We present two applications of a multigrid preconditioning technique originally developed for certain classes of inverse problems and later successfully applied to a variety of PDE-constrained optimization problems.

The first part of the talk focuses on linear-quadratic optimal control problems constrained by elliptic equations with uncertain coefficients. These problems are addressed using a "first discretize, then optimize" approach, with the elliptic PDE discretized via either the stochastic Galerkin (SG) or stochastic collocation (SC) method. This leads to the solution of a potentially large-scale linear system representing the reduced first-order optimality conditions. We show that the multilevel preconditioning technique, originally designed for optimal control of deterministic elliptic PDEs, extends naturally to problems with uncertainty. It maintains optimal behavior with respect to mesh refinement, that is, the quality of the preconditioner improves at the optimal rate as the mesh is refined. Furthermore, under certain assumptions, its effectiveness is shown to be robust with respect to two key parameters that drastically affect problem size: polynomial degree and stochastic dimension.

In the second part of the talk, we apply a similar technique to the inverse problem of identifying the diffusivity in a heat transfer model based on various state measurements. Using synthetic experiments involving heat sources and multiple measurements of temperature and heat fluxes, the goal is to determine the diffusivity tensor, assumed isotropic. Due to the problem’s inherent ill-posedness, we formulate it as a regularized least-squares problem, which is nonlinear and non-convex. We show how the multigrid preconditioning technique can be efficiently integrated with the Gauss-Newton method, and we highlight its inapplicability - in its current form - to the full Newton method.

BIO: Andrei Draganescu is a Professor in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County, which he joined in 2006, after completing a two-year postdoctoral appointment in the Optimization and Uncertainty Estimation Department at the Sandia National Laboratories in Albuquerque, New Mexico. He earned his Ph.D. in mathematics from the University of Chicago in 2004, with a dissertation directed by Todd Dupont and Ridgway Scott. His current research interests include discrete maximum principles for finite element methods, first order optimization, and numerical methods for the optimal control of partial differential equations, with emphasis on multilevel algorithms.