the Lloyd’s Register Foundation programme on data-centric engineering at the Alan Turing Institute, United Kingdom
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 2209-2233
  • MSC (2020): Primary 65D30, 46E22; Secondary 65D12, 41A25
  • DOI: https://doi.org/10.1090/mcom/3659
  • MathSciNet review: 4280298