Karniadakis G., Zhang Z. Numerical methods for stochastic partial differential equations with white noise

New York: Springer, 2017. — 391 p.Prologue
Why random and Brownian motion (white noise)?
What color is the noise?
Solutions
Modeling with SPDEs
Specific topics of this book
Brownian motion and stochastic calculus
Gaussian processes and their representations
Brownian motion and white noise
Some properties of Brownian motion
Regularity of Brownian motion
Approximation of Brownian motion
Brownian motion and stochastic calculus
Stochastic chain rule: Ito formula
Integration methods in random space
Monte Carlo method and its variants Multilevel Monte Carlo method
Quasi-Monte Carlo methods
Wiener chaos expansion method
Stochastic collocation method
Smolyak's sparse grid
Application to SODEs
Bibliographic notes
Suggested practice
Numerical methods for stochastic differential equations
Basic aspects of SODEs
Existence and uniqueness of strong solutions
Solution methods
The integrating factor method
Moment equations of solutions
Numerical methods for SODEs
Derivation of numerical methods based on numerical integration
Strong convergence
Weak convergence
Linear stability
Summary of numerical SODEs
Basic aspects of SPDEs
Functional spaces
Solutions in different senses
Solutions to SPDEs in explicit form
Linear stochastic advection-diffusion-reactionequations
Existence and uniqueness
Conversion between Ito and Stratonovichformulation
Numerical methods for SPDEs
Direct semi-discretization methods for parabolicSPDEs
Second-order equations
Fourth-order equations
Wong-Zakai approximation for parabolic SPDEs
Preprocessing methods for parabolic SPDEsSplitting methods
Integrating factor (exponential integrator) techniques
What could go wrong? Examples of stochastic Burgers and Navier-Stokes equations
Stability and convergence of existing numericalmethods
Weak convergence
Pathwise convergence
Stability
Summary of numerical SPDEs
Summary and bibliographic notes
Suggested practice
Numerical Stochastic Ordinary Differential Equations
Numerical schemes for SDEs with time delay using the Wong-Zakai approximation
Wong-Zakai approximation for SODEs
Wong-Zakai approximation for SDDEs
Derivation of numerical schemes
A predictor-corrector scheme
The midpoint scheme
A Milstein-like scheme
Linear stability of some schemes
Stability region of the forward Euler scheme
Stability analysis of the predictor-corrector scheme
Stability analysis of the midpoint scheme
Numerical results
Summary and bibliographic notes
Suggested practice
Balanced numerical schemes for SDEs with non-Lipschitzcoefficients
A motivating example
Fundamental theorem
On application of Theorem