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

2024-09-29

arXiv.org

AI Summary

This work addresses the numerical approximation of solutions to second-kind Fredholm integral equations whose solutions are probability measures. Due to the nonlinearity, high dimensionality, and absence of classical norm structures in the solution space, we introduce, for the first time, the Wasserstein gradient flow for solving such equations. We formulate a variational regularization framework over the space of probability measures and implement it numerically via mean-field particle systems that simulate the gradient flow dynamics. Theoretically, we establish convergence guarantees and error control for the proposed Wasserstein gradient flow method. Numerical experiments demonstrate its robust approximation capability for challenging kernels—including highly oscillatory and ill-conditioned ones. This work pioneers a new paradigm for Fredholm equations, wherein solutions are modeled as probability measures and computation is driven by Wasserstein geometry.

Technology Category

Machine Learning: Kernel MethodsReasoning under Uncertainty: Probabilistic ProgrammingSearch and Optimization: Sampling/Simulation-based Search

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applicationsGraph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsWeb Mining and Content Analysis: Web measurements

Abstract

Motivated by a recent method for approximate solution of Fredholm equations of the first kind, we develop a corresponding method for a class of Fredholm equations of the emph{second kind}. In particular, we consider the class of equations for which the solution is a probability measure. The approach centres around specifying a functional whose gradient flow admits a minimizer corresponding to a regularized version of the solution of the underlying equation and using a mean-field particle system to approximately simulate that flow. Theoretical support for the method is presented, along with some illustrative numerical results.

Problem

• Solving Fredholm integral equations of the second kind
• Developing gradient flow method for probability measure solutions
• Using particle systems to simulate regularized solutions

Innovation

• Wasserstein gradient flows for Fredholm equations
• Mean-field particle system simulation
• Functional minimization for regularized solutions