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- W3186538844 abstract "This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values $(z(y_i))_{1leq ileq N}$ of the solution $z$ at a set of $N$ random and independent points $(y_i)_{1leq ileq N}$ are approximated by the solution $(z_{N,i})_{1leq ileq N}$ of a discrete $N$-dimensional linear system obtained by replacing the integral with the empirical average over the samples $(y_i)_{1leq ileq N}$. 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 $(z_{N,i})_{1leq ileq N}$ towards the point values $(z(y_i))_{1leq ileq N}$ 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 $(y_i)_{1leq ileq N}$. 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 $Nrightarrow +infty$ 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." @default.
- W3186538844 created "2021-08-02" @default.
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- W3186538844 date "2022-06-01" @default.
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- W3186538844 title "Analysis of a Monte-Carlo Nystrom Method" @default.
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- W3186538844 doi "https://doi.org/10.1137/21m1432338" @default.
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