Unendliche Körper mit Quantorenelimination
Tuesday, 4.7.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Heintze-Karcher inequality and Alexandrov’s theorem for capillary hypersurfaces
Tuesday, 4.7.23, 16:00-17:00, Raum 127, Ernst-Zermelo-Str. 1
Heintze-Karcher’s inequality is an optimal geometric inequality for embedded closed hypersurfaces, which can be used to prove Alexandrov’s soap bubble theorem on embedded closed CMC hypersurfaces in the Euclidean space. In this talk, we introduce a Heintze-Karcher-type inequality for hypersurfaces with boundary in the half-space. As application, we give a new proof of Wente’s Alexandrov-type theorem for embedded CMC capillary hypersurfaces. Moreover, the proof can be adapted to the anisotropic case, which enable us to prove an Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces.\nThis is based on joint works with Xiaohan Jia, Guofang Wang and Xuwen Zhang.\n
On a Mystery in Machine Learning
Friday, 7.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
In classical regression modelling, the complexity of the model, e.g. measured by the number of parameters, is smaller than the amount of training data. The prediction error exhibits a U-shaped behaviour. The (first) descent is due to decreasing bias, the ascent due to increasing variance. In modern machine learning, often the number of parameters by far exceeds the number of training data points. Intuitively, one could expect that the prediction error explodes with increasing model complexity due to overfitting. Belkin et al. (2019) observed that this is not the case. Instead, the prediction error decreases again when surpassing a certain threshold in model complexity, in some case even below the minimum of the classical, U-shaped regime. A phenomenon the authors denominated as double descent. To understand double descent, we study the simplest setting of linear regression and show that it can be explained by investigating the singular values of the design matrix. Finally, we give an outlook for the non-linear model setting.\n\nBelkin, M.; Hsu, D.; Ma, S.; Mandal, S.: Reconciling modern machine-learning practice and the classical bias–variance trade-off. In: Proceedings of the National Academy of Sciences 116 (2019), jul, Nr. 32, 15849–15854.
Fakultätsfest und Abschlussfeier
Friday, 7.7.23, 14:00-15:00, Großer Hörsaal Physik, Hermann-Herder-Straße 3a
Stability properties of the \(L^2\)-projection mapping to finite element spaces on adaptively generated meshes
Tuesday, 11.7.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nThe \(L^2\)-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Its Sobolev stability is known to be the key to discrete stability and quasi-optimality estimates for parabolic problems. For adaptively generated meshes the proof of Sobolev stability is challenging and requires conditions on how strongly the mesh size varies.\n\nHence, for the newest vertex bisection and its generalisation to higher dimensions by Maubach and Traxler we present optimal estimates on the mesh grading. Previously, grading estimates have been available only for 2D mesh refinement strategies. For such adaptively generated meshes we discuss Sobolev stability of the \(L^2\)-projection mapping to Lagrange finite element spaces under certain conditions on the polynomial degree and on the space dimension. In particular, the \(L^2\)-projection is \(W^{1,2}\)-stable for any polynomial degree, for any space dimension smaller than \(7\).\n\nThis is joint work with Lars Diening and Johannes Storn (Bielefeld University).\n
On the Six Functor Formalism for Nori Motivic Sheaves
Friday, 14.7.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
A Principal-Agent Framework for Optimal Incentives in Renewable Investments
Friday, 14.7.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
We investigate the optimal regulation of energy production reflecting the long-term goals of the Paris climate agreement.\n\nWe analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is served either by a monopoly or by two competing agents. The regulator wishes to encourage green investments to limit carbon emissions, while simultaneously reducing intermittency of the total energy production. We find that the regulation of a competitive market is more efficient than the one of the monopoly as measured with the certainty equivalent of the Principal’s value function. This higher efficiency is achieved thanks to a higher degree of freedom of the incentive mechanisms which involves cross-subsidies between firms. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario. In addition, we expand the monopolistic and competitive setup to a more general class of tractable Principal-Multi-Agent incentives problems when both the drift and the volatility of a multi-dimensional diffusion process can be controlled by the Agents. We follow the resolution methodology of Cvitanić et al. (2018) in an extended linear quadratic setting with exponential utilities and a multi-dimensional state process of Ornstein-Uhlenbeck type. We provide closed-form expression of the second-best contracts. In particular, we show that they are in rebate form involving time-dependent prices of each state variable.
Spinoren, kalibrierte Untermannigfaltigkeit und Instantonen
Monday, 17.7.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
In der Modulraumtheorie gibt es eine tiefe, weitgehend unverstandene Dualität zwischen den Instanton-Zusammenhängen auf Hauptfaserbündeln über einer Mannigfaltigkeit und den kalibrierten Untermannigfaltigkeiten, welche als Modelle für „singuläre“ Instantonen auftreten. Während des Vortrags werde ich mithilfe von Dirac-Operatoren und Spinoren eine entsprechende Dualität (im adiabatischen Limes) der linearisierten Deformationstheorien dieser Modulräume herstellen. Das Hauptergebnis hat Anwendungen auf die Konstruktion von Orientierungsdaten in der Donaldson-Thomas Theorie.
Friday, 21.7.23, 13:30-14:30, Raum 226, Hermann-Herder-Str. 10
On the role of discrete Green’s operator preconditioning in FFT-based computational homogenization methods
Tuesday, 25.7.23, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
Solving computational homogenization problems on fine grids leads to systems of linear equations with millions to billions of unknowns, which favour iterative solvers over direct solvers.\nHowever, the number of iterations of iterative solvers can grow with increasing system size. To\novercome this issue, the so-called FFT-based solvers use the discrete Green’s operator preconditioning, which makes the condition number of the resulting linear system independent of the\nsystem/grid size [1, 2]. We studied the discrete Green’s operator preconditioning from a linear\nalgebra viewpoint and showed that all individual eigenvalues of such preconditioned systems can\nbe bounded purely from the knowledge of the material data of the problems, both original and\nreference. We developed a simple algorithm to compute these bounds [3, 4]. In my talk, I will\ndiscuss the theoretical aspects of these results and practical applications of the discrete Green’s\noperator preconditioning to periodic homogenisation problems discretised on regular grids [5].\n\n\nReferences\n\n[1] Moulinec, H. and Suquet, P., A fast numerical method for computing the linear and nonlinear\nmechanical properties of composites, Comptes Rendus de l’Acad´emie des sciences. S´erie II.\nM´ecanique, physique, chimie, astronomie, 318 (1994) 1417–1423\n\n[2] Schneider, M., A review of nonlinear FFT-based computational homogenization methods,\nActa Mechanica, 29 (2021) DOI 10.1007/s00707-021-02962-1,\n\n[3] Ladeck´y, M. and Pultarov´a, I. and Zeman, J., Guaranteed two-sided bounds on all eigenvalues of preconditioned diffusion and elasticity problems solved by the finite element method,\nApplications of Mathematics, 66, 21–42 (2020) DOI 10.21136/AM.2020.0217-19\n\n[4] Pultarov´a, I., Ladeck´y, M., Two-sided guaranteed bounds to individual eigenvalues of preconditioned finite element and finite difference problems, Numerical Linear Algebra Applications\n28 (2021) e2382. DOI 10.1002/nla.2382.\n\n[5] Ladeck´y, M., Leute, J.R., Falsafi, A., Pultarov´a, I., Pastewka, L., Junge, T., and Zeman,\nJ. An Optimal Preconditioned FFT-accelerated Finite Element Solver for Homogenization.\nApplied Mathematics and Computation 446 (2023) 127835 DOI 10.1016/j.amc.2023.127835