On the fast computation of high dimensional volume potentials

TL;DR

This paper introduces a high-order, fast computational method for high-dimensional volume potentials that scales linearly with dimension, demonstrated by numerical experiments achieving high approximation accuracy.

Contribution

It presents a novel high-order approximation technique using basis functions from approximate approximation theory, effective in very high dimensions.

Findings

Achieves approximation order O(h^8) for Newton potential in high dimensions.

Computational time scales linearly with the space dimension.

Provides new integral representations for advection-diffusion and heat potentials.

Abstract

A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order O(h^8) for the Newton potential in high dimensions, for example, for n= 200 000, are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.