"Stochastic spectral methods for Bayesian inference in inverse problems"

Youssef Marzouk
Sandia National Laboratories, Livermore, CA

The Bayesian approach to inverse problems provides a foundation for
inference from noisy and limited data, a natural mechanism for
regularization in the form of prior information, and a quantitative
assessment of uncertainty in the inverse solution. With computationally
intensive forward models such as PDEs, however, the cost of repeated
likelihood evaluations may render a Bayesian approach prohibitive. This
difficulty is compounded by high dimensionality, as when the unknown is a
spatiotemporal field.

We present new algorithmic developments for Bayesian inference in this
context, showing strong connections with the forward propagation of
uncertainty. In particular, we introduce a stochastic spectral formulation
that accelerates the Bayesian solution of inverse problems via rapid
evaluation of a surrogate posterior. We also pursue dimensionality reduction
for the inference of spatiotemporal fields, using truncated Karhunen-Lo�ve
representations of Gaussian process priors. These approaches are
demonstrated on scalar transport problems arising in contaminant source
inversion and in the inference of inhomogeneous transport properties.