A Fast Summation Method for translation invariant kernels

TL;DR

This paper introduces the Empirical Interpolation Fast Multipole Method (EIFMM), a novel algorithm for efficiently approximating translation-invariant kernels with adaptive error control, improving performance for inhomogeneous kernels.

Contribution

The paper develops EIFMM, a new FMM variant using the Empirical Interpolation Method that adapts to the kernel and provides built-in error estimation for better accuracy and efficiency.

Findings

EIFMM achieves faster computations for translation-invariant kernels.

The method adaptively selects interpolation points based on kernel evaluations.

EIFMM provides reliable error estimates, optimizing the number of interpolation points.

Abstract

We derive a Fast Multipole Method (FMM) where a low-rank approximation of the kernel is obtained using the Empirical Interpolation Method (EIM). Contrary to classical interpolation-based FMM, where the interpolation points and basis are fixed beforehand, the EIM is a nonlinear approximation method which constructs interpolation points and basis which are adapted to the kernel under consideration. The basis functions are obtained using evaluations of the kernel itself. We restrict ourselves to translation-invariant kernels, for which a modified version of the EIM approximation can be used in a multilevel FMM context; we call the obtained algorithm Empirical Interpolation Fast Multipole Method (EIFMM). An important feature of the EIFMM is a built-in error estimation of the interpolation error made by the low-rank approximation of the far-field behavior of the kernel: the algorithm selects the optimal number of interpolation points required to ensure a given accuracy for the result, leading to important gains for inhomogeneous kernels.