Computational Aspects of the Monte-Carlo Method Vector Algorithm for Solving a Nonlinear Integral Equation

Abstract

It is usually impossible to obtain an exact solution of a nonlinear integral equation. The numerical solution of such equations is associated with significant difficulties. If a function that is a solution to a nonlinear equation depends on a large number of variables, then the complexity of the methods associated with the approximate replacement of the integral by the sum turns out to be very large. In the linear case this difficulty also occurs and is overcome in many cases by using the Monte Carlo method. To solve an integral equation with polynomial nonlinearity, a vector estimate is proposed constructed on realizations of a homogeneous branching Markov process. The advisability of using the proposed estimate is examined. The proposed estimate is based on the integration of part of the variables. Since such integration can only reduce the variance, the variance of the vector estimate will always be no more than the variance of the estimate proposed by the author earlier. In addition, the majorant conditions for the vector estimate are weaker than for the usual estimate. Using the example of quadratic nonlinearity, it is shown that performing summation to complete constructing a tree leads to a large amount of computation. Therefore, you must first construct a tree, and then perform the summation, starting with the absorption points on the trajectory. If we start from the beginning of the trajectory, then the memory required to memorize intermediate results will increase exponentially with an increase in the number of particles. Thus, it is impractical to perform calculations in direct order. The above reasoning is obviously true in the case of an equation of a higher degree. The results obtained expand the Monte Carlo method for solving a special class of problems.