Random Walks on Boundary for Solving PDEs

Abstract

The monograph presents new probabilistic representations for classical boundary value problems of mathematical physics. When comparing with the well known probabilistic representations in the form of Wiener and diffusion path integrals, the trajectories of random walks in our representations are simulated on the boundary of the domain as Markov chains generated by the kernels of the boundary integral equations equivalent to the original boundary value problem. The Monte Carlo methods based on the walk on boundary processes have a series of advantages: (1) high-dimensional problems can be solved, (2) the method is grid-free and gives the solution simultaneously in arbitrary points, (3) external and internal boundary value problems are solved using one and the same random walk on boundary process, (4) when comparing with the classical probabilistic representations, there is no ε-error generated by the approximations in the ε-boundary, and (5) parallel implementation of the walk on boundary algorithms is straightforward and much easier. This is the first book devoted to the walk on boundary algorithms. First introduced by K. Sabelfeld for solving the interior and exterior boundary value problems for the Laplace and heat equations, the method was then extended to all the main boundary value problems of the potential and elasticity theories. For specialists in applied and computational mathematics, applied probabilists, for students and postgraduates studying new numerical methods for solving PDEs.

A Parallelized 3d Floating Random-Walk Algorithm for the Solution of the Nonlinear Poisson-Boltzmann Equation

“…In a 1954 paper [13], J. R. Curtiss wrote: "So far as the author is aware, the extension of Monte Carlo methods to non-linear processes has not yet been accomplished and may be impossible." Stochastic approaches to solving nonlinear equations (in particular the NPB equation) that have been suggested in literature [14] involve an iterative solution of a series of linear problems. In our proposed approach, an approximate (yet accurate) expression for the Green's function for the nonlinear problem is obtained through perturbation theory, which gives rise to an integral formulation that is valid for the entire nonlinear problem.…”

The random walk on the boundary method for calculating capacitance

“…This density function corresponds to a uniform distribution of successive points, y n+1 , in the solid angle with vertex y n . This is the so-called isotropic "random walk on the boundary" [12] process.…”

“….}. Then, for some integrable functions f ∈ L(∂G) and h ∈ L * (∂G) the direct and adjoint estimators, respectively, are defined as [12] (h, K n f ) = EQ n h(y n ) = EQ * n f (y n ) .…”

“…Elimination of the WOS bias due to the boundary with GFFP, the simulation-tabulation technique [10], and last passage variants of these Monte Carlo algorithms [11] further extend the capabilities of stochastic computational methods when applied to electrostatics problems. The progress achieved so far stimulated our investigation of the "random walk on the boundary algorithm" [12] as applied to capacitance and charge density calculations. The idea of using this method on these types of problems dates back to [13].…”

Random Walks for Solving Boundary-Value Problems with Flux Conditions

“…For elliptic equations, the most efficient and commonly used Monte Carlo methods are the walk-on-spheres (WOS) algorithm [1][2][3], and the random walk on the boundary algorithm [4]. The WOS algorithm provides the tool for efficient simulation of exit points of the diffusion process to the domain's boundary.…”