A Particle Method for Solving Fredholm Equations of the First Kind

TL;DR

This paper introduces a novel particle method for solving Fredholm equations of the first kind, addressing issues of discretization and regularity assumptions, and demonstrating superior performance in practical inverse problems.

Contribution

A new particle-based algorithm that approximates the EMS scheme, enabling adaptive discretization and smooth solutions without strong regularity assumptions.

Findings

Outperforms standard EMS in realistic inverse problems

Provides a Monte Carlo approximation with adaptive discretization

Achieves smooth solutions for ill-posed integral equations

Abstract

Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-section images of the brain from positron emission tomography.