Optimization of Weighted Monte Carlo Methods

1. 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.