arXiv:2103.12129·v1·math.NA
Tensor-Train Numerical Integration of Multivariate Functions with Singularities
Lev I. Vysotsky (1,2 and 3)🇷🇺 · Alexander V. Smirnov (4 and 3)🇷🇺 · Eugene E. Tyrtyshnikov (5 and 4) ((1) HSE University, (2) Faculty of Computational Mathematics and Cybernetics of Moscow State University, (3) Moscow Center for Fundamental and Applied Mathematics, (4) Research Computing Center of Moscow State University, (5) Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences)🇷🇺
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.
Comments: 12 pages, 1 PostScript figure