Bani K. Mallick
Texas A&M
Bayesian inference and uncertainty quantification in large scale inverse
problems
We consider a Bayesian approach to to nonlinear inverse problems in
which the unknown quantity is a random field (spatial or temporal). The
Bayesian approach contains a natural mechanism for regularization in the
form of prior information, can incorporate information from from
heterogeneous sources and provide a quantitative assessment of
uncertainty in the inverse solution. The Bayesian setting casts the
inverse solution as a posterior probability distribution over the model
parameters. Karhunen-Lo\'eve expansion is used for dimension reduction
of the random field. Furthermore, we use a hierarchical Bayes model to
inject multiscale data in the modeling framework. In this Bayesian
framework, we have shown that this inverse problem is well-posed by
proving that the posterior measure is Lipschitz continuous with respect
to the data in total variation norm. Computation challenges in this
construction arise from the need for repeated evaluations of the forward
model (e.g. in the context of MCMC) and are compounded by high
dimensionality of the posterior. We develop two-stage reversible jump
MCMC which has the ability to screen the bad proposals in the first
inexpensive stage. Numerical results are presented by analyzing
simulated as well as real data.