A Robust Solver for Distributed Optimal Control for Stokes Flow

Dipl.-Ing. Markus Kollmann

Nov. 8, 2011, 3:30 p.m. S2 046

In this talk we consider the following optimal control problem:

Minimize $J(u, f) = \frac{1}{2} \left \| u-u_d \right \|_{L^2(\Omega)}^2 + \frac {\alpha}{2} \left \| f \right \|_{L^2(\Omega)}^2 $ (1)

subject to

$-\Delta u + \nabla p = f$ in $\Omega$
div $u = 0$ in $\Omega$
$u = 0$ on $\Gamma = \partial \Omega$
$f_a < f < f_b$ a.e. in $\Omega$

where $\Omega$ is an open and bounded domain in $\mathbb{R}^d$ for $d \in \{1, 2, 3\}$, $\Gamma$ denotes the boundary, $\alpha > 0$ is a cost parameter, $f_a$, $f_b$ are the lower and upper bounds for the control variable $f$, respectively. The first order system of necessary and sufficient optimality conditions of (1) is nonlinear. A semi-smooth Newton iteration is applied in order to linearize the system. In every Newton step a linear saddle point system has to be solved (after discretization). The symmetric linear system is solved by a preconditioned MINRES method using block-diagonal preconditioners. For the case without inequality constraints on the control, we present a preconditioner which leads to convergence rates robust in the discretization parameter $h$ and the cost parameter $\alpha$. For the case with inequality constraints, we present a preconditioner which leads to convergence rates robust in the discretization parameter. Numerical examples are given which illustrate the theoretical results.