## Invited Session Tue.1.MA 415

#### Tuesday, 10:30 - 12:00 h, Room: MA 415

**Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods** [...]

### Adaptive methods in PDE constrained optimization

**Chair: Stefan Ulbrich**

**Tuesday, 10:30 - 10:55 h, Room: MA 415, Talk 1**

**Winnifried Wollner**

Adaptive finite element discretizations in structural optimization

**Abstract:**

In this talk we will consider a prototypical example from structural optimization.

Namely the well known compliance minimization of a variable thickness sheet, i.e.,

given a domain *Ω ⊂ ***R**^{2}, we consider

min_{q ∈ L2, u ∈ H1D} l(u)

subject to the constraints

** **

(q ,σ(∇ u),∇ φ) = l(φ) ∀, φ ∈ H^{1}_{D}(Ω;**R**^{2}),

0 < q_{min} ≤ q ≤ q_{max},

∫_{Ω} q ≤ V_{max},

where *H*^{1}_{D}(Ω;**R**^{2}) denotes the usual *H*^{1}-Sobolev space with certain Dirichlet boundary conditions,

and *σ(∇ u)* denotes the usual (linear) Lam{é}-Navier stress tensor.

As it is well known that the effort for the optimization is directly linked to the number of unknowns

present in the discretization we will derive an a posteriori error estimator in order to drive local mesh

refinement with respect to a given target quantity.

Finally we will give an outlook to possible extensions.

**Tuesday, 11:00 - 11:25 h, Room: MA 415, Talk 2**

**Ronald H. Hoppe**

Adaptive space-time finite element approximations of parabolic optimal control problems

**Abstract:**

We consider adaptive space-time finite element approximations of parabolic

optimal control problems with distributed controls based on an approach

where the optimality system is stated as a fourth order elliptic boundary

value

problem. The numerical solution relies on the formulation of the fourth

order

equation as a system of two second order ones which enables the

discretization

by P1 conforming finite elements with respect to simplicial triangulations

of

the space-time domain. The resulting algebraic saddle point problem is

solved

by preconditioned Richardson iterations featuring preconditioners

constructed

by means of appropriately chosen left and right transforms. The space-time

adaptivity is realized by a reliable residual-type a posteriori error

estimator

which is derived by the evaluation of the two residuals associated with

the

underlying second order system. Numerical results are given that

illustrate

the performance of the adaptive space-time finite element approximation.

% The results are based on joint work with F. Ibrahim, M. Hintermüller, M.

% Hinze, and Y. Iliash.