ABSTRACT:
In many inverse problems, the quantity that we would like to be able to understand is often a function. For instance, it may be the initial condition for a PDE, such as the Navier-Stokes equation. The Bayesian approach to these problems gives rise to a probability distribution on function space, which not only allows us to identify likely functions that could give rise to the data, but also allows us to quantify the uncertainty in the problem. One approach to characterising these probability distributions is to use Markov chain Monte Carlo (MCMC) methods. However, these methods can be very costly, and the convergence rates converge to zero as the mesh on the function of interest is refined. In this talk, we will introduce the function space MCMC methods. These methods are well-defined on function space, and as such, their convergence rates are independent of the approximation used for the function. Then we will apply these methods to a shape registration problem with applications in the biomedical sciences. This is joint work with Andrew Stuart, Gareth Roberts, David White, Colin Cotter and F.-X. Vialard.
ADDITIONAL INFORMATION:
Simon’s interests lie in the interface between applied probability and numerical analysis. He is particularly interested in various applications of data assimilation, combining prior beliefs about a dynamical system with observations, in order to characterise properties of that system. Approaching this from a Bayesian viewpoint results in the need for statistical algorithms in order to sample from complex probability ditributions, and as such I am engaged in the design and implementation of Marko chain Monte Carlo algorithms for inverse problems. He is also interested in the development of methodologies for efficient modelling of multiscale stochastic biochemical systems. He is currently looking to apply the philosophy with which one can approach other inverse problems to the observations of multiscale biochemical networks, that have been made possible by the DNA microarray and other technologies. Simon is also collaborating with an obstetrician on mathematical models of placental development.