Large Inverse Problems and Infinite Dimensional Sampling: Stochastic PDE for Sampling Conditioned SD

There are a variety of important applications which may be formulated as inverse problems where the object of interest is the time-dependent solution of a dynamical system. Examples include the sampling of rare events in molecular dynamics, data assimilation in the ocean/atmosphere sciences, signal processing and data interpolation in econometrics. The natural setting for such inverse problems is a statistical one, leading to infinite dimensional sampling problems. The aim of the talk is to describe a unifying approach to such problems, via the development of MCMC methods in infinite dimensions, and using stochastic PDEs in particular.
The talk will contain a lengthy introduction to the subject through the study of three applications: (i) vacancy diffusion in molecular dynamics; (ii) the determination of the velocity field in an ocean from the motion of tracers in the fluid; (iii) and non-Gaussian, nonlinear signal processing.
All of these applications can be cast as sampling problems for conditioned SDEs (diffusion processes). In all these examples the object to sample is time continuous process, and is hence infinite dimensional. We describe an abstract MCMC method for sampling such problems, based on generalizing Metropolis adjusted Langevin algorithms to infinite dimensions. This leads naturally to the study of stochastic reaction-diffusion equations which, in their invariant measure, sample from the required distribution. Furthermore, the study of preconditioning in this context leads to some interesting new infinite dimensional semilinear evolution equations. We give an overview of the mathematics underlying the algorithms developed, describing the analytical, computational and statistical challenges arising in this new subject area.
(This talk will be followed by a reception)