Multigrid Methods for Elliptic Optimal Control Problems with Neumann Boundary Control

Dr. Stefan Takacs

June 9, 2009, 3:30 p.m. MZ 005B

In this talk we will discuss multigrid methods for solving the discretized optimality system for elliptic optimal control problems. We will concentrate on the following model problem with Neumann boundary control:

$ J(y, u) = \frac{1}{2} \left\Vert y − y_D \right\Vert_{L^2(Ω)}^2 + \frac{\gamma}{2} \left\Vert u \right\Vert_{L^2 (\partial Ω)} \to \mathrm{min} $
$ −\Delta y + y = 0 ~ \mathrm{in} ~ \Omega, \qquad \frac{\partial y}{\partial n} = u ~ \mathrm{on} ~ \partial \Omega $

The proposed approach is based on the formulation of Karush-Kuhn-Tucker system in terms of the state y, the control u and the adjoined state p. For the model problem, the approximation property is shown similar to the case of distributed control, see [3] and [1]. We will propose on the one hand an Uzawa type smoother and on the other hand a smoother that is based on the normal equation of the Kuhn-Tucker system. For both methods rigorous analysis is available. We will compare both methods in numerical experiments.

Of course, the results can be generalized to other problems, especially the observation can be restricted to the boundary or to some part of the domain Ω.

[1] S. C. Brenner. Multigrid methods for parameter dependent problems. RAIRO, Modélisation Math. Anal. Numér, 30:265 – 297, 1996.
[2] J. Schöberl and W. Zulehner. On Schwarz-type smoothers for saddle point problems. Numer. Math., 95:377 – 399, 2003.
[3] R. Simon and W. Zulehner. On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems. Numer. Math., 111:445 – 468, 2009.