Error Analysis of Biased Stochastic Algorithms for the Second Kind Fredholm Integral Equation

Abstract

In this work error analysis of biased stochastic algorithms for the second kind of Fredholm integral equation is considered. There are unbiased and biased stochastic algorithms, but the latter algorithms are more interesting, because there are two errors in there solutions—stochastic and systematic errors. An almost optimal Monte Carlo algorithm for integral equations in a combination with the idea of balancing of both systematic and stochastic errors is analysed. An optimal ratio between the number of realizations of the random variable and the number of iterations in the algorithm is studied. We considered two examples of integral equations that are widely used in computational physics and environmental sciences. We have shown that the almost optimal Monte Carlo algorithm based on balancing of the errors gives excellent results.