The stochastic Maxwell equations play an essential role in many fields, including fluctuational electrodynamics, statistical radiophysics, integrated circuits, and stochastic inverse problems.

This book provides some recent advances in the investigation of numerical approximations of the stochastic Maxwell equations via structure-preserving algorithms. It presents an accessible overview of the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behaviors of the stochastic Maxwell equations. A friendly introduction to the simulation of the stochastic Maxwell equations with some structure-preserving algorithms is provided using MATLAB for the reader's convenience.

The objects considered in this book are related to several fascinating mathematical fields: numerical analysis, stochastic analysis, (multi-)symplectic geometry, large deviations principle, ergodic theory, partial differential equation, probability theory, etc. This book will appeal to researchers who are interested in these topics.

Provides the study of intrinsic properties possessed by stochastic Maxwell equations Constructs structure-preserving algorithms of stochastic Maxwell equations Presents the analysis on regularities and error estimates for the stochastic structure-preserving algorithms

Autorentext
Chuchu Chen is an associate professor at the Chinese Academy of Sciences. She obtained her Ph.D. in 2015 at the Chinese Academy of Sciences. Her research interest is in the numerical analysis of stochastic partial differential equations, especially in the structure-preserving algorithms for stochastic Hamiltonian PDEs including the stochastic Maxwell equations and the stochastic Schrödinger equation, the analysis of the long-time dynamical behaviors including the ergodicity and intermittency of stochastic numerical methods, the influence of numerical discretizations on the statistical properties like the hitting probability and density function of stochastic PDEs. Jialin Hong is a professor at the Chinese Academy of Sciences. He obtained his Ph.D. in 1994 at Jilin University. He works in various directions including structure-preserving algorithms for dynamical systems involving symplectic and multi-symplectic methods for Hamiltonian ODEs andPDEs, Lie group methods and applications, numerical dynamics including chaos, bifurcations for discrete systems, numerical methods for stochastic ordinary differential systems, stochastic partial differential equations, and backward stochastic differential equations, almost periodic dynamical systems, and ergodic theory. Lihai Ji is an associate professor at the Institute of Applied Physics and Computational Mathematics. He obtained his Ph.D. in 2013 at the Chinese Academy of Sciences. He works in stochastic partial differential equations and their numerical algorithms. He has been investigating the construction and analysis of various energy-preserving algorithms, positive-preserving algorithms, stochastic symplectic and multi-symplectic algorithms for the stochastic Lotka-Volterra model and stochastic Hamiltonian PDEs including the stochastic Maxwell equations, the stochastic Schrödinger equation, and the coupled stochastic Schrödinger equation.

Inhalt

Introduction.- Solution Theory of Stochastic Maxwell Equations.- Intrinsic Properties of Stochastic Maxwell Equations.- Structure-Preserving Algorithms for Stochastic Maxwell Equations.- Convergence Analysis of Structure-Preserving Algorithms.- Implementation of Numerical Experiments.- Appendix A: Basic Identities and Inequalities.- Appendix B: Semigroup, Sobolev Space, and Differential Calculus.- Appendix C: Estimates Related to Maxwell Operators.- Appendix D: Some Results of Stochastic Partial Differential Equations.- References.