# Optimal control, topology and shape design using multigrid and augmented Lagrangians

## Ing. Dalibor Lukáš PhD

**Feb. 20, 2007, 3:30 p.m. HF 136**

The talk consists of two parts. First, we describe a method of augmented Lagrangians for solution to equality constrained quadratic programming. The method has been recently developed and analyzed by Z. Dostal (Ostrava) and the complexity in terms of outer iterations has been proven to depend only on the smallest eigenvalue of the Hessian of the quadratics. Our approach does not require a unique Lagrange multipliers. After a proper multigrid preconditioning of the inner PCG-iterations, the overall complexity is also optimal. We will present numerical applications to the Stokes, stationary Navier-Stokes and optimal control problems. At the end of the first part we will present a recent application to simultaneous topology optimization in magnetostatics.

The second part of the talk is devoted to a multigrid preconditioning for black-box (nested) shape optimization in magnetostatics. We will present a standard approach in which the shape controls perturbation of the discretization grid, which however restricts the design space on finer levels. As a remedy to the latter we propose a fixed grid (fictitious domain-like) approach, where the shape controls PDE-coefficients of the intersected grid elements. The arising linear systems are solved using an algebraic multigrid package developed by Johannes Kraus (RICAM Linz).