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11.03.2019 17:00 Alexander Leichner (Fraunhofer ITWM):
A Voxel-based Algorithm for Contact using an Implicit Boundary RepresentationGebäude 33, Raum 2401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

Nowadays, computational contact mechanics has developed into an effective simulation tool for collisions between solids in various engineering problems. During the last decades, the development of alternative solvers operating on regular grids has been driven forward, too. Hence, the question has arisen if a voxel-based method for contact problems can be established. In contrast to a standard polygonal discretization, the numerical treatment of those based on a structured mesh is not equally perfected.

The common procedures rely on boundary aligned meshes which support the numerical enforcement of the contact conditions. Based on that geometrical approximation, the finite element method proves itself to be suitable to handle with those. Concerning computational performance, voxel-based procedures are generally competitive with the common ones. In particular, direct image processing is possible eliminating the need for expensive mesh generations.

Nevertheless, the main disadvantage of regular grids is the poor quality of the boundary approximation in general. Even with fine resolutions, surfaces can not be reproduced as accurately as by comparable unstructured meshes. In addition, the application of surface projections is not as straightforward as with the standard discretization.

For all mentioned issues we propose the usage of the implicit boundary representation: Instead of the precise localization of surfaces we only require isovalues of level set functions. The boundaries are then identified as surface contours resulting in a Eulerian description of the kinematics. In combination with Nitsche's approach, we obtain an appropriate scheme for handling contact problems in connection with regular grids. Our proposed Nitsche-based method additionally relies on a novel reference surface correcting the biased choice of the contact surface through the usual master-slave view.

The final algorithm is a composition of methods operating from different perspectives: On the one hand, we apply in particular the level set method and fast marching method for the active set search in the Eulerian point of view. On the other hand, the finite element formulation based on structured meshes is applied according to the Lagrangian description. This variation of perspectives allows for a simple and efficient implementation and accurate results on regular grids. We validate this at the end of the talk by showing optimal convergence and presenting complex examples with multiple contacts.

12.03.2019 14:30 Prof. Dr. Ehrhard Behrends (Freie Universität Berlin):
Mathematik zaubertMI HS 1 (Boltzmannstr. 3, 85748 Garching)

Es ist weitgehend bekannt, dass es Zaubertricks mit mathematischem Hintergrund gibt. Doch hat sich bei vielen noch nicht herumgesprochen, dass es um weit mehr geht als um Zahlenmanipulationen. Ergebnisse aus vielen mathematischen Teilgebieten geben Anlass zu Zaubertricks: Kombinatorik, Zahlentheorie, Codierungstheorie, Stochastik, ...

In dem Vortrag soll ein breites Spektrum vorgestellt werden. Zum Beispiel werden die Fibonaccizahlen, quadratische Reste, Normalteiler und Wartezeiten zum Einsatz kommen. Die auch für Laien zugänglichen Aspekte sind in meinem Buch „Der mathematische Zauberstab“ veröffentlicht worden. Die Ergebnisse, die eine mathematische Vorbildung erfordern, finden sich in „Mathematik und Zaubern - ein Einstieg für Mathematiker“.

Mit Ihrem eigenen Kartenspiel können Sie einige Tricks vor Ort sofort ausprobieren!

13.03.2019 14:00 Georgios Piliouras:
Online Optimization in Zero-Sum Games: A Dynamical Systems ApproachMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Zero-Sum games are basic staples of game theory. Their study is also closely connected with machine learning challenges such as training Generative Adversarial Networks (GANs). Online learning algorithms, such as online gradient descent and multiplicative weights are standard tools of optimization theory and are known to provide optimal performance guarantees in adversarial environments. We study systems where such online algorithms compete against each other in zero-sum games. We prove that these systems exhibit Poincaré recurrence and can be interpreted as Hamiltonian dynamics. We discuss implications of these results for discrete-time dynamics and we present some open questions.

Bio: Dr. Piliouras is an assistant professor at the Singapore University of Technology and Design (SUTD). He received his PhD in Computer Science from Cornell University in 2010. He has held postdoc positions at the Georgia Institute of Technology (GaTech, ECE Dept.) and California Institute of Technology (Caltech, Dept. of Computing and Mathematical Sciences). He has held visiting positions at UC Berkeley and DeepMind and he is a collaborator of the Ethereum Foundation. He is the recipient of a Singapore NRF Fellowship (2018) and a Simons/UC Berkeley Fellowship (2015).