A stochastic algorithm to solve multiple dimensional Fredholm integral equations of the second kind (Q462963)

This article focuses on a new stochastic algorithm that is able to solve multiple dimensional Fredholm integral equations of the second kind. The authors describe the solution of the integral equation by a Neumann series expansion. Each term of this series expansion is interpreted as an expectation. The expectation is then approximated by a continuous Markov chain Monte Carlo method. Next, an algorithm is proposed to simulate such a continuous Markov chain with a probability density function that arises from an important sampling technique. The authors establish some theoretical results in a normed space. Precisely, they provide a convergence condition and an error estimation of the new proposed method. It is noteworthy that the method itself has a simple structure and can be easily parallized due to the fact that many independent sample paths are used for estimating the solution. Additionally, numerical results in two dimensions are presented to confirm the efficiency and the accuracy of the new stochastic algorithm.