Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows

TL;DR

This paper introduces a grid-free, scalable algorithm for solving Fredholm integral equations of the first kind using Wasserstein gradient flows and particle approximations, with proven convergence and practical applications.

Contribution

It proposes a novel particle-based algorithm leveraging Wasserstein gradient flows for regularized Fredholm equations, avoiding domain discretization and improving scalability.

Findings

Algorithm converges to the unique solution

Outperforms classical discretization methods

Effective in applications like density deconvolution and epidemiology

Abstract

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.