Department Colloquium in Winterterm 

05 February, 16:00 - 17:00

Prof. Dr. Arnold Reusken, RWTH Aachen

In this presentation we give an overview of several aspects related to the analysis and numerical simulation of elliptic and parabolic PDEs on (evolving) surfaces. We consider both scalar valued and vector valued PDEs. Surface partial differential equations are used in models from, for example, computational fluid dynamics and computational biology. Two examples of such applications are briefy addressed. Concerning the analysis of surface PDEs, a few results related to well-posedness of certain weak formulations are explained. We discuss a class of recently developed finite element discretization methods for an accurate numerical simulation of surface PDEs. The key idea of these methods is explained and numerical simulation results are presented.

22 January,  16:00 - 17:00 

Prof. Dr. Jan Maas, IST Austria

At the end of the 1990s it was discovered by Jordan/ Kinderlehrer/Otto that the diffusion equation is a gradient flow in the space of probability measures, where the driving functional is the Boltzmann-Shannon entropy, and the dissipation mechanism is given by the 2-Wasserstein metric from optimal transport. This result has been the starting point for striking developments at the interface of analysis, probability, and metric geometry.

In this talk I will review joint work with Eric Carlen, in which we introduced new optimal transport metrics that yield gradient flow descriptions for dissipative quantum systems with detailed balance. This approach yields functional inequalities related to convergence to equilibrium in several examples.

27 November,  16:00 - 17:00 

Prof. Dr. Lars Grüne, University of Bayreuth

Model predictive control (MPC) is a popular control method, in which a feedback control for a problem on a variably or infinitely long time horizon is computed from the successive numerical solution of optimal control problems on relatively short time horizons. It can thus be seen as a model reduction technique in time for optimal control. Clearly, an optimal control problem must have a certain amount of redundancy for MPC to work properly. In the first part of this talk, we will show that the so-called turnpike property from optimal control provides the desired redundancy.

In the second part we address computational issues when applying MPC to PDEs. Here we exploit a particular feature of MPC, i.e., that typically the optimal control problems are solved on overlapping horizons, implying that only a small portion of the computed optimal control function is actually used. This suggests that an adapted discretization in time and/or space may offer a large benefit for MPC of PDEs. We explain the theoretical justification of this approach based on novel sensitivity results for the optimal control of general evolution equations. Then the efficiency of the proposed method is illustrated by numerical experiments.
The talk is based on joint work with Roberto Guglielmi (L'Aquila), Matthias Müller (Hannover), Manuel Schaller and Anton Schiela (both Bayreuth), Marleen Stieler (BASF Ludwigshafen)

30 October, 16:00 - 17:00

Prof. Dr. Bruno Després, Université de Paris 6, John von Neumann Professor

The design of high order methods with strong control of the maximum principle is still a core problem in scientific computing. That is why a natural theoretical question is  to have good representations of polynomial with bounds. This topic ranges from scientific computing and numerical analysis to convex analysis and purely algebraic considerations. I will describe  recent ideas based the Lukacs Theorem and algebraic representations as sums of squares (SOS): it results in new efficient Newton-Raphson and convex  algorithms for  calculations of SOS. The case of polynomial with two bounds can be treated with a (surprising) quaternion algebra. An application to the discretization of the transport equation will be described. 

16 October, 16:00 - 17:00 

Prof. Patrick Dondl, Albert-Ludwigs-Universität Freiburg

Additive manufacturing (AM) is a rapidly emerging technology that has the potential to produce personalized scaffolds for tissue engineering applications with unprecedented control of structural and functional design. Particularly for bone defect regeneration, the complex coupling of biological mechanisms to the scaffolds’ properties has led to a widespread trial-and-error approach. To mitigate this, shape or topology optimization can be a useful tool to design a scaffold architecture that matches the desired design targets, albeit at high computational cost. Here, we consider two complementary approaches: first, an efficient macroscopic optimization routine based on a simple one-dimensional time-dependent model for bone regeneration in the presence of a bioresorbable polymer scaffold is developed. The result of the optimization procedure is a scaffold porosity distribution which maximizes the stiffness of the scaffold and regenerated bone system over the regeneration time, so that the propensity for mechanical failure is minimized. Second, we consider a periodic microstructure optimization problem for scaffold architectures based on a domain-splitting.

Joint work with K. Bhattacharya (Caltech), M. v. Griensven (TU Munich), P. Poh (Charité Berlin), M. Rumpf (Bonn), S. Simon (Bonn), D. Valainis (TUM).