# Analysis/Numerical Mathematics/Optimization

The research activities in the areas falling under the general theme of Analysis, Numerical Mathematics and Optimization at the UDE are closely intertwined. The research groups cover a broad range of topics, starting from rather “pure” research questions, for example on geometrically motivated partial differential equations, through aspects of mathematical modelling and numerical treatment of specific applications as in nonlinear solid mechanics, to their optimisation and optimal control.

Participation in many large-scale research programmes funded by the DFG in the areas of Optimization and Numerical Mathematics underlines the topical nature of our research projects. In this context, participation of Prof. Rüdiger Schultz’s group in the second phase of CRC Transregio 154 on “Mathematical Modeling, Simulation and Optimisation using the Example of Gas Networks”, as well as in the expiring Research Training Group RTG 1855 at TU Dortmund University, should be mentioned here. The group is also actively involved with funded projects in several DFG Priority Programmes (SPP) at different stages of implementation. In SPP 1984, “Hybrid and Multimodal Energy Systems: Systems Theory and Methods for the Transformation and Operation of Complex Networks”, which started in 2017, the Schultz group is also taking part with a project on stochastic optimisation with uncertainties.

SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”, to which we are contributing with three projects, started in 2016. In this context, the groups of Professors Christian Clason and Arnd Rösch are concerned with the parameter identification for nonlinear differential equations with nondifferentiable terms which appear as models for phase transitions and are strongly related to variational inequalities. Among its results, this project led to the development of an iterative method for a class of such problems based on generalised derivatives. Closely related are optimal control problems, which are also examined in this joint project with Arnd Rösch. Stress-based methods for variational inequalities in solid mechanics are investigated in the group of Prof. Gerhard Starke in collaboration with colleagues from Universitá della Svizzera italiana in Lugano. Here, new results were obtained on a posteriori error estimation for the discretisation of quasi-variational inequalities to describe frictional contact of elastic bodies. A multiphysics problem associated with high-temperature superconductivity is the focus of Prof. Irwin Yousept in his project. It is concerned with the theoretical and numerical treatment of a coupled PDE model consisting of non-smooth heat equations and hyperbolic Maxwell variational inequalities of the second kind. This PDE model describes the magnetisation process and hysteresis loss of type-II superconductors under the strong influence of extremely low temperature sources and external magnetic fields. The mathematical theory developed for this phenomenon is not only useful for more precise numerical simulations of the multiphysical process itself but also for the optimisation of electromagnetic processes in high-tech superconductivity applications.

Finally, SPP 1748 “Reliable Simulation Techniques in Solid Mechanics: Development of Non-Standard Discretization Methods, Mechanical and Mathematical Analysis” is already in its second three-year funding period. Here again, the Starke group is taking part with a project on stress reconstruction for hyperelastic material models, which is carried out jointly with Prof. Fleurianne Bertrand (Junior Professor at HU Berlin, formerly also a member of the Starke group) and Prof. Jörg Schröder (Chair of Mechanics in the UDE’s civil engineering department). The ultimate goal of this project is to investigate just how far reconstruction-based error estimators may be generalised to nonlinear elasticity models with polyconvex energy functions.

All of this participation in broader research programmes profits from the orientation on methodological competence in mathematical research. This makes use of the fact that closely related methods sharing the same basic idea can often be employed in completely different areas of application. An example relating to a number of the projects described above is given by Newton’s method for non-smooth problems. Appropriate modifications of Newton’s method with weakened prerequisites on differentiability which do not require a smoothing regularisation are the subject of numerous research activities in the area of numerical optimisation. This theme was also part of the IFIP TC 7 conference on “System Modelling and Optimization”, which took place in July 2018 in Essen with Christian Clason in charge of its organisation. With more than 250 participants, this meeting was far beyond the scope of the usual size of events at our Faculty. While the professors involved in the conference organisation were busy ensuring the orderly course of events, the junior researchers from the areas of Optimization and Numerical Mathematics were able to contribute directly to its success by organising mini-symposia and with presentations in programme sections.

Besides these coordinated research activities, projects on an individual basis also play an important role in the areas of Optimization, Numerical Mathematics and Analysis. Among them are numerous projects funded by the DFG as individual grants, but also highly topical research in the form of dissertation projects funded through the core Faculty budget or national or international cooperation.

