Stochastic Methods for Boundary Value Problems
Stochastic Methods for Boundary Value Problems

Author: Karl K. Sabelfeld

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2016-09-26

Total Pages: 208

ISBN-13: 3110479451

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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

Stochastic Methods for Boundary Value Problems
Stochastic Methods for Boundary Value Problems

Author: Karl K. Sabelfeld

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2016-09-26

Total Pages: 235

ISBN-13: 3110479168

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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

Stochastic versus Deterministic Systems of Differential Equations
Stochastic versus Deterministic Systems of Differential Equations

Author: G. S. Ladde

Publisher: CRC Press

Published: 2003-12-05

Total Pages: 269

ISBN-13: 0203027027

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This peerless reference/text unfurls a unified and systematic study of the two types of mathematical models of dynamic processes-stochastic and deterministic-as placed in the context of systems of stochastic differential equations. Using the tools of variational comparison, generalized variation of constants, and probability distribution as its met

Boundary Value Problems and Markov Processes
Boundary Value Problems and Markov Processes

Author: Kazuaki Taira

Publisher: Springer Science & Business Media

Published: 2009-06-30

Total Pages: 196

ISBN-13: 3642016766

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This is a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. It presents new developments in the theory of singular integrals.

Monte Carlo Methods
Monte Carlo Methods

Author: Karl Karlovich Sabelʹfelʹd

Publisher: Springer

Published: 1991-10-04

Total Pages: 314

ISBN-13:

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This book deals with Random Walk Methods for solving multidimensional boundary value problems. Monte Carlo algorithms are constructed for three classes of problems: (1) potential theory, (2) elasticity, and (3) diffusion. Some of the advantages of our new methods as compared to conventional numerical methods are that they cater for stochasticities in the boundary value problems and complicated shapes of the boundaries.

Boundary Value Problems and Markov Processes
Boundary Value Problems and Markov Processes

Author: Kazuaki Taira

Publisher:

Published: 2020

Total Pages: 502

ISBN-13: 9783030487898

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This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject. The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory.

Stochastic Numerics for Mathematical Physics
Stochastic Numerics for Mathematical Physics

Author: Grigori Noah Milstein

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 612

ISBN-13: 3662100630

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Stochastic differential equations have many applications in the natural sciences. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce solution of multi-dimensional problems for partial differential equations to integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. The authors propose many new special schemes, some published here for the first time. In the second part of the book they construct numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

Optimal Stopping and Free-Boundary Problems
Optimal Stopping and Free-Boundary Problems

Author: Goran Peskir

Publisher: Springer Science & Business Media

Published: 2006-11-10

Total Pages: 515

ISBN-13: 3764373903

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This book discloses a fascinating connection between optimal stopping problems in probability and free-boundary problems. It focuses on key examples and the theory of optimal stopping is exposed at its basic principles in discrete and continuous time covering martingale and Markovian methods. Methods of solution explained range from change of time, space, and measure, to more recent ones such as local time-space calculus and nonlinear integral equations. A chapter on stochastic processes makes the material more accessible. The book will appeal to those wishing to master stochastic calculus via fundamental examples. Areas of application include financial mathematics, financial engineering, and mathematical statistics.