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.
Table of Contents1. Mathematical Models of Weighted Monte Carlo Methods.- 1.1 Simple Facts from Functional Analysis.- 1.2 Simple Facts from Convergence Theory for Random Functions.- 1.3 Integral Equations of the Transfer Theory and Monte Carlo Methods.- 1.4 Other Integral Equations Solved by Monte Carlo Methods.- 1.5 Monte Carlo Methods for Calculating Integrals.- 1.6 Unbiasedness and Variance of Monte Carlo Methods.- 1.7 Weighted Estimates for Bilinear Functionals.- 1.8 Calculation of the Derivatives of the Linear Functionals and the Weak Convergence of the Functional Estimates.- 2. Using Information About the Solution.- 2.1 Importance Sampling Technique.- 2.2 Weighted Path Estimates in the Transfer Theory.- 2.3 Estimation of the Variance D?x for Importance Sampling Technique.- 2.4 Using the Asymptotic Solution to the One-Velocity Transfer Equation.- 3. Nonlinear Theory of Optimization for Solving Integral Equations.- 3.1 Formulation of the Problem.- 3.2 Investigation of the Master Equation.- 3.3 A Model Problem.- 3.4 Asymptotic Optimization of the Radiative Transfer.- 3.5 Asymptotic Optimization in a Special Class of Densities.- 3.6 Minimization of the Variance of the Collision Estimates.- 4. Minimax Weighted Estimates.- 4.1 Statement of the Problem. The Basic Lemma.- 4.2 The Minimax Estimates for the Integrals.- 4.3 Optimization of Estimates for the Integral Equations.- 4.4 Minimax Choice of the First Step in the Markov Chain.- 5. Vector Monte Carlo Algorithms.- 5.1 Variance Vector Algorithms.- 5.2 Uniform Optimization of Weighted Monte Carlo Estimates in the Transfer Theory.- 5.3 Vector Algorithm Related to a Stratified Sampling with Respect to One Variable.- 5.4 Accuracy of the Monte Carlo Method for Solving the Vector Transfer Equation.- 5.5 Vector Estimates for Triangular Matrix Kernel.- 5.6 Vector Estimates for the Resolvent Iterations.- 5.7 Vector Representations of Bilinear Estimates.- 5.8 Vector Algorithm for Evaluating the Effective Fission Coefficient.- 5.9 Variance Reduction for the Vector Estimates.- 5.10 Asymptotic Investigation of a Monte Carlo Method Combined with the Method of Finite Sums.- 6. Randomization of Weighted Algorithms.- 6.1 Randomized Estimation for Statistical Moments of the Solution.- 6.2 Lower Bound of the Variance. Averaging Exponential Kernels.- 6.3 Special Models of Non-Gaussian Random Fields Related to Stationary Point Fluxes.- 6.4 Simulation of Homogeneous Gaussian Fields by Randomization of the Spectral Representation.- 6.5 Stochastic Problems of Radiative Transfer Theory.- 6.6 A Stochastic Elasticity Problem.- 6.7 Simulation of Admixture Diffusion in Stochastic Velocity Fields.- 7. The Method of Multiple Splitting.- 7.1 Optimization of the Splitting Method.- 7.2 Optimization of the Splitting Technique for Calculating the Transmission Probability.- 7.3 Numerical Calculation of the Optimal Splitting Parameters.- 7.4 Uniform Optimization of the Splitting Method.- 7.5 Randomized Splitting Method.- 7.6 Splitting of the Collision Estimate.- 8. Transformation of Equations and Weighted Estimates.- 8.1 The Averaging Transformation.- 8.2 Translations.- 8.3 Some Relations Between the Variances.- 8.4 Notions on the Functional Convergence of the Estimates.- 9. Monte Carlo Methods and Perturbation Theory.- 9.1 Vector Weighted Monte Carlo Methods.- 9.2 Differentiation of Integral Equations with Respect to a Parameter.- 9.3 Calculation of Perturbations.- 9.4 Calculation of Derivatives.- 9.5 Calculation of Perturbations in the Transfer Theory.- 9.6 Calculation of Derivatives of Solutions to Boundary Value Problems by the Monte Carlo Method.- Appendix. Models of Random Variables.- A.1 Simulation of Random Variables.- A.2 Simulation of Random Vectors.- References.