Geometric Numerical Integration for Hamiltonian Monte Carlo and Extrapolation

Abstract

Hamiltonian ordinary differential equations (ODEs) occur in many applications (celestial mechanics, molecular dynamics, sampling) and their efficient numerical integration is thus of great interest. This thesis reviews the important concepts of classical numerical integration (stability and order) and their pitfalls and relevance when translated to geometric numerical integration, the alternative paradigm most appropriate to the study of numerical integrators for Hamiltonian problems, which possess the key geometric property of symplecticity. The success of integrators which conserve this property is elaborated through numerical experiments and the now well-established theory of backward error analysis, before various approaches to the optimal design of symplectic integrators are reviewed. The application of extrapolation methods to symplectic integration and a review of the operation of Hamiltonian Monte Carlo (HMC) and results characterising its performance round out the introduction. The remainder of the thesis consists of three chapters, each corresponding to a published research article - on generalised extrapolation methods, preconditioning for alternative integrators for HMC, and the optimal stability interval of a family of integrators for semilinear second-order ODEs - and a conclusion with proposals for future work stemming from the results of the thesis.