Stochastic Numerics for Mathematical Physics

This book is a revised and expanded edition of stochastic numerics, covering new topics such as mean-square and weak approximations, conditional probabilistic representations, multi-level Monte Carlo method, ergodic limits, and numerical methods for FBSDEs. It is designed for researchers and graduate students in numerical analysis, applied probability, physics, chemistry, engineering, mathematical biology, and financial mathematics.

Format: Paperback / softback
Length: 736 pages
Publication date: 05 December 2022
Publisher: Springer Nature Switzerland AG

This book represents a significant and comprehensive revision, reflecting significant advancements in stochastic numerics since its initial publication in 2004. The revised edition encompasses a wide range of new topics, particularly focusing on mean-square and weak approximations for nonglobally Lipschitz coefficients of Stochastic Differential Equations (SDEs), including the concept of rejecting trajectories. Additionally, conditional probabilistic representations and their practical applications in variance reduction through regression methods are explored. The multi-level Monte Carlo method is introduced, along with computational techniques for ergodic limits and geometric integrators used in molecular dynamics. Numerical methods for FBSDEs, approximation of parabolic SPDEs, and nonlinear filtering problems based on the method of characteristics are also discussed.

SDEs find widespread applications in natural sciences and finance, and the integration of probabilistic representations and Monte Carlo techniques enables the solution of multi-dimensional problems for partial differential equations. This approach leads to powerful computational mathematics, which is extensively presented in the treatise. Numerous special schemes for SDEs are introduced, providing practical solutions for various applications.

In the second part of the book, numerical methods for solving complex problems for partial differential equations arising in practical applications, both linear and nonlinear, are constructed. These methods are grounded in rigorous reasoning, ensuring their validity. Moreover, most of the methods are accompanied by numerical algorithms that are readily executable in practice, making the book an invaluable resource for researchers and graduate students in numerical analysis, applied probability, physics, and chemistry.

Weight: 1151g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030820428
Edition number: 2nd ed. 2021