Efficient, high-dimensional sampling using stochastic differential equations

Sampling is the problem of computing expectations with respect to a specified target measure (which we assume to have a smooth density).  I will discuss sampling algorithms that have arisen in our work in molecular dynamics but which have much wider potential applications for example in statistical science.  The starting point is numerical discretisation of Langevin and Brownian (overdamped Langevin) dynamics with a focus on order of accuracy in the discretisation parameter with respect to approximation of ergodic averages, but as I will explain accuracy of averages in the long time limit does not necessarily translate into efficiency in the context of a finite computational budget.    I will therefore also describe some new methods we are developing to accelerate convergence in systems with multimodal landscapes, specifically 'infinite swap simulated tempering' (ISST) which extends the state variables to include a variable temperature, and ensemble quasi-Newton (EQN) which rescales the dynamical model of a walker (particle) collection to accelerate entropically restricted transitions.   Joint work with  Jianfeng Lu (Duke), Anton Martinsson (UoE), Charlies Matthews (UoE), Eric Vanden-Eijnden (NYU), and Jonathan Weare (NYU).