Young Researchers' Minisymposia 2019

The port-Hamiltonian (pH) modeling approach is a powerful tool, which allows to read important properties of a dynamical system directly from its structure. By extending the Hamiltonian system formulation by ports, one is able to describe phenomena where energy is dissipated or exchanged. In fact, the pH formulation leads to an energy balance stating that the temporal change of the Hamiltonian, which is often the total energy, is only affected by dissipation and by the exchange of energy with the environment or with other systems. From this energy balance it directly follows that the Hamiltonian represents a storage function as in the definition of passivity as well as a Lyapunov function. Consequently, pH systems are both stable and passive. Moreover, the pH approach is perfectly suited for networks and multiphysics problems, since a power-preserving interconnection of such systems results again in a pH system. Due to the mentioned properties, pH formulations are used in various applications including fluid dynamics, solid mechanics, electromagnetism, and chemical reactions. All these disciplines have energy as a common language and therefore they fit perfectly into the pH framework. Another important aspect is that a pH structure can be formulated for ordinary differential equations and differential-algebraic equations as well as for partial differential equations. As a consequence, the pH structure can be used for preserving stability and passivity after discretization and model reduction. This leads to the fields of structure-preserving discretization and model reduction, which experienced much research effort in the past decade. Also control concepts based on a pH formulation have been thoroughly investigated in the past years, especially in relation to the topic of passivity-based control. The aim of this minisymposium is to bring together young scientists from the field of pH modeling and control. Due to the interdisciplinarity of this field, a special emphasis is on addressing researchers from different scientific disciplines including applied mathematics and engineering. Topics covered in the minisymposium are modeling in various application fields, structure-preserving model order reduction, and control of pH systems.

Organizers: Robert Altmann (Uni Augsburg), Philipp Schulze (TU Berlin)

In recent years, the interest in numerical methods based on Polytopal Meshes has blown up significantly. The high flexibility gained by such general meshes is advantageous, for instance, in adaptive mesh refinements and in handling complex geometries arising from real life applications. A short and incomplete list of Polytopal Methods includes Extended and Polygonal Finite Element Methods (FEM), Discontinuous Galerkin FEM, Mimetic Finite Differences, Virtual Element Methods, Hybrid High Order Methods, Hybrid Discontinuous Galerkin Methods, BEM-based FEM, and Weak Galerkin Methods. The length of the list illustrates that the appeal of such methods is actually enormous. The aim of this mini-symposium is to give an opportunity to young researchers working on this topic to share their recent advances to the community. Importantly, in addition to the fact that many different approaches are going to be presented herein, we stress that also the range of applications of such methods to different areas of Numerical Analysis (multiscale problems, hp methods, wave equations, time-dependent problems, fractured porous media, anisotropic polytopal elements ...) is wide and extensive. We deem that this variety of applications can enable the construction of a common platform for young experts working in this branch of Applied Mathematics and that this mini-symposium can pave the way to future scientific collaborations.

Organizers: Lorenzo Mascotto (Uni Wien), Steffen Weißer (Uni Saarland)

Zur Modellierung von Schädigungs- und Ermüdungseffekten wird meist eine Schädigungsvariable d eingeführt, die den lokalen Grad der durch Mikrorisse verursachten Reduktion von Steifigkeit beschreibt. Ein charakteristisches Spannungs-Dehnungs-Diagramm zeigt daher abnehmende Spannungen bei zunehmenden Dehnungen. Dies ist ein Hinweis darauf, dass die verwendete freie Energie nicht konvex ist. Bei der simulativen Auswertung entsprechender Modelle zeigt sich diese Nicht-Konvexität durch einen starken Einfluss der numerischen Diskretisierung des mechanischen Körpers auf die erzielten Resultate. Beispielsweise tritt bei Verwendung der Finite-Elemente-Methode eine Evolution der Schädigungsvariable d ausschließlich in einzelnen finiten Elementen auf. Weiterhin hängt das globale Materialverhalten, ausgedrückt in Kraft-Verformungs-Kurven, ebenfalls sehr stark von der finite Elemente Diskretisierung ab. Ohne entsprechende Korrekturmechanismen können Schädigungsmodelle daher nicht verwendet werden, um ein Materialverhalten mechanischer Körper zuverlässig abzubilden. Es existieren unterschiedliche Verfahren zur Regularisierung der freien Energie angewendet werden, die den angesprochenen Effekten entgegentreten. Beispielhaft seien die Gradientenbestrafung und viskose Regularisierungsverfahren genannt. Im vorgeschlagenen Minisymposium sollen die neuesten Entwicklungen in diesem spannenden und wichtigen Feld der Mechanik präsentiert und diskutiert werden. Dazu wird einleitend ein Wissenschaftler aus dem Bereich der Materialwissenschaften einen Einblick in die experimentell beobachteten Effekte geben, die zu der oben genannten (makroskopischen) Reduktion von Steifigkeit führen. Nach dieser Motivation, werden Nachwuchswissenschaftler aus dem Bereich der Mechanik von unterschiedlichen Standorten ihre neuesten Ergebnisse zur Modellierung von Schädigung präsentieren und einen besonderen Fokus auf die Regularisierung legen. Es wurden Beiträge zu mikromorphen Ansätze, zu viskoser Regularisierung, zur Phasenfeld-Methodik und zur Gradientenregularisierung zugesagt. Damit ist ein weites Feld der Regularisierung abgedeckt. Außerdem werden unterschiedliche numerische Verfahren wie die Finite-Elemente-Methode und die Phasenfeld-Methode betrachtet. Zusammen mit der Sicht eines Experimentators wird das vorgeschlagene Minisymposium daher einen umfassenden und aktuellen Überblick über die physikalischen Ursachen, die akkurate Modellierung und die effiziente numerische Simulation von Schädigung geben.

