Tony Shardlow

An Introduction to Computational Stochastic PDEs

This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of the art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modeling and materials science.

Splitting for Dissipative Particle Dynamics

We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total momentum is conserved. We introduce splitting methods and examine the behavior of these methods experimentally. The performance of the methods, particularly temperature control, is compared to the modified velocity Verlet method used in many previous papers.

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).

Weak Convergence of a Numerical Method for a Stochastic Heat Equation

Weak convergence with respect to a space of twice continuously differentiable test functions is established for a discretisation of a heat equation with homogeneous Dirichlet boundary conditions in one dimension, forced by a space-time Brownian motion. The discretisation is based on finite differences in space and time, incorporating a spectral approximation in space to the Brownian motion.

Postprocessing for Stochastic Parabolic Partial Differential Equations

We investigate the strong approximation of stochastic parabolic partial differential equations with additive noise. We introduce postprocessing in the context of a standard Galerkin approximation, although other spatial discretizations are possible. In time, we follow [G. J. Lord and J. Rougemont, IMA J. Numer. Anal., 24 (2004), pp. 587-604] and use an exponential integrator. We prove strong error estimates and discuss the best number of postprocessing terms to take. Numerically, we evaluate the efficiency of the methods and observe rates of convergence. Some experiments with the implicit Euler-Maruyama method are described.

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.

WEAK CONVERGENCE OF THE EULER SCHEME FOR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS

We study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is nonanticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

Computing the geodesic interpolating spline

We examine non-rigid image registration by knotpoint mat-break ching. We consider registering two images, each with a set of knotpoints marked, where one of the images is to be registered to the other by a nonlinear warp so that the knotpoints on the template image are exactly aligned with the corresponding knotpoints on the reference image. We explore two approaches for computing the registration by the Geodesic Interpolating Spline. First, we describe a method which exploits the structure of the problem in order to permit efficient optimization and second, we outline an approach using the framework of classical mechanics.

Langevin Equations for Landmark Image Registration with Uncertainty

Registration of images parameterised by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows one landmark set to the other. This is sometimes known as the geodesic interpolating spline and can be solved via a Hamiltonian boundary-value problem to give a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks.

Use of Fin Equation to Calculate Nusselt Numbers for Rotating Disks

Conduction in thin discs can be modelled using the finequation, and there are analytical solutions of this equation for a circular disc with a constant heat-transfer coefficient. However, convection (particularly free convection) in rotating-disc systems is a conju gate problem: the heat transfer in the fluid and the solid are coupled, and the relative effects of conduction and convection are related to the Biot number, Bi, which in turn is related to the Nusselt number. In principle, if the radial distribution of the disc temperature is known then Bi can be determined numerically. But the determina tion of heat flux from temperature measurements is an example of an inverse problem where small uncertainties in the temperatures can create large uncertainties in the computed heat flux. In this pa per, Bayesian statistics are applied to the inverse solution of the cir cular fin equation to produce reliable estimates of Bi for rotating discs, and numerical experiments using simulated noisy tempera ture measurements are used to demonstrate the effectiveness of the Bayesian method. Using published experimental temperature measurements, the method is also applied to the conjugate problem of buoyancy-induced flow in the cavity between corotating com pressor discs.