2022/09/16 by Francesca R. Crucinio, Valentin De Bortoli, Crucinio, Francesca R. +5
Mathematics · Physics and Astronomy · #45B05 #65C35 #65K10 #65Rxx #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Methodology (stat.ME) #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2209.09936
openalex publication_date 2022/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.