Fast discrete convolution in $\mathbb{R}^2$ using Sparse Bessel Decomposition

TL;DR

This paper introduces an efficient algorithm for fast discrete convolution in 2D using Sparse Bessel Decomposition, extending previous 3D methods, with detailed complexity analysis and numerical validation.

Contribution

It extends the Sparse Cardinal Sine Decomposition to 2D, providing a detailed complexity analysis and demonstrating efficiency through large-scale numerical tests.

Findings

Efficient computation of boundary integral equations in 2D.

Successful handling of large systems up to 10^7 points.

Numerical results confirm the method's speed and accuracy.

Abstract

We describe an efficient algorithm for computing the matrix vector products that appear in the numerical resolution of boundary integral equations in 2 space dimension. This work is an extension of the so-called Sparse Cardinal Sine Decomposition algorithm by Alouges et al., which is restricted to three-dimensional setups. Although the approach is similar, significant differences appear throughout the analysis of the method. Bessel decomposition, in particular, yield longer series for the same accuracy. We propose a careful study of the method that leads to a precise estimation of the complexity in terms of the number of points and chosen accuracy. We also provide numerical tests to demonstrate the efficiency of this approach. We give the compression performance for a $N \times N$ linear system for several values $N$ up to $10^7$ and report the computation time for the off-line and on-line parts of our algorithm. We also include a toy application to sound canceling to further illustrate the efficiency of our method.