Optimization of Weighted Monte Carlo Methods

Weighted Monte Carlo algorithms are extremely useful when direct simulation techniques are inapplicable or ineffective. The methods presented in this book help to minimize computer time and memory required in constructing statistical models for systems described by integral equations. Approximate solutions of integral and differential equations serve as weighted functionals of special Markov chains. Variances of these solutions are minimized by (nonlinear) "importance" functions for the determination of which the author presents an asymptotic approach. Key points: Optimization of randomized algorithms for estimating probabilistic characteristics of equations with random parameters and applications; computational models for random fields and numerical simulations; vector Monte Carlo algorithms for solving systems of integral equations; a special approach to the application of perturbation theory based on this method.

The Monte Carlo method is based on the munerical realization of natural or artificial models of the phenomena under considerations. In contrast to classical computing methods the Monte Carlo efficiency depends weakly on the dimen sion and geometric details of the problem. The method is used for solving complex problems of the radiation transfer theory, turbulent diffusion, chemi cal kinetics, theory of rarefied gases, diffraction of waves on random surfaces, etc. The Monte Carlo method is especially effective when using multi-processor computing systems which allow many independent statistical experiments to be simulated simultaneously. The weighted Monte Carlo estimates are constructed in order to diminish errors and to obtain dependent estimates for the calculated functionals for different values of parameters of the problem, i.e., to improve the functional dependence. In addition, the weighted estimates make it possible to evaluate special functionals, for example, the derivatives with respect to the parameters. There are many works concerned with the development of the weighted estimates. In Chap. 1 we give the necessary information about these works and present a set of illustrations. The rest of the book is devoted to the solution of a series of mathematical problems related to the optimization of the weighted Monte Carlo estimates."