Monte Carlo Methods in Boundary Value Problems

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.- 1.2 Random Walks and Approximations of Random Processes.- 2. Monte Carlo Algorithms for Solving Integral Equations.- 2.1 Algorithms Based on Numerical Analytical Continuation.- 2.2 Asymptotically Unbiased Estimates Based on Singular Approximation of the Kernel.- 2.3 The Eigen-value Problem for the Integral Operators.- 2.4 Alternative Constructions of the Resolvent: Modifications and Numerical Experiments.- 3. Monte Carlo Algorithms for Solving Boundary Value Problems of the Potential Theory.- 3.1 The Walk on Boundary Algorithms for Solving Interior and Exterior Boundary Value Problems of the Potential Theory.- 3.2 Walk Inside the Domain Algorithms.- 3.3 Numerical Solution of Some Test and Applied Problems of Potential Theory in Deterministic and Stochastic Formulation.- 4. Monte Carlo Algorithms for Solving High-Order Equations and the Elasticity Problems.- 4.1 Biharmonic Problem.- 4.2 Metaharmonic Equations.- 4.3 Spatial Problems of the Elasticity Theory.- 4.4 Application to Stochastic Elasticity Problems.- 5. Monte Carlo Algorithms for Solving Diffusion Problems.- 5.1 Walk on Boundary Algorithms for the Heat Equation.- 5.2 The Walk Inside the Domain Algorithms.- 5.3 Particle Diffusion in Random Velocity Fields.- 5.4 Applications to Diffusion Problems.- References.