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## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Verantwortliche Dozenten: | Alle Professoren der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen |
---|---|

Koordination: | Dr. Matthias Voigt |

Termine: | Di 16-18 Uhr in MA 313 und nach Vereinbarung |

Inhalt: | Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen |

Kontakt: | kolloquium-mnd@math.tu-berlin.de |

## Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Datum date | Zeit time | Raum room | Vortragende(r) speaker | Titel title | Einladender invited by |
---|---|---|---|---|---|

23.10.2018 | 16:15 | MA 313 | Ronald W. Hoppe (Universität Augsburg / University of Houston) Rolf D. Grigorieff | Numerical Solution of Second and Fourth Order Total Variation Flow Problems | Kolloquium zum 80. Geburtstag von Rolf D. Grigorieff |

06.11.2018 | 16:15 | MA 313 | Robert Calderbank (Duke University) | Golay, Heisenberg and Weyl | G. Kutyniok |

13.11.2018 | 16:15 | MA 313 | David Šiška (University of Edinburgh) | Exponential Convergence of Policy Improvement Algorithm for Controlled Diffusions | E. Emmrich |

04.12.2018 | 16:15 | MA 313 | María Soledad Aronna (Escola de Matemática Aplicada Rio de Janeiro) | On second order optimality conditions for control-affine problems | F. Tröltzsch |

08.01.2019 | 16:15 | MA 313 | Herbert Egger (TU Darmstadt) | On the systematic approximation of evolution problems with dissipation, Hamiltonian, or gradient structure | V. Mehrmann |

15.01.2019 | 16:15 | MA 313 | |||

22.01.2019 | 16:15 | MA 313 | Volker Mehrmann (TU Berlin) | Stability through structure for port-Hamiltonian differential-algebraic systems | B. Zwicknagl |

29.01.2019 | 16:15 | MA 313 | Paul Kotyczka (TU München) | Discrete-time port-Hamiltonian systems: A definition based on symplectic integration | V. Mehrmann |

05.02.2019 | 16:15 | MA 313 | Raphael Kruse (TU Berlin) | On randomized time-stepping methods for non-autonomous evolution equations with time-irregular coefficients | M. Voigt |

12.02.2019 | 16:15 | MA 313 | Heiner Olbermann (UC Louvain) | Paper crumpling - at the crossroads of differential geometry, calculus of variations and materials science | B. Zwicknagl |

# Abstracts zu den Vorträgen

**Ronald W. Hoppe ****(Universität Augsburg / University of Houston)**

Numerical Solution of Second and Fourth Order Total Variation Flow Problems

**María Soledad Aronna (Escola de Matemática Aplicada Rio de Janeiro): **

*On second order optimality conditions for control-affine problems *

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control and nonlinear with respect to the state. Assuming the presence of control constraints, I will present a second order sufficient condition for optimality and a numerical scheme in the form of a shooting method.

**Robert Calderbank (Duke University): **

*Golay, Heisenberg and Weyl*

Sixty years ago, efforts by Marcel Golay to improve the sensitivity of far infrared spectrometry led to the discovery of pairs of complementary sequences. We will describe how these sequences are finding new application in active sensing, where the challenge is how to see faster, to see more finely where necessary, and to see with greater sensitivity, by being more discriminating about how we look.

**Biography: **Robert Calderbank is Director of the Information Initiative at Duke University, where he is Professor of Mathematics and of Electrical and Computer Engineering. Prior to joining Duke in 2010, he directed the Program in Applied and Computational Mathematics at Princeton University. Prior to joining Princeton in 2004 he was Vice President for Research at AT&T, in charge of what may have been the first industrial research lab where the primary focus was Big Data.

Dr. Calderbank is well known for contributions to voiceband modem technology, to quantum information theory, and for co-invention of space-time codes for wireless communication. His research papers have been cited almost 50,000 times and his inventions are found in billions of consumer devices. Professor Calderbank was elected to the National Academy of Engineering in 2005 and has received a number of awards, including the 2013 IEEE Hamming Medal for his contributions to information transmission, and the 2015 Claude E. Shannon Award.

**David Siska (University of Edinburgh)**

*Exponential Convergence of Policy Improvement Algorithm for Controlled Diffusions*

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control and nonlinear with respect to the state. Assuming the presence of control constraints, I will present a second order sufficient condition for optimality and a numerical scheme in the form of a shooting method.

