پیاده روی های تصادفی بر روی مرز برای حل معادلات دیفرانسیل جزئی ۲۰۱۳
Random Walks on Boundary for Solving PDEs 2013

1. Introduction

2. Random walk algorithms for solving integral equations

2.1. Conventional Monte Carlo scheme

2.2. Biased estimators

2.3. Linear-fractional transformations and relations to iterative processes

2.4. Asymptotically unbiased estimators based on singular approximation of the kernel

2.5. Integral equation of the first kind

3. Random Walk on Boundary algorithms for solving the Laplace equation

3.1. Newton potentials and boundary integral equations of the electrostatics

3.2. The interior Dirichlet problem and isotropic Random Walk on Boundary process

3.3. Solution of the Neumann problem

3.4. Random estimators for the exterior Dirichlet problem

3.5. Third boundary value problem and alternative methods of solving the Dirichlet problem

3.6. Inhomogeneous problems

3.7. Calculation of the derivatives near the boundary

3.8. Normal derivative of a double layer potential

4. Walk on Boundary algorithms for the heat equation

4.1. Heat potentials and Volterra boundary integral equations

4.2. Nonstationary Walk on Boundary process

4.3. The Dirichlet problem

4.4. The Neumann problem

4.5. Third boundary value problem

4.6. Unbiasedness and variance of the Walk on Boundary algorithms

4.7. The cost of the Walk on Boundary algorithms

4.8. Inhomogeneous heat equation

4.9. Calculation of derivatives on the boundary

5. Spatial problems of elasticity

5.1. Elastopotentials and systems of boundary integral equations of the elasticity theory

5.2. First boundary value problem and estimators for singular integrals

5.3. Other boundary value problems for the Lame equations and regular integral equations

6. Variants of the Random Walk on Boundary for solving the stationary potential problems

6.1. The Robin problem and the ergodic theorem

6.2. Stationary diffusion equation with absorption

6.3. Stabilization method

6.4. Multiply connected domains

7. Random Walk on Boundary in nonlinear problems

7.1. Nonlinear Poisson equation

7.2. Boundary value problem for the Navier-Stokes equation

39. Bibliography