Random Walks on Boundary for Solving PDEs

Abstract

Part 1 Random walk algorithms for solving integral solutions: conventional Monte Carlo scheme biased estimators linear-fractional transformations and relations to iterative processes asymptotically unbiased estimators based on singular approximation of the kernel integral equation of the first kind. Part 2 Random walk on boundary algorithms for solving the Laplace equation: Newton potentials and boundary integral equations of the electrostatics the interior Dirichlet problem and isotropic random walk on boundary process solution of the Neumann problem random estimators for the exterior Dirichlet problem third boundary value problem and alternative methods of solving the Dirichlet problem in-homogeneous problems calculation of the derivatives near the boundary normal derivative of a double-layer potential. Part 3 Walk on boundary algorithms for the heat equation: heat potentials and Voltrerra boundary integral equations non-stationary walk on boundary process the Dirichlet problem the Neumann problem third boundary value problem unbiasedness and variance of the walk on boundary algorithms the cost of the walk on boundary algorithms in-homogeneous heat equation calculation of derivatives on the boundary. Part 4 Spatial problems of elasticity: elastopotentials and systems of boundary integral equations of the elasticity theory first boundary value problem and estimators for singular integrals other boundary value problems for the Lame equations and regular integral equations. Part 5 Variants of the random walk on boundary for solving the stationary potential problem: the Robin problem and the ergodic theorem stationary diffusion equation with absorption stabilization methods multiply connected domains. Part 6 Random walk on boundary in nonlinear problems: nonlinear Poisson equation boundary value problem for the Navier-Stokes equation.