**Herbert Egger (TU Darmstadt)**

*On the systematic approximation of evolution problems with dissipation, Hamiltonian, or gradient structure.*

A general framework for the numerical approximation of evolution problems is presented that allows

to preserve an underlying dissipative, Hamiltonian, or gradient structure. The approach is based on rewriting the evolution problem in a particular form that complies with the underlying structure and its variational formulation. The underlying structure is then preserved automatically under Galerkin projection in space, which allows to deduce important structural properties for appropriate discretization schemes including projection based model reduction methods.

For the time-discretization, we consider two different approaches depending on the underlying geometric

structure, i.e., discontinuous Galerkin and Petrov-Galerkin approximations. Again, the basic structure of the problem is inherited automatically by the proposed discretization schemes.

The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Several examples will be discussed for illustration and some connections to other discretization approaches will be revealed.

**Volker Mehrmann (TU Berlin)**

*Stability through structure for port-Hamiltonian differential-algebraic systems*

To analyse stability of constrained dynamical systems is an important taks in many applications. In the context of energy based modeling, stability and also passivity of the system can be read off directly from the structure of the system as a port-Hamiltonian system.

This approach works directly in linear stability analysis but also in the general nonlinear system. We present the concept of port-Hamiltonian differential-algebraic systems and analyze the properties, including the construction of Lyapunov functions and the computation of the distance to instability/non-passivity for large scale systems. The results are illustrated for several applications including the analysis of brake squeal.

**Paul Kotyczka (TU München)**

*Discrete-time port-Hamiltonian systems: A definition based on symplectic integration*

We introduce a new definition of discrete-time port-Hamiltonian systems, which is based on the discretization of the underlying continuous-time Dirac structure using the collocation method. Discrete-time dynamics is added by means of an s-stage symplectic numerical integration scheme.

The preservation of the structural energy balance, expressed in terms of a discrete-time Dirac structure, can be considered a natural extension of symplecticity of geometric integration schemes to open systems. Substituting discrete-time constitutive equations in the balance equation, an error between supplied and stored energy occurs, whose order corresponds to the symplectic integration scheme. For implicit Gauss-Legendre schemes, applied to linear port-Hamiltonian systems, the error vanishes and the discrete energy balance is exact. Numerical experiments illustrate our results.

**Raphael Kruse (TU Berlin)**

*On randomized time-stepping methods for non-autonomous evolution equations with time-irregular coefficients*

In this talk, we consider the numerical approximation of Carathéodory-type ODEs and of nonlinear and non-autonomous evolution equations whose coefficients may be irregular or discontinuous with respect to the time variable. In this non-smooth situation, it is difficult to construct numerical algorithms with a positive convergence rate. In fact, it can be shown that any deterministic

algorithm depending only on point evaluations may fail to converge. Instead, we propose to apply randomized Runge–Kutta methods to such time-irregular evolution equations as, for instance, a randomized version of the backward Euler method. We obtain positive convergence rates with respect to the mean-square norm under considerably relaxed temporal regularity conditions. An important ingredient in the error analysis consists of a well-known variance reduction technique for Monte Carlo methods, the stratified sampling. We demonstrate the practicability of the new algorithm in the case of a fully discrete approximation of a parabolic PDE.

This talk is based on joint work with Monika Eisenmann (TU Berlin), Mihály Kovács and Stig Larsson (both Chalmers University of Technology) as well as Yue Wu (U Edinburgh).

**Heiner Olbermann (UC Louvain)**

*Paper crumpling - at the crossroads of differential geometry, calculus of variations and materials science*

When crumpling a thin elastic sheet (think: a piece of paper), one observes the emergence of an intricate pattern of regions where the deformation energy focuses. The rigorous analysis of these patterns turns out to be a very difficult and interesting problem, linking nonlinear elasticity, the calculus of variations and differential geometry. In this talk, I will give a short overview of how these questions can be located in a larger mathematical context, and then present the current state of the art, including some of my own results.

## Rückblick

- Kolloquium ModNumDiff Sommer 2018
- Kolloquium ModNumDiff Winter 2017/2018
- Kolloquium ModNumDiff Sommer 2017
- Kolloquium ModNumDiff Winter 2016/17
- Kolloquium ModNumDiff Sommer 2016
- Kolloquium ModNumDiff Winter 2015/16
- Kolloquium ModNumDiff Sommer 2015
- Kolloquium ModNumDiff Winter 2014/15
- Kolloquium ModNumDiff Sommer 2014
- Kolloquium ModNumDiff Winter 2013/14
- Kolloquium ModNumDiff Sommer 2013
- Kolloquium ModNumDiff Winter 2012/13
- Kolloquium ModNumDiff Sommer 2012
- Kolloquium ModNumDiff Winter 2011/12
- Kolloquium ModNumDiff Sommer 2011