The Fast and Free Memory Method for the efficient computation of convolution kernels

TL;DR

The paper presents the Fast Free Memory (FFM) method, a kernel-independent algorithm inspired by the Fast Multipole Method, enabling efficient convolution computations with linear storage and quasi-linear complexity, suitable for large-scale problems.

Contribution

The paper introduces the FFM algorithm, providing a complete description, complexity analysis, and demonstrating its application to large-scale scattering problems.

Findings

Able to evaluate convolution products with up to one billion entries

Successfully applied to solve large boundary integral equations with millions of unknowns

Achieves linear storage and quasi-linear computational complexity

Abstract

We introduce the Fast Free Memory method (FFM), a new fast method for the numerical evaluation of convolution products. Inheriting from the Fast Multipole Method, the FFM is a descent-only and kernel-independent algorithm. We give the complete algorithm and the relevant complexity analysis. While dense matrices arise normally in such computations, the linear storage complexity and the quasi-linear computational complexity enable the evaluation of convolution products featuring up to one billion entries. We show how we are able to solve complex scattering problems using Boundary Integral Equations with dozen of millions of unknowns. Our implementation is made freely available within the Gypsilab framework under the GPL 3.0 license.