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

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.

💡 Research Summary

The paper addresses the problem of solving a class of Fredholm integral equations of the second kind whose solution is a probability measure. The canonical form of the equation is
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