This book provides an accessible overview concerning the stochastic numerical methods inheriting long-time dynamical behaviours of finite and infinite-dimensional stochastic Hamiltonian systems. The long-time dynamical behaviours under study involve symplectic structure, invariants, ergodicity and invariant measure. The emphasis is placed on the systematic construction and the probabilistic superiority of stochastic symplectic methods, which preserve the geometric structure of the stochastic flow of stochastic Hamiltonian systems.
The problems considered in this book are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, stochastic ordinary and partial differential equations, and rough path theory. This book will appeal to researchers who are interested in these topics.
Author(s): Jialin Hong, Liying Sun
Series: Lecture Notes in Mathematics, 2314
Publisher: Springer
Year: 2023
Language: English
Pages: 306
City: Singapore
1
Preface
Acknowledgments
Contents
Notation and Symbols
978-981-19-7670-4_1
1 Stochastic Hamiltonian Systems
1.1 Stratonovich Stochastic Differential Equations
1.2 Stochastic Hamiltonian Systems
1.2.1 Stochastic Symplectic Structure
1.2.2 Stochastic Variational Principle
1.2.3 Stochastic θ-Generating Function
1.3 Non-canonical Stochastic Hamiltonian Systems
1.3.1 Stochastic Forced Hamiltonian Systems
1.3.2 Stochastic Poisson Systems
1.4 Rough Hamiltonian Systems
1.5 End Notes
978-981-19-7670-4_2
2 Stochastic Structure-Preserving Numerical Methods
2.1 Stochastic Numerical Methods
2.1.1 Strong and Weak Convergence
2.1.2 Numerical Methods via Taylor Expansions
2.1.3 Stochastic Runge–Kutta Methods
2.2 Stochastic Symplectic Methods
2.2.1 Symplectic Methods via Padé Approximation
2.2.2 Symplectic Methods via Generating Function
2.2.3 Stochastic Galerkin Variational Integrators
2.2.4 Stochastic (Rough) Symplectic Runge–Kutta Methods
2.3 Application of Symplectic Methods to Stochastic Poisson Systems
2.4 Superiority of Stochastic Symplectic Methods via Large Deviation Principle (LDP)
2.4.1 LDP for Numerical Methods
2.4.2 Asymptotical Preservation for LDP
2.5 Stochastic Pseudo-Symplectic Methods and Stochastic Methods Preserving Invariants
2.5.1 Stochastic Pseudo-Symplectic Methods
2.5.2 Stochastic Numerical Methods Preserving Invariants
2.6 End Notes
978-981-19-7670-4_3
3 Stochastic Modified Equations and Applications
3.1 Stochastic Modified Equations
3.1.1 Modified Equation via Backward Kolmogorov Equation
3.1.2 Modified Equation via Stochastic Generating Function
3.1.3 Modified Equation for Rough Symplectic Methods
3.2 Stochastic Numerical Methods Based on Modified Equation
3.2.1 High Order Methods via Backward KolmogorovEquation
3.2.2 High Order Symplectic Methods via GeneratingFunction
3.3 Conformal Symplectic and Ergodic Methods for Stochastic Langevin Equation
3.4 End Notes
978-981-19-7670-4_4
4 Infinite-Dimensional Stochastic Hamiltonian Systems
4.1 Infinite-Dimensional Stochastic Hamiltonian Systems
4.2 Stochastic Maxwell Equation
4.2.1 Stochastic Exponential Integrator
4.2.2 Stochastic Symplectic Runge–Kutta Discretizations
4.3 Stochastic Schrödinger Equation
4.3.1 Stochastic Symplectic Runge–Kutta Discretizations
4.3.2 Approximating Large Deviation Rate Functions via Spectral Galerkin Method
4.4 Stochastic Wave Equation with Cubic Nonlinearity
4.4.1 Spectral Galerkin Method and Exponential Integrability
4.4.2 Exponentially Integrable Full Discretization
4.5 End Notes
1 (1)
Appendix A
A.1 Stochastic Stratonovich Integral
A.2 Large Deviation Principle
A.3 Exponential Integrability
A.4 Stochastic Multi-Symplectic Structure
References