The Schultz group, besides participating in numerous special programmes already mentioned above, engaged in very successful cooperation with mathematicians from Bonn University as well as with the Fraunhofer Institute for Molecular Biology and Applied Ecology in Schmallenberg. As one of the organisers in charge, Rüdiger Schultz delivered Workshop 1834 “New Directions in Stochastic Optimisation” at the Mathematical Research Institute Oberwolfach jointly with Professors Darinka Dentcheva (Stevens Institute of Technology, Hoboken), Jesús De Loera (University of California, Davis) and Georg Ch. Pflug (University of Vienna).

The “Inverse Problems” research group of Christian Clason is concerned mainly with the parameter identification in nonlinear differential equations with non-smooth terms. In cooperation with Tuomo Valkonen (Escuela Politécnica Nacional, Quito, Ecuador), work is under way on iterative methods for non-differentiable non-convex minimisation with partial differential equations. In this context, the analytical and numerical proof of optimal convergence rates for primal-dual methods in infinite-dimensional Hilbert spaces was achieved for the first time last year.

The research focus of Prof. Paola Pozzi’s group is on analytical and numerical problems associated with geometrical partial differential equations. In cooperation with external researchers, the group achieved analytical results on the long-time existence of the elastic flow for curves and results on the short time existence for the anisotropic curvature flow. These and other conclusions obtained in recent years are currently being employed in order to explore sophisticated questions relating to networks of curves. It is also important to note that the project proposal “Flow of elastic networks” in cooperation with Anna Dall’Acqua (University of Ulm) and Chun-Chi Lin (National Taiwan Normal University) was approved for funding by the DFG and MoST (Ministry of Science and Technology in Taiwan) respectively in 2018. Finally, ongoing cooperation continued with Prof. Björn Stinner (Warwick) on the numerical analysis of coupled systems.

In the “Nonlinear Analysis and Modelling” research group of Prof. Patrizio Neff, the many examples of cooperation, among others with mathematicians and engineering researchers at INSA Lyon, University of Cape Town and KIT, continued and were extended. The related research efforts comprise different areas of generalised solid mechanics, such as development of a novel relaxed micromorphic model for description of so-called “band gap” phenomena, as well as basic questions associated with applied variational calculus and generalised convexity. In this context, funding for the project “Modelling and mathematical analysis of geometrically nonlinear Cosserat shells with higher order and residual effects” was granted by the DFG in 2018.

In the “Geometrical Analysis” research group of Prof. Andreas Gastel, the DFG project “Elliptic systems of higher order with critical nonlinearity“ was completed. The result is a rather general continuity theorem for solutions to such systems in the critical dimension. It also sought applications for methods from geometrical analysis, particularly in the Cosserat elasticity and in models of theoretical physics.

In the last two years, Prof. Christoph Scheven has established contact with the research group of Juha Kinnunen at Aalto University in Espoo, Finland. As part of this collaboration, a breakthrough in the regularity theory for equations of porous media type has been achieved. Prof. Scheven has also obtained results on existence and regularity for nonlinear parabolic equations in international cooperation with researchers from Florence, Naples, Salzburg and Seoul.

The research group of Prof. Georg Weiss on the analysis of partial differential equations investigated singularities in free boundary problems in international collaboration (including with Nina Uraltseva). Free boundaries are surfaces that are not fixed a priori between different phases such as the surface of a water wave or the surface of an ice crystal. The recent award of the Fields Medal to Alessio Figalli has given free boundary research a tremendous additional boost. Apart from free boundaries, the group is studying travelling waves in ramified domains as part of a doctoral thesis motivated by research on flow in leaf veins and diffusion of medical spray in lungs. In collaboration with Sagun Chanillo from our partner institution Rutgers University, Georg Weiss is working on analysis of the free surface of neutron stars.

The group of Prof. Petra Wittbold continued its successful research collaboration with colleagues from the University of Pau and the TU Berlin on the topic of nonlinear and nonlocal partial differential equations with stochastic perturbations. In Prof. Ulrich Dierkes’s group, investigation of analytical aspects of geometrical differential equations and geometrically motivated variational problems continued. Typically, this involves equations which describe generalised surfaces with prescribed curvature components or the minimisation of certain energy functionals or surface areas. Three dissertation projects on these topics were completed, one of which was in collaboration with the group of Paola Pozzi, and a scientific exchange with the Universidad da Granada was initiated and developed.