Method Monte-Carlo for solving of non-linear integral equations

TL;DR

This paper introduces a Monte-Carlo method for solving well-posed non-linear second Fredholm and Volterra integral equations, establishing confidence regions and proving convergence rates matching classical methods.

Contribution

The paper presents a novel Monte-Carlo approach for non-linear integral equations with confidence region construction and convergence analysis.

Findings

Method achieves convergence rate comparable to classical techniques.

Constructs confidence regions for solutions in uniform norm.

Applicable to well-posed second Fredholm and Volterra equations.

Abstract

We offer in this short report a simple Monte-Carlo method for solving a well-posed non-linear integral equations of second Fredholm's and Volterra's type and built a confidence region for solution in an uniform norm, applying the grounded Central Limit Theorem in the Banach space of continuous functions. We prove that the rate of convergence our method coincides with the classical one