Fast cubature of high dimensional biharmonic potential based on Approximate Approximations

TL;DR

This paper introduces new high-dimensional cubature formulas for the biharmonic potential that are fast, accurate, and effective even in extremely high dimensions, demonstrated by numerical tests up to 10^7 dimensions.

Contribution

The authors develop novel formulas for high-dimensional biharmonic potentials acting on Gaussians, enabling high-order, fast cubature in very high dimensions.

Findings

Formulas achieve O(h^8) approximation rate

Effective in dimensions up to 10^7

Numerical tests confirm accuracy and efficiency

Abstract

We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast and effective also in very high dimensions. Numerical tests show that the formulas are accurate and provide the predicted approximation rate (O(h^8)) up to the dimension 10^7.