Approximation of integral operators using convolution-product expansions

TL;DR

This paper introduces convolution-product expansions for efficiently approximating linear integral operators with varying impulse responses, enabling fast algorithms with optimal approximation and adaptivity.

Contribution

It develops novel convolution-product expansion techniques with explicit approximation rates, complexity analysis, and wavelet-based implementations for integral operators with time-varying kernels.

Findings

Explicit approximation rates for various expansions

Wavelet-based implementations with optimal performance

Reduced complexity and storage compared to standard methods

Abstract

We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computation-ally intensive problem necessary for many practical problems. We analyze a technique called convolution-product expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.