Balancing of Systematic and Stochastic Errors in Monte Carlo Algorithms for Integral Equations

Abstract

The problem of balancing of both systematic and stochastic error is very important when Monte Carlo algorithms are used. A Monte Carlo method for integral equations based on balancing of systematic and stochastic errors is presented. An approach to the problem of controlling the error in non- deterministics methods is presented. The problem of obtaining an optimal ratio between the number of realizations $N$ of the random variable and the mean value $k$ of the number of steps in each random trajectory is discussed. Lower bounds for $N$ and $k$ are provided once a preliminary given error is given. Meaningful numerical examples and experiments are presented and discussed. Experimental and theoretical relative errors are presented. Monte Carlo algorithms with various initial and transition probabilities are compared. An almost optimal Monte Carlo algorithm is discussed and it is proven that it gives more reliable results.