Abstract

Many problems in the physical sciences require the determination of an unknown field from a finite set of indirect measurements. Examples include oceanography, oil recovery, water resource management and weather forecasting. The Bayesian approach to these problems is natural for many reasons, including the under-determined and ill-posed nature of the inversion, the noise in the data and the uncertainty in the differential equation models used to describe complex mutiscale physics.

In this talk I will describe the advantages of formulating Bayesian inversion on function space in order to solve these problems. I will overview theoretical results concerning well-posedness of the posterior distribution, approximation theorems for the posterior distribution, and specially constructed MCMC methods to explore the posterior distribution when the prior is a Gaussian random field. I will also highlight the widespread use by practitioners of various ad hoc algorithms such as the Ensemble Kalman Filter, and the need for mathematical and statistical analysis of these ad hoc algorithms.

Introductory reading and references may be found in:

http://arxiv.org/abs/1202.0709 [arxiv.org] http://arxiv.org/abs/1209.2736 [arxiv.org]