Random Walks On Boundary For Solving Pdes

Download or read book Random Walks on Boundary for Solving PDEs written by Karl K. Sabelfeld and published by Walter de Gruyter. This book was released on 2013-07-05 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Random Walks on Boundary for Solving PDEs written by K. K. Sabelfeld and published by . This book was released on 1994 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Random Walk and the Heat Equation written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2010-11-22 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.

Download or read book Random Walks on Boundary for Solving PDEs written by N. A. Simonov and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Methods for Boundary Value Problems written by Karl K. Sabelfeld and published by Walter de Gruyter GmbH & Co KG. This book was released on 2016-09-26 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Large-Scale Scientific Computing written by Ivan Lirkov and published by Springer. This book was released on 2018-01-10 with total page 607 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the thoroughly refereed post-conference proceedings of the 11th International Conference on Large-Scale Scientific Computations, LSSC 2017, held in Sozopol, Bulgaria, in June 2017. The 63 revised short papers together with 3 full papers presented were carefully reviewed and selected from 63 submissions. The conference presents results from the following topics: Hierarchical, adaptive, domain decomposition and local refinement methods; Robust preconditioning algorithms; Monte Carlo methods and algorithms; Numerical linear algebra; Control and optimization; Parallel algorithms and performance analysis; Large-scale computations of environmental, biomedical and engineering problems. The chapter 'Parallel Aggregation Based on Compatible Weighted Matching for AMG' is available open access under a CC BY 4.0 license.

Download or read book Spherical and Plane Integral Operators for PDEs written by Karl K. Sabelfeld and published by Walter de Gruyter. This book was released on 2013-10-29 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents integral formulations for partial differential equations, with the focus on spherical and plane integral operators. The integral relations are obtained for different elliptic and parabolic equations, and both direct and inverse mean value relations are studied. The derived integral equations are used to construct new numerical methods for solving relevant boundary value problems, both deterministic and stochastic based on probabilistic interpretation of the spherical and plane integral operators.