# Department Colloquium in Winterterm

## Talks in Winter Semester 2019/20

At the Colloquium of the Department of Mathematics international researchers report on their work. The colloquium will be held on Wednesdays in "Hörsaal 3" (MI 00.06.011).

We invite all interested parties.

**These are the lecture dates in Winter Semester 2019/20: **

- 16 October 2019: Patrick Dondl (16:00)
- 30 October 2019: Bruno Després (16:00)
- 27 November 2019: Lars Grüne (16:00)
- 22 January 2020: Jan Maas (16:00)
- 05 February 2020: Arnold Reusken (16:00)

### Topics at the department colloquium

#### Partial differential equations on surfaces: Analysis, numerical methods and applications

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.

#### Gradient flows and entropy inequalities in dissipative quantum systems

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.

#### Model predictive control: when does it work and how can it be

done efficiently?

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)

#### Positive polynomials and numerical approximation

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.

#### Optimization of additively manufactured polymer scaffolds for bone tissue engineering

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).

Further information can be found on the overview page of the Colloquium of the Department of Mathematics.