Description
This monograph is devoted to random walk based stochastic algorithms for solving high-dimensional boundary value problems of mathematical physics and chemistry. It includes Monte Carlo methods where the random walks live not only on the boundary, but also inside the domain. A variety of examples from capacitance calculations to electron dynamics in semiconductors are discussed to illustrate the viability of the approach.
The book is written for mathematicians who work in the field of partial differential and integral equations, physicists and engineers dealing with computational methods and applied probability, for students and postgraduates studying mathematical physics and numerical mathematics.
Contents:
Introduction
Random walk algorithms for solving integral equations
Random walk-on-boundary algorithms for the Laplace equation
Walk-on-boundary algorithms for the heat equation
Spatial problems of elasticity
Variants of the random walk on boundary for solving stationary potential problems
Splitting and survival probabilities in random walk methods and applications
A random WOS-based KMC method for electron–hole recombinations
Monte Carlo methods for computing macromolecules properties and solving related problems
Bibliography
Chapter
2.3 Linear-fractional transformations and their relations to iterative processes
2.4 Asymptotically unbiased estimators based on singular approximations
2.5 Integral equation of the first kind
3 Random walk-on-boundary algorithms for 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 BVP and alternative methods of solving the Dirichlet problem
3.6 Inhomogeneous problems
3.7.1 Walk on boundary for the continuity problem
3.8 Calculation of the solution derivatives near the boundary
3.9 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
4.6 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 BVP and estimators for singular integrals
5.3 Other BVPs for the Lamé equations and regular integral equations
6 Variants of the random walk on boundary for solving stationary potential problems
6.1 The Robin problem and the ergodic theorem
6.1.1 Monte Carlo estimator for computing capacitance
6.1.2 Computing charge density
6.2 Stationary diffusion equation with absorption
6.3 Multiply connected domains
6.5 Nonlinear Poisson equation
7 Splitting and survival probabilities in random walk methods and applications
7.2 Survival probability for a sphere and an interval
7.3 The reciprocity theorem for particle collection in the general case of Robin boundary conditions
7.4 Splitting and survival probabilities
7.4.1 Splitting probabilities for a finite interval and nanowire growth simulation
7.4.2 Survival probability for a disc and the exterior of circular cylinder
7.4.3 Splitting probabilities for concentric spheres and annulus
7.5.1 The random WOS and hemispheres algorithm
7.6 Conclusion and discussion
8 A random WOS-based KMC method for electron–hole recombinations
8.2 The mean field equations
8.3 Monte Carlo Algorithms
8.3.1 Random WOS for the diffusion simulation
8.3.2 Radiative and nonradiative recombination in the absence of diffusion
8.3.3 General case of radiative and nonradiative recombination in the presence of diffusion
8.4 Simulation results and comparison
8.5 Summary and conclusion
9 Monte Carlo methods for computing macromolecules properties and solving related problems
9.1 Diffusion-limited reaction rate and other integral parameters
9.1.1 Formulation of the problem
9.1.2 Capacitance calculations
9.2 Walk in subdomains and efficient simulation of Brownian motion exit points
9.3 Monte Carlo algorithms for boundary-value conditions containing the normal derivative
9.3.1 WOS algorithm for mixed boundary-value conditions
9.3.2 Mean-value relation at a point on the boundary
9.3.3 Construction of the algorithm and its convergence
9.4.2 Integral representation at a boundary point
9.4.3 Estimate for the boundary value
9.4.4 Construction of the algorithm and its convergence
9.5 Computing macromolecule energy
9.5.1 Mathematical model and computational results
Stochastic Methods for Boundary Value Problems
：Numerics for High-dimensional PDEs and Applications
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