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Analysis of a Monte-Carlo Nystrom method

by F. Feppon and H. Ammari

(Report number 2021-19)

Abstract
This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values \((z(y_i))_{1\leqslant i\leqslant N}\) of the solution \(z\) at a set of \(N\) random and independent points \((y_i)_{1\leqslant i\leqslant N}\) are approximated by the solution \((z_{N,i})_{1\leqslant i\leqslant N}\) of a discrete, \(N\)-dimensional linear system obtained by replacing the integral with the empirical average over the samples \((y_i)_{1\leqslant i\leqslant N}\). 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 \((z_{N,i})_{1\leqslant i\leqslant N}\) towards the point values \((z(y_i))_{1\leqslant i\leqslant N}\) in a mean-square sense at a rate \(O(N^{-\frac{1}{2}})\). For 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 \((y_i)_{1\leqslant i\leqslant N}\). In this context, our result can equivalently be 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 1D examples and for solving a 2D Lippmann-Schwinger equation.

Keywords: Monte-Carlo method, Nystrom method, Foldy-Lax approximation, point scatterers, effective medium.

@Techreport{FA21_961,
  author = {F. Feppon and H. Ammari},
  title = {Analysis of a Monte-Carlo Nystrom method},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-19.pdf },
  year = {2021}
}

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