Optimal Control of Thermoviscoplasticity
Dienstag, 8.5.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Elastoplastische Verformungen spielen eine wichtige Rolle in industriellen Verformungsprozessen. Oft finden diese unter nicht isothermen Bedingungen statt. Daher ist die Optimierung solcher Probleme nicht nur von mathematischem Interesse sondern insbesondere interessant für industrielle Anwendungen.\n\nIn meinem Vortrag werde ich die Analysis der Existenz von globalen Lösungen eines Optimalsteuerproblems basierend auf einem thermovisko(elasto)plastischen Modell und ihre Differenzierbarkeitseigenschaften diskutieren. Ich werde insbesondere auf die Schwierigkeiten eingehen, die durch die nichtlineare Koppelung der Wärmeleitungsgleichung mit den mechanischen Gleichungen des Models, entstehen.\n\nSchließlich werde ich einige numerische Beispiele präsentieren.
Norm-Resolvent Convergence in Perforated Domains
Dienstag, 12.6.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Heuristic Solvers for Edge Clique Cover Graph Problems Based on Deep Neural Networks
Dienstag, 19.6.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Combinatorial optimization over graphs often presents NP-hard problems, which require considerable manual effort for deriving problem-specific heuristic solvers. Recent research suggests learning of such solvers with deep neural networks and yields high experimental performance on various problems. Yet a theoretical discussion on learnability from a statistical perspective is omitted. We propose traditional feedforward neural networks (FNN) and recurrent neural networks (RNN) that adapt to graph size, in order to learn heuristic solvers for the NP-hard edge clique cover number (ECCN) problem. Both types of architectures are examined in the framework of statistical learning theory, which allows derivation of problem-independent sample complexity bounds for the respective networks. We find that, whereas iterating through all graphs with n vertices takes O(2^{n^2}), FNN-based solvers require at most O(n^3 ln(n)), and RNN-based solvers at most O(n^6) samples to provide reliable heuristic solvers. Experimental evaluation with random graphs on the ECCN problem con firrms a high solution quality, especially of RNNs. On dense graphs, an accuracy of 82.7% is reached, which outperforms the state-of-the-art heuristic from the operations research literature by 15.6 percentage points.
Dienstag, 3.7.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications
Freitag, 13.7.18, 12:00-13:00, Bibliothek Angewandte Mathematik, R216, RZ, Hermann-Herder-Strasse 10
We propose a new variational model in weighted Sobolev spaces with\nnon-standard weights and applications to image processing. We show that\nthese weights are, in general, not of Muckenhoupt type and therefore the\nclassical analysis tools may not apply. For special cases of the\nweights, the resulting variational problem is known to be equivalent to\nthe fractional Poisson problem. The trace space for the weighted Sobolev\nspace is identified to be embedded in a weighted \(L^2\) space. We propose\na finite element scheme to solve the Euler-Lagrange equations, and for\nthe image denoising application we propose an algorithm to identify the\nunknown weights. The approach is illustrated on several test problems\nand it yields better results when compared to the existing total\nvariation techniques.\n
wird noch bekanntgegeben
Montag, 6.8.18, 11:30-12:30, Raum 226, Hermann-Herder-Str. 10