An accurate and quadrature-free evaluation of multipole expansion of functions represented by multiwavelets

TL;DR

This paper introduces a quadrature-free, highly accurate method for converting functions represented by multiwavelets into their multipole expansions, crucial for fast multipole methods, especially for oscillatory and near-singular kernels.

Contribution

The authors develop a novel, quadrature-free approach using series expansions of Bessel functions and solid harmonics for efficient, accurate multipole-multiwavelet conversion.

Findings

Achieves machine precision accuracy in conversion

Provides a fast and reliable computational scheme

Handles near-singular and oscillatory kernels effectively

Abstract

We present formulas for accurate numerical conversion between functions represented by multiwavelets and their multipole/local expansions with respect to the kernel of the form, $e^{\lambda r}/r$. The conversion is essential for the application of fast multipole methods for functions represented by multiwavelets. The corresponding separated kernels exhibit near-singular behaviors at large $\lambda$. Moreover, a multiwavelet basis function oscillates more wildly as its degree increases. These characteristics in combination render any brute-force approach based on numerical quadratures impractical. Our approach utilizes the series expansions of the modified spherical Bessel functions and the Cartesian expansions of solid harmonics so that the multipole-multiwavelet conversion matrix can be evaluated like a special function. The result is a quadrature-free, fast, reliable, and machine precision accurate scheme to compute the conversion matrix with predictable sparsity patterns.