Analysis of a Monte-Carlo Nystrom Method

Abstract

This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values $(z(y_i))_{1\leq i\leq N}$ of the solution $z$ at a set of $N$ random and independent points $(y_i)_{1\leq i\leq N}$ are approximated by the solution $(z_{N,i})_{1\leq i\leq N}$ of a discrete $N$-dimensional linear system obtained by replacing the integral with the empirical average over the samples $(y_i)_{1\leq i\leq N}$. Under the unique assumption that the integral equation admits a unique solution $z(y)$, we prove the invertibility of the linear system for sufficiently large $N$ with probability one, and the convergence of the solution $(z_{N,i})_{1\leq i\leq N}$ towards the point values $(z(y_i))_{1\leq i\leq N}$ in a mean-square sense at a rate $O(N^{-\frac{1}{2}})$. For a particular choices of kernels, the discrete linear system arises as the Foldy--Lax approximation for the scattered field generated by a system of $N$ sources emitting waves at the points $(y_i)_{1\leq i\leq N}$. In this context, our result can be equivalently considered as a proof of the well-posedness of the Foldy--Lax approximation for systems of $N$ point scatterers, and of its convergence as $N\rightarrow +\infty$ in a mean-square sense to the solution of a Lippmann--Schwinger equation characterizing the effective medium. The convergence of Monte-Carlo solutions at the rate $O(N^{-1/2})$ is numerically illustrated on one-dimensional examples and for solving a two-dimensional Lippmann--Schwinger equation.