Tensor product approximations of high dimensional potentials

TL;DR

This paper introduces a tensor product approximation method for efficiently computing high-dimensional volume potentials, significantly reducing computational resources by transforming convolutions into one-dimensional integrals with low-rank tensor representations.

Contribution

It combines approximate approximations with structured tensor methods to enable efficient high-order cubature formulas for high-dimensional potentials.

Findings

Reduces high-dimensional convolutions to one-dimensional integrals

Achieves low-rank tensor approximations for harmonic and Yukawa potentials

Significantly decreases computational resources required for high-dimensional volume potentials

Abstract

The paper is devoted to the efficient computation of high-order cubature formulas for volume potentials obtained within the framework of approximate approximations. We combine this approach with modern methods of structured tensor product approximations. Instead of performing high-dimensional discrete convolutions the cubature of the potentials can be reduced to a certain number of one-dimensional convolutions leading to a considerable reduction of computing resources. We propose one-dimensional integral representions of high-order cubature formulas for n-dimensional harmonic and Yukawa potentials, which allow low rank tensor product approximations.