Tensor-Train Numerical Integration of Multivariate Functions with Singularities

TL;DR

This paper introduces a tensor-train based numerical integration method for multivariate functions with singularities, effectively addressing the curse of dimensionality and providing theoretical accuracy analysis and open-source implementation.

Contribution

It presents a novel tensor-train decomposition approach for multivariate integration that handles singularities and improves computational efficiency.

Findings

Effective handling of singularities in multivariate integration

Theoretical analysis confirms accuracy of the method

Open-source implementation available

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.