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


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 ⊂ R2, we consider

minq ∈ L2, u ∈ H1D l(u)

subject to the constraints

(q ,σ(∇ u),∇ φ) = l(φ)   ∀, φ ∈ H1D(Ω;R2),

0 < qmin ≤ q ≤ qmax,

Ω q ≤ Vmax,

where H1D(Ω;R2) denotes the usual H1-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


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
problem. The numerical solution relies on the formulation of the fourth
equation as a system of two second order ones which enables the
by P1 conforming finite elements with respect to simplicial triangulations
the space-time domain. The resulting algebraic saddle point problem is
by preconditioned Richardson iterations featuring preconditioners
by means of appropriately chosen left and right transforms. The space-time
adaptivity is realized by a reliable residual-type a posteriori error
which is derived by the evaluation of the two residuals associated with
underlying second order system. Numerical results are given that
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.


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