Organizers: Philipp Junker (Ruhr-Uni Bochum), Xiaoying Zhuang (Uni Hannover)

The mathematical analysis of deep learning aims at uncovering the underlying principles that facilitate its groundbreaking success across a diverse range of classification and regression tasks. The goal is to obtain rigorous analytical answers to some of the following key questions: Is it possible to quantify the expressive power and other desirable properties of a deep learning architecture in terms of parameters such as depth, width, or the choice of filters in convolutional networks? Fitting deep neural networks to a given set of training data is typically a highly non-convex task. Nonetheless, optimization methods such as stochastic gradient descent via backpropagation often yield advantageous results in applications. Is there an explanation of how and why the training of a deep neural network succeeds in practice despite a high degree of non-convexity? Deep neural networks often have hundreds of thousands of free parameters. Still, they appear to be resistant to overfitting and generalize surprisingly well in a wide range of classification tasks. Is it possible to analytically explain this surprising and maybe most significant property of deep neural networks? Finally, deep neural networks, like most machine learning approaches, are often black boxes in the sense that they do not yield an explanation as to how a specific task was solved. This raises the question whether it is possible to develop techniques that can determine which properties of an input signal are in fact responsible for a specific classification decision yielded by a deep neural network.Recent fundamental mathematical research indeed provides first answers to these questions. A detailed study of the approximation properties of deep neural networks can already provide explanations for many observed phenomena such as the efficiency of deep neural networks over shallow counterparts. Moreover, it was shown that specific types of deep learning architectures possess desirable invariance properties, such as stability with respect to small deformations. These results are in particular interesting, as they indicate that deep neural networks implicitly consider low-dimensional approximations of high-dimensional input signals, which could explain how deep neural networks succeed at avoiding the curse of dimensionality. Other recent results concern the conditions under which a local minimum of the considered loss function is also a global minimum. These findings offer first insights as to why optimization of deep neural networks seems to succeed in practice even though the considered optimization problem is of a highly non-convex nature. Parallel to providing first analytical insights, applied mathematicians are increasingly using deep learning methods as tools for their research. Deep neural networks can, for instance, be applied to learn dictionaries that yield sparse and efficient representations of certain signal classes. The impact of deep neural networks is also increasingly growing in the field of inverse problems. There, deep neural networks can be used to obtain novel regularizing functionals that are not based on analytical reasoning but are directly being learned from observed sets of data. Furthermore, deep neural networks can also be used to obtain solutions for highly ill-posed inverse problems.Despite the somewhat clear separation of the theory of deep learning into multiple seemingly separate questions, a real answer to understanding the efficiency of the method can likely only be gained by adopting a holistic point of view, i.e., by understanding the interactions of these questions with each other. This mini-symposium, therefore, aims at bringing together young applied mathematicians who are in one form or another working on one of the issues described above to share their latest advances, discuss possible future directions, and to foster collaboration.

Organizers: Rafael Reisenhofer (Uni Bremen), Philipp Petersen (TU Berlin)

In spite of the recent advances in the field of experimental characterization of elastomers, the description of their multi-physics phenomena (such as electroelasticity, magnetoelasticity, mechanoluminescence and strain-induced crystallization) remains a major challenge for science. The complex multi-physics coupling phenomena in elastomers, for example between the deformation field and an electromagnetic or temperature field have not been fully understood. The aim of this mini-symposium is to bring together young researchers from the polymer community and give them the opportunity to present and exchange their latest research ideas. Its topics include, but are not limited to, multi-scale and multi-physics constitutive modeling and finite element analysis for rubber-like materials.

Organizers: Vu Ngoc Khiêm (RWTH Aachen), Markus Mehnert (Uni Erlangen-Nürnberg)