Stochastic Methods for Boundary Value Problems : Numerics for High-dimensional PDEs and Applications.

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Bibliographic Details
Author / Creator:	Sabelfeld, Karl K.
Imprint:	Berlin/Boston, GERMANY : De Gruyter, 2016. ©2016
Description:	1 online resource (208)
Language:	English
Subject:	Boundary value problems -- Numerical solutions. Stochastic analysis. Random walks (Mathematics) Boundary value problems -- Numerical solutions. Random walks (Mathematics) Stochastic analysis.
Format:	E-Resource Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/11408344

Hidden Bibliographic Details
Other authors / contributors:	Simonov, N. A.
ISBN:	3110479451 9783110479454 9783110479454 9783110479164 3110479168 3110479060 9783110479065
Digital file characteristics:	text file PDF
Notes:	Includes bibliographical references. In English. Print version record.
Summary:	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.
Other form:	Print version: 9783110479065 3110479060
Standard no.:	10.1515/9783110479454

Table of Contents:

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 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 Continuity BVP 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 ; 4.4 The Neumann problem ; 4.5 Third BVP ; 4.6 Unbiasedness and variance of the walk-on-boundary algorithms.

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