Applied Math Colloquium: Dr. Alen Alexanderian
NC State
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.