Random Walks on Boundary for Solving PDEs
Published on 1. October 1994
Book
Hardback
141 pages
978-90-6764-183-8 (ISBN)
Description
01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.
This monograph presents new probabilistic representations for classical boundary value problems of mathematical physics and is the first book devoted to the walk on boundary algorithms. Compared to the well-known Wiener and diffusion path integrals, the trajectories of random walks in this publication are simlated 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 book opens with an introduction for solving the interior and exterior boundary values for the Laplace and heat equations, which is followed by applying this method to all main boundary value problems of the potential and elasticity theories.
This monograph presents new probabilistic representations for classical boundary value problems of mathematical physics and is the first book devoted to the walk on boundary algorithms. Compared to the well-known Wiener and diffusion path integrals, the trajectories of random walks in this publication are simlated 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 book opens with an introduction for solving the interior and exterior boundary values for the Laplace and heat equations, which is followed by applying this method to all main boundary value problems of the potential and elasticity theories.
More details
Edition
Reprint 2012
Language
English
Place of publication
Zeist
Netherlands
Publishing group
Brill
Target group
College/higher education
Professional and scholarly
US School Grade: College Graduate Student
Weight
370 gr
ISBN-13
978-90-6764-183-8 (9789067641838)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Karl K. Sabelfeld | Nikolai A. Simonov
Random Walks on Boundary for Solving PDEs
E-Book
07/2013
1st Edition
De Gruyter
€169.95
Available for download
Karl K. Sabelfeld | Nikolai A. Simonov
Random Walks on Boundary for Solving PDEs
Book
01/1994
1st Edition
De Gruyter
€259.00
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Content
Introduction
RANDOM WALK ALGORITHMS FOR SOLVING INTEGRAL EQUATIONS
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
RANDOM WALK ON BOUNDARY ALGORITHMS FOR SOLVING THELAPLACE 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
Third boundary value problem and alternative methods of solving the Dirichlet problem
Inhomogeneous problems
Calculation of the derivatives near the boundary
Normal derivative of a double layer potential
WALK ON BOUNDARY ALGORITHMS FOR THE HEAT EQUATION
Heat potential and Volterra boundary integral equations
Nonstationary 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
Inhomogeneous heat equation
Calculation of derivatives on the boundary
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
VARIANTS OF THE RANDOM WALK ON BOUNDARY FOR SOLVING THE STATIONARY POTENTIAL PROBLEMS
The Robin problem and the ergodic theorem
Stationary diffusion equation with absorption
Stabilization method
Multiply connected domains
RANDOM WALK ON BOUNDARY IN NON LINEAR PROBLEMS
Nonlinear Poisson equation
Boundary value problem for the Navier--Stokes equation
Bibliography
RANDOM WALK ALGORITHMS FOR SOLVING INTEGRAL EQUATIONS
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
RANDOM WALK ON BOUNDARY ALGORITHMS FOR SOLVING THELAPLACE 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
Third boundary value problem and alternative methods of solving the Dirichlet problem
Inhomogeneous problems
Calculation of the derivatives near the boundary
Normal derivative of a double layer potential
WALK ON BOUNDARY ALGORITHMS FOR THE HEAT EQUATION
Heat potential and Volterra boundary integral equations
Nonstationary 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
Inhomogeneous heat equation
Calculation of derivatives on the boundary
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
VARIANTS OF THE RANDOM WALK ON BOUNDARY FOR SOLVING THE STATIONARY POTENTIAL PROBLEMS
The Robin problem and the ergodic theorem
Stationary diffusion equation with absorption
Stabilization method
Multiply connected domains
RANDOM WALK ON BOUNDARY IN NON LINEAR PROBLEMS
Nonlinear Poisson equation
Boundary value problem for the Navier--Stokes equation
Bibliography