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07.03.2018 16:00 Martin Siebenborn (Universität Hamburg):
Shape optimization algorithms for interface identificationMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In many applications, which are modeled by partial differential equations (PDEs), there is a small number of materials or parameters distinguished by interfaces to be identified. While classical approaches in the field of optimal control yield continuous solutions, a spatially distributed, binary variable is closer to the desired application. It is thus favorable, to treat the shape of the interface between an active and inactive control as the variable. Moreover, since the involved materials may form complex contours, high resolutions are required in the underlying finite element method. We investigate a combination of classical PDE constrained optimization methods and a rounding strategy based on shape optimization for the identification of interfaces. The goal is to identify the location of pollution sources in fluid flows represented by a control that is either active or inactive. While the most intuitive approach would be to treat this as a combinatorial problem and to decide cell by cell whether it is polluted or not, computational complexity and mesh-dependent solutions are major drawbacks. These issues can be circumvented by the algorithm we present here. Moreover, it is shown how the topology of an inaccurate initial guess for the pollutant locations can be corrected during the shape optimization, which is a typical problem in that field.

References [1] M. Siebenborn. A shape optimization algorithm for interface identification allowing topological changes. arxiv.org/abs/1711.02535, 2018. [2] M. Siebenborn and K. Welker. Algorithmic aspects of multigrid methods for optimization in shape spaces. SIAM Journal on Scientific Computing, 39(6):B1156–B1177, 2017.

12.03.2018 15:00 Dr. Sebastian Riedel (Technische Universität Berlin):
Rough differential equations with unbounded driftMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider rough differential equations with a possibly unbounded drift term. It turns out that the usual one-sided growth conditions are not sufficient to guarantee non-explosion of the solution in finite time. We then provide a further condition under which non-explosion can be assured. Some applications in stochastic analysis are discussed. Joint work with M. Scheutzow (Berlin).

14.03.2018 16:30 Prof. Dr. Vincent Tassion (ETH Zürich):
The phase transition for Boolean percolationB 252 (Theresienstr. 39, 80333 München)

Based on joint works with Daniel Albergh, Hugo Duminil-Copin, Aran Raoufi and Augusto Teixeira.

Consider Poisson-Boolean percolation in R^d. Around every point of a Poisson point process of intensity lambda, draw independently a ball with random radii. I will discuss the sharpness of the phase transition in lambda for this process. In particular, I will present sharp bounds for the cluster of the origin to have radius greater than n in the subcritical regime, and how these bounds depend on the radii distributions. >