DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**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

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491