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Tensor-Train Numerical Integration of Multivariate Functions with Singularities

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Abstract

Numerical integration is a classical problem emerging in many fields of science. Multivariate integration cannot be approached with classical methods due to the exponential growth of the number of quadrature nodes. We propose a method to overcome this problem. Tensor-train decomposition of a tensor approximating the integrand is constructed and used to evaluate a multivariate quadrature formula. We show how to deal with singularities in the integration domain and conduct theoretical analysis of the integration accuracy. The reference open-source implementation is provided.

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Notes

  1. They appear as ingredients in quantum chromodynamics (QCD) calculations and in supersymmetric \(N=4\) Yang–Mills theory within perturbation theory. These Feynman integrals are found in many important functions of these theories, in particular, various anomalous dimensions and form factors.

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Funding

The work is supported by Russian Ministry of Science and Higher Education, agreement no. 075-15-2019-1621.

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Authors and Affiliations

  1. HSE University, Moscow, Russia

    L. I. Vysotsky

  2. Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    L. I. Vysotsky

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    L. I. Vysotsky, A. V. Smirnov & E. E. Tyrtyshnikov

  4. Research Computing Center, Moscow State University, 119991, Moscow, Russia

    A. V. Smirnov & E. E. Tyrtyshnikov

  5. Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences, 119991, Moscow, Russia

    E. E. Tyrtyshnikov

Authors
  1. L. I. Vysotsky
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  2. A. V. Smirnov
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  3. E. E. Tyrtyshnikov
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Corresponding authors

Correspondence to L. I. Vysotsky, A. V. Smirnov or E. E. Tyrtyshnikov.

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(Submitted by V. V. Voevodin)

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Vysotsky, L.I., Smirnov, A.V. & Tyrtyshnikov, E.E. Tensor-Train Numerical Integration of Multivariate Functions with Singularities. Lobachevskii J Math 42, 1608–1621 (2021). https://doi.org/10.1134/S1995080221070258

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