On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems

Dr. René Simon

Jan. 23, 2007, 3:30 p.m. T 1010

In this talk we consider additive (and multiplicative) Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step requires the solution of several small local saddle point problems. In a previous work (by Joachim Schöberl and Walter Zulehner) the general construction of such patch smoothers for mixed problems were discussed. It was shown that, under suitable conditions, the additive Schwarz-type iteration fulfills the so-called smoothing property, an important part of a multigrid convergence proof, and the theory was applied to the Stokes problem.

Here we consider a certain class of optimization problems from optimal control. A natural property of the corresponding Karush-Kuhn-Tucker (KKT) system, a 2-by-2 block system which characterizes the solution of the optimization problems, is the positivity of the (1,1) block only on the kernel of the (2,1) block. Therefore, a straight forward application of this construction to KKT systems for elliptic optimal control problems fails. We extend the results for the Stokes problem to PDE-constrained optimization problems and present a patch smoother, which allows a rigorous convergence analysis of the corresponding multigrid method.