Pacific Institute for the Mathematical Sciences - PIMS

Stochastic PDEs for Sampling Conditioned SDEs

Abstract: 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.

CSC/PIMS Distinguished Speaker Series in Applied and Computational Mathematics 2006

Reception will follow the talk.
Dr. Stuart received his PhD at Oxford University and in addition to
Warwick University has held positions at MIT, Bath University, and
Stanford. He is a leading numerical analyst whose work has been at the
forefront of the development of computational analysis of evolving
systems. He has investigated the relationships between dynamical
systems and their computational models, and has contributed theoretical
and practical advances in that area, as well as to the study of
differential equations. His work is internationally renowned and has
been recognised by the award of numerous prizes.