Fast cubature of volume potentials over rectangular domains

TL;DR

This paper introduces high-order cubature formulas for efficiently computing volume potentials over rectangular domains, achieving high accuracy and scalability in very high dimensions using tensor product representations.

Contribution

It develops a novel tensor product-based cubature method for high-dimensional volume potentials using approximate approximation basis functions.

Findings

Achieves $O(h^6)$ accuracy in high dimensions

Efficient cubature in dimensions up to 10^8

Numerical tests confirm high accuracy and efficiency

Abstract

In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order $O(h^6)$ up to dimension $10^8$.