# The Bayesian Approach to Inverse Problems

## Document Type

Presentation Abstract

## Presentation Date

7-14-2014

## Abstract

Many problems in the physical sciences require the determination of an unknown function 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. The object of interest in the Bayesian approach is the posterior probability distribution on the unknown field [1].

In its purest form the Bayesian approach presents a computationally formidable task as it results in the need to probe a probability measure on an infinite dimensional space; furthermore the likelihood is defined through the solution of a partial differential equation. In this talk I will discuss three computational approaches designed to make this computational task feasible. I will describe Monte Carlo-Markov chain methods tailored to scale well under mesh refinement in the computational model [2]. I will discuss ensemble Kalman filter methods which may be viewed as derivative free optimization methods [3]. And I will describe approximation of the posterior probability distribution by a Gaussian measure, looking for the closest approximation with respect to the Kullback-Leibler divergence [4]; furthermore I will show how the approximate Gaussians can be used to speed-up MCMC sampling of the posterior distribution [5], linking up with ideas from [2].

[1] A.M. Stuart. "Inverse problems: a Bayesian perspective." Acta Numerica 19(2010) and http://arxiv.org/abs/1302.6989

[2] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White, "MCMC methods for functions: modifying old algorithms to make them faster". Statistical Science 28(2013). http://arxiv.org/abs/1202.0709

[3] M.A. Iglesias, K.J.H. Law and A.M. Stuart, "Ensemble Kalman Methods for Inverse Problems." Inverse Problems, 29(2013) 045001. http://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart99.pdf

[4] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Kullback-Leibler Approximations for measures on infinite dimensional spaces." http://arxiv.org/abs/1310.7845

[5] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Algorithms for Kullback-Leibler approximation of probability measures in infinite dimensions." In preparation.

## Recommended Citation

Stuart, Andrew, "The Bayesian Approach to Inverse Problems" (2014). *Colloquia of the Department of Mathematical Sciences*. 458.

https://scholarworks.umt.edu/mathcolloquia/458

## Additional Details

Monday, July 14, 2014 at 3:10 p.m. in Math 103