Minimization-based sampling from the posterior distribution for inverse problems with Gaussian prior distributions

Inverse problems for subsurface ow are typically characterized by large numbers of para- meters (e.g. coef cients of PDEs describing ow and transport) and fairly large numbers of observations that are indirectly and nonlinearly related to the parameters. One fairly effective method for characterizing the uncertainty in predictions of future behavior is

to generate samples of model parameters from the posterior probability distribution via minimization of a stochastic cost function. This method is known to sample correctly for Bayesian data assimilation problems with Gaussian prior distributions, linear observation operators and additive Gaussian observation errors. Sampling is only approximate when the observation operator is nonlinear, but experience has shown that it is often quite good, even when the posterior probability distribution is multimodel. In practice, the only cor- rection that we make to sampling is to apply model diagnostics to eliminate samples that appear to be stuck in a local minimum of the cost function.

I will show how the approximate methodology is applied to large-scale problems using ensemble Kalman lter-like methods, and will describe attempts to improve sampling by computation of importance weighting.