Monte Carlo Methods

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 Equations.- 2.2.3 General Case of the Kernel Approximation.- 2.3 The Eigen-value Problem for the Integral Operators.- 2.3.1 Calculation of Eigen-values on the Basis of the Transformation ? = ?(?).- 2.3.2 Calculation of Eigen-values by Asymptotically Unbiased Estimates.- 2.4 Alternative Constructions of the Resolvent: Modifications and Numerical Experiments.- 2.4.1 Continuation by the Mittag-Leffler Method Combined with the Transformation ? = i(i + p)/(i - p).- 2.4.2 Generalized Summation Methods.- 2.4.3 Transformation and Convergence Acceleration of Series. Euler Summation.- 2.4.4 Padé Approximation of the Resolvent and Approximation by Continued Fractions.- 2.4.5 Methods of Regularization of Analytical Continuation for Solving Integral Equations.- 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.- 3.1.1 Boundary Integral Equations.- 3.1.2 Interior Dirichlet and Exterior Neumann Problems.- 3.1.3 Interior Neumann, Exterior Dirichlet, and the Third Boundary Value Problems.- 3.1.4 Dirichlet and Neumann Problems for the Helmholtz Equation.- 3.1.5 The Variance, the Error and the Cost of the Walk on Boundary Algorithms.- 3.2 Walk Inside the Domain Algorithms.- 3.2.1 General Scheme for Constructing Monte Carlo Estimates on the Walk Inside the Domain Processes.- 3.2.2 Non-homogeneous Equations and Global Walk on Spheres Algorithm for Calculating the Solution and Derivative Fields.- 3.2.3 The Walk on Small Spheres and on Other Standard Domains.- 3.3 Numerical Solution of Some Test and Applied Problems of Potential Theory in Deterministic and Stochastic Formulation.- 3.3.1 Numerical Experiments: Solution of Test Problems of Potential Theory.- 3.3.2 Calculation of the Capture Coefficient of Highly Dispersed Aerosols (3D).- 4. Monte Carlo Algorithms for Solving High-Order Equations and the Elasticity Problems.- 4.1 Biharmonic Problem.- 4.1.1 Vector Walk on Circles Algorithm for Solving the Plate Bending Problem for Simply Supported Plates.- 4.1.2 Plates with Arbitrary Boundaries.- 4.1.3 Direct and Adjoint Algorithms for Calculating the Fields of Solution and Derivatives.- 4.2 Metaharmonic Equations.- 4.2.1 Mean Value Theorems for Metaharmonic Equations.- 4.2.2 Vector Walk on Spheres Algorithm.- 4.2.3 Scalar Algorithms.- 4.3 Spatial Problems of the Elasticity Theory.- 4.3.1 Walk on Boundary Algorithms for the Lamé Equation.- 4.3.2 Walk on Spheres Algorithm for the First Boundary Value Problem.- 4.4 Application to Stochastic Elasticity Problems.- 4.4.1 The Bending Problem for a Plate Lying on an Elastic Base.- 4.4.2 Random Loads.- 5. Monte Carlo Algorithms for Solving Diffusion Problems.- 5.1 Walk on Boundary Algorithms for the Heat Equation.- 5.1.1 Generalization of Isotropic Walk on Boundary Processes to the Nonstationary Case.- 5.1.2 The Variance and Cost of the Walk on Boundary Algorithms.- 5.1.3 Diffusion Equation in a Half-space. Direct Monte Carlo Scheme.- 5.1.4 Adjoint Scheme.- 5.1.5 Nonhomogeneous Case.- 5.1.6 Calculation of Derivatives.- 5.2 The Walk Inside the Domain Algorithms.- 5.2.1 Cauchy Problem.- 5.2.2 Use of the Laplace Transform.- 5.3 Particle Diffusion in Random Velocity Fields.- 5.3.1 Particle Diffusion in Local-Isotropic Velocity Fields.- 5.3.2 Calculation of Statistical Characteristics of a Cloud.- 5.3.3 Statistical Model of the Turbulent Velocity Field for a Horizontally Homogeneous Boundary Layer.- 5.4 Applications to Diffusion Problems.- 5.4.1 Distribution of the First Passage Time for Particles Moving in Classical Isotropic Random Velocity Fields.- 5.4.2 Spread of Clouds of Particles of Aerosol Insecticide in Arboreal Canopies.- 5.4.3 Diffuse Deposition of Polydispersed Aerosol Particles in Pipes.- 5.4.4 Simulation of the Growth of Nuclei Highly Dispersed Aerosol Particles.- References.