Optimized Stochastic Approaches Based on Sobol Quasirandom Sequences for Fredholm Integral Equations of the Second Kind

In this paper three possible approaches to compute linear functionals of the solution of the Fredholm integral equation of the second kind are under consideration: a biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series; transformation of this problem into the problem of computing a finite number of integrals, and an unbiased stochastic approach. In the second case several Monte Carlo algorithms for numerical integration have been applied including optimized stochastic approaches developed in our previous studies. The unbiased stochastic approach has been applied to a multidimensional numerical example. A comprehensive analysis about the reliability and the efficiency of the algorithms has been done.