Analysis of a Monte-Carlo Nystrom Method
نویسندگان
چکیده
This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby values $(z(y_i))_{1\leq i\leq N}$ solution $z$ at set $N$ random and independent points $(y_i)_{1\leq are approximated by $(z_{N,i})_{1\leq discrete $N$-dimensional linear system obtained replacing with empirical average over samples N}$. Under unique assumption that equation admits $z(y)$, we prove invertibility sufficiently large probability one, convergence towards point in mean-square sense rate $O(N^{-\frac{1}{2}})$. For particular choices kernels, arises as Foldy--Lax approximation scattered field generated sources emitting waves In this context, our result can be equivalently considered proof well-posedness systems scatterers, its $N\rightarrow +\infty$ to Lippmann--Schwinger characterizing effective medium. The solutions $O(N^{-1/2})$ is numerically illustrated on one-dimensional examples two-dimensional equation.
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ژورنال
عنوان ژورنال: SIAM Journal on Numerical Analysis
سال: 2022
ISSN: ['0036-1429', '1095-7170']
DOI: https://doi.org/10.1137/21m1432338