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ALOP-Colloquium with Prof. Dr. Luise Blank
June 29 / 16:00 - 18:00
On Monday, June 29. 2020 at 16:00 c.t. Prof. Dr. Luise Blank with the Universitaet Regensburg will present her recent work at our colloquium entitled
Optimization with convex constraints and an application in topology optimization
This talk focuses on projection type methods for convexly constrained optimization problems. We shortly introduce well known projection methods using the gradient in ﬁnite dimensions and summarize known results. Varying the underlying scalar product allows to include second order information to speed up the method. E.g. the projection of the Newton direction leads to a quadratic order of convergence. The method can also be interpreted as solving a sequence of optimization problems where the cost function is quadratically approximated. Results in function spaces are important to obtain methods with iteration numbers independent ot the discretization level. They also indicate which underlying metric shall be used for the projection. E.g. pde constraint problems provide often only diﬀerentiability of the cost functional in the L∞ metric. While the generalization to Hilbert spaces is straight forward the extension to Banach spaces is more involved due to the missing scalar product. We present a global convergence result for the variable metric projection type (VMPT-)method, which allows application to nonreﬂexive Banach spaces and expand the possible choices of the metric. Moreover, we give examples which fulﬁll the requirements on the spaces and the variable metric.
As an application we consider a structural topology optimization problem. The model is based on a diﬀuse interface ansatz using phase ﬁeld variables. The necessary regularization of the cost functional with the perimeter is substituted by the Ginzburg-Landau energy. We obtain a minimization problem with pde constraints and simplex constraints for the controls. The elimination of the state using the nonlinear control-to-state operator yields a convexly constrained optimization problem, where we can apply the VMPT-algorithm in L∞ ∩ H1. In the numerical results one can clearly see that choosing an appropriate inner product, namely H1, leads –in contrast to L2– to mesh independency of the iteration numbers. Including additionally second order information speeds up the method drastically.
The presentation will take place remotely via Zoom.
If you wish to receive a link and invitation to this presentation, please send an email to shawATuni-trier.de