Fast Adaptive Algorithms in the Non-Standard Form for Multidimensional Problems

TL;DR

This paper introduces a fast, adaptive multiresolution algorithm for applying integral operators with radially symmetric kernels in multiple dimensions, leveraging separated representations for efficiency and broad applicability.

Contribution

The paper presents a novel multiresolution algorithm that efficiently handles a wide class of radially symmetric kernels, extendable to higher dimensions and applicable across various fields.

Findings

Comparable in speed to the Fast Multipole Method

Provides controllable accuracy at reasonable computational cost

Applicable to multiple operators with radially symmetric kernels

Abstract

We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class $(-\Delta+\mu^{2}I)^{-\alpha}$, where $\mu\geq0$ and $0<\alpha<3/2$, and illustrate the algorithm for the Poisson and Schr\"{o}dinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has an advantage of being easily extendable to higher dimensions.