Applied Math Colloquium: Dr. Alen Alexanderian

to subsurface flow 

Abstract: We consider inverse problems that seek to infer an infinite-dimensional

parameter from measurement data observed at a set of sensor locations and from

the governing PDEs. We focus on the problem of optimal placement of sensors 

that result in minimized uncertainty in the inferred parameter field. This can be

formulated as an optimal experimental design problem.  We present a method for

computing optimal sensor placements for Bayesian linear inverse problems

governed by PDEs with model uncertainties.  Specifically, given a statistical

distribution for the model uncertainties we seek to find sensor placements that

minimize the expected value of the posterior covariance trace; i.e., the

expected value of the A-optimal criterion. The expected value is approximated

using Monte Carlo leading to an objective function consisting of a finite sum

of trace operators and a sparsity inducing penalty. Minimization of this

objective requires many PDE solves in each step, making the problem extremely

challenging.  We will discuss strategies for making the problem computationally

tractable. These include reduced order modeling and exploiting

low-dimensionality of the measurements, in the problems we target.  

We present numerical results for inference of the initial condition in a 

subsurface flow problem with inherent uncertainty in the velocity field.