MINIMAX MONTE-CARLO METHODS FOR SOLVING INTEGRAL-EQUATIONS OF THE 2ND KIND

A Markov chain minimizing the maximum of the weighting function that determines the variance of the standard Monte-Carlo estimators of linear functionals of the solution of an integral equation of the 2nd kind is constructed. The results are extended to vector estimators associated with the simultaneous solution of many integral equations by simulation of a single Markov chain. The uniform optimization of "track estimators" and local estimators for the solution of the transport problem are also considered. It is shown how an approximate optimization can be constructed using asymptotic solutions. An example for a simple radiative transfer model with "delta-scattering" is considered.