Improved Monte-Carlo method for solving of integral Fredholm's equations of a second kind, with confidence regions in the uniform norm

TL;DR

This paper presents a modified Monte-Carlo method for solving second-kind Fredholm integral equations, demonstrating optimal convergence rates and constructing confidence regions in the uniform norm.

Contribution

It introduces a novel modification to the Monte-Carlo method that achieves optimal convergence and provides confidence regions in the uniform norm.

Findings

Proves the optimal convergence rate of the modified method.

Constructs asymptotic and non-asymptotic confidence regions.

Validates the method's effectiveness in solving Fredholm equations.

Abstract

We offer in this article some modification of Monte-Carlo method for solving of a linear integral Fredholm's equation of a second kind (Fredholm's well posed problem). We prove that the rate of convergence of offered method is optimal under natural conditions still in an uniform norm, and construct an asymptotic as well as non-asymptotic confidence region, again in the uniform norm.