Monte Carlo Methods in Boundary Value Problems [Paperback]

This book deals with Random Walk Methods for solving multidimensional boundary value problems. Monte Carlo algorithms are constructed for three classes of problems: (1) potential theory, (2) elasticity, and (3) diffusion. Some of the advantages of our new methods as compared to conventional numerical methods are that they cater for stochasticities in the boundary value problems and complicated shapes of the boundaries.This book deals with Random Walk Methods for solving multidimensional boundary value problems. Monte Carlo algorithms are constructed for three classes of problems: (1) potential theory, (2) elasticity, and (3) diffusion. Some of the advantages of our new methods as compared to conventional numerical methods are that they cater for stochasticities in the boundary value problems and complicated shapes of the boundaries.1. General Schemes for Constructing Scalar and Vector Monte Carlo Alogorithms for Solving Boundary Value Problems.- 1.1 Random Walks on Boundary and Inside the Domain Algorithms.- 1.1.1 Monte Carlo Algorithms.- 1.1.2 Scalar and Vector Walk Inside the Domain Algorithms.- 1.1.3 Walk on Boundary Algorithms.- 1.1.4 Probabilistic Representations in the Form of Continual Integrals.- 1.2 Random Walks and Approximations of Random Processes.- 1.2.1 Walk Inside the Domain Processes.- 1.2.2 Walk on Boundary Processes.- 1.2.3 Approximation of Wiener Processes.- 1.2.4 Simulation of Random Fields.- 1.2.5 Stochastic Problems and Double Randomization.- 2. Monte Carlo Algorithms for Solving Integral Equations.- 2.1 Algorithms Based on Numerical Analytical Continuation.- 2.1.1 Statement of the Problem and the Main Definitions.- 2.1.2 Analytical Continuation of Neumann Series Based on the Spectral Parameter Transformation.- 2.1.3 Transformations of the Type ? = ?(?) = a0 + a1? + a2?2 +.- 2.2 Asymptotically Unbiased Estimates Based on Singular Approximation of the Kernel.- 2.2.1 Finite-Dimensional Case and One-Point Approximation.- 2.2.2 Systems of Integral l£5