Applied and Computational Math Seminar: Multigrid preconditioning for PDE-Constrained optimization: two new applications

Speaker: Andrei Draganescu, UMBC
Title: Multigrid preconditioning for PDE-Constrained optimization: two new applications

Abstract: We present two new applications of a multigrid preconditioning technique that was originally developed for certain classes of inverse problems, and then applied successfully to optimal control of partial differential equations. The first part of the talk will focus on optimal control problems constrained by elliptic equations with stochastic coefficients. Assuming a generalized polynomial chaos expansion for the stochastic components, our approach uses a stochastic Galerkin finite element discretization for the PDE, thus leading to a discrete optimization problem. The key aspect is solving the potentially very large linear systems arising when solving the system representing the first-order optimality conditions. We show that the multilevel preconditioning technique from the optimal control of deterministic elliptic PDEs has a natural extension to the stochastic case, and exhibits a similar optimal behavior with respect to the mesh size, namely the quality of the preconditioner increases with decreasing mesh-size at the optimal rate. Moreover, under certain assumptions, we show that the quality is robust also with respect the two additional parameters that influence the dimension of the problem radically: polynomial degree and stochastic dimension. In the second part of the talk we apply a similar technique to an optimization-based non-overlapping domain decomposition method for elliptic partial differential equations developed by Gunzburger, Heinkenschloss, and Lee (2000). While it is not surprising that, for a fixed partition in subdomains, the preconditioner leads to the expected behavior of increasing quality (lower number of iterations) as the resolution increases, it is remarkable that the quality of the preconditioner is relatively robust with respect to the number and configuration of subdomains. 

Time: Friday, September 21, 2018, 1:30-2:30pm

Place: Exploratory Hall, Room 4106