Analysis/Numerical Analysis/Optimization

Analysis/Numerical Analysis/Optimization covers a broad range of mathematical topics from the fields of differential geometry, analysis and numerical analysis of partial differential equations and optimization. The research focuses to a greater or lesser extent on applied problems from the natural and engineering sciences, but it also ­considers fundamental intrinsic mathematical questions in fields such as geometric analysis.

Over the past two years, various new appointments have made it possible to restructure the area of Numerical Analysis with varying degrees of reference to Optimization. The appointment of Professors Christian Clason, Johannes Kraus and Irwin Yousept brings three further excellent scientists to our Faculty. Christian Clason researches inverse problems and is an internationally recognised expert in his field, among other things in his capacity as Chair of the Working Group TC7.4 “Inverse Problems and Imaging” of the ­International Federation for Information Processing (IFIP). Johannes Kraus’ field is fast solvers for discretisation of partial differential equations, a subject on which he was invited to give a keynote address at the international conference on “Domain Decomposition Methods” DD22 in Lugano. The research group of Irwin Yousept works on numerical analysis of partial differential equations in the context of optimal control. One main area of his research currently deals with questions relating to electromagnetic fields. At the GAMM annual conference in 2014, he was awarded the Richard-von-Mises Prize for exceptional scientific achievement in Applied Mathematics and Mechanics. Even before joining our Faculty he played a successful role in securing the DFG Collaborative Research Centre SFB/TRR 154 “Mathematic Modelling, Simulation and Optimization Using the Example of Gas Networks”, a project with which he continues to be associated.

In Analysis we were able to gain an experienced and established researcher for our Faculty as of the winter semester 2014/15 in Georg Weiss, whose field is the theory of nonlinear partial differential equations. His work at the interface between pure and applied mathematics attracted a great deal of attention in relation to problems with free boundaries. He employs analytical and geometric methods for a large range of issues motivated by application, including the study of water waves, magnetohydrodynamics and rotating stars.

Following his appointment to Essen, Prof. Gerhard Starke and his research group have been able to continue and build on a long-standing and successful collaboration with Prof. Jörg Schröder and the Institute of Mechanics. In a joint project, which is being funded by the DFG under its Individual Grants Programme, a stress-based mixed finite element method for numerical simulation of hyperelastic materials was developed and analysed for the first time. Within the framework of the co-initiated DFG Priority Programme 1748 “Reliable Simulation Techniques in Solid Mechanics”, a joint proposal by the two research groups to develop new finite element methods for finite elastoplasticity is being funded over the next three years. This project, like the Priority Programme as a whole, aims to ­extend the application and understanding of modern discretisation methods for simulation in solid mechanics, for example as a way of predicting material behaviour under loads. Over the past two years, new findings have been made in this area on the approximation quality of edge and surface-based finite element techniques ­under prescribed conditions on curved bends or surfaces.

Research in Prof. Rüdiger Schultz’s group on the optimisation of gas transport in pipeline ­systems and on stochastic optimisation was further consolidated in the reporting period. Here a total of four doctorates were brought to successful conclusion. The work on two subprojects of the Research Training Group 1855/1 “Discrete optimization of technical systems under uncertainty” of the TU Dortmund University has been launched successfully. International regard for the findings of the joint Mercator project with Prof. Clausen (TU Dortmund University) on “Resource-saving route planning in the Ruhr metropolis using stochastic optimisation” was reflected among other things in an invitation to the project researcher to spend several months at the prestigious Centre de recherche sur les transports (CRT) of the University of Montreal. Project cooperation with the Essen-based gas transport company Open Grid Europe continued at all ­levels, extending from pure mathematics research to going live with developed software components. Prof. Schultz received special recognition of his work on the theory and industrial applications of stochastic optimisation in his appointment as an external subproject leader in the SFB Transregio TRR 154 “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks” launched by the DFG in Erlangen, Berlin and Darmstadt as from 1 October 2014.

During the reporting period, the research group of Prof. Arnd Rösch made further key findings in a priori and a posteriori error analysis of optimal control problems in partial differential equations. The results were yielded partly by the projects funded under the now expired DFG Priority Programme 1253 “Optimization with partial differential equations”. In the period under review, Bui Trong Kien, an experienced Humboldt postdoctoral fellow, joined the research group for a total of 18 months with the project “Qualitative studies for optimal control problems governed by ordinary differential equations and partial differential equations”. As from November 2014, Prof. Rösch is working with his colleagues Prof. Christian Meyer (TU Dortmund University) and Prof. Klaus Hackl (RU Bochum) on a MERCUR project on “Optimal control of mechanical degradation processes”.
Research in the group of Prof. Andreas Gastel was conducted on problems of the calculus of variations using functionals from gauge theory and higher-order geometric functionals. Since autumn 2014, a research project on the latter topic has been financed by the DFG under its Individual Grants Programme. In collaboration with Prof. Christoph Scheven, both subareas have been combined in a single work on higher-order gauge functionals.

Prof. Christoph Scheven has further been researching degenerate parabolic problems and geometrically motivated flows. In the reporting ­period he initiated cooperation with Prof. Verena Bögelein from Salzburg and Dr. Teemu Lukkari of Jyväskylä (Finland) on obstacle problems of the porous media type, which has already produced results on the existence and regularity of solutions.
Research in the group of Prof. Paola Pozzi was conducted on analytical and numerical questions relating to geometric partial differential equations and the regularity theory of geometric functionals. In collaboration with Prof. Martin Rumpf (Bonn) and Prof. Anna Dall’Aqua (Ulm), interesting ­results were achieved on topics including modelling and numerical analysis of the anisotropic Willmore flow and on the long-time existence of the flow of elastic curves. New insights were also gained into geometric curvature energies with possible applications in biophysical repulsion processes.

The research group of Prof. Ulrich Dierkes investigates analytical aspects of differential equations and variation problems motivated by geometrical or physical questions. These include existence or regularity questions relating to minimal or suspended roof-type surfaces, geometric evolution problems or free boundary value problems. Interesting sandwich theorems in relation to geometric measure theory have been proven in a doctoral project funded by the German ­National Academic Foundation. Two further dissertations deal with what are known as Bernstein sets theorems for nonlinear equations or systems in higher dimensions, and with difficult regularity analysis in singular variation problems and the free boundaries occurring in the process, which have already yielded interesting results. Coopera­tion also continues with the research group of Prof. Pozzi on projects relating to numerical geometric analysis, in particular numerical analysis for minimal surfaces with free boundaries. The researchers have already achieved important and promising results in this context. There are therefore plans to continue and intensify this successful collaboration.

In Prof. Petra Wittbold’s research group, the successful collaboration with the University of Warsaw under the institute partnership funded by the Humboldt Foundation (2013–2015) continued and produced new findings on conservation laws with discontinuous flow. During a period of research at the University in Pau at the invitation of and funded by the French CNRS, various studies on the topic of “Stochastic perturbations of conservation laws” were launched, the first of which has already been published. In November 2013, Prof. Wittbold and colleagues from Bielefeld and the TU Berlin organised a workshop entitled ­“Recent Trends in Differential Equations: Analysis and Discretization Methods” in Berlin. It was also here in March 2013 that the fifth “Spring School on Evolution Equations” took place, which was organised jointly with Prof. Etienne Emmrich (TU Berlin).

The achievements of the research group of Prof. Patrizio Neff included the discovery of new connections between differential geometry and nonlinear elasticity theory, which have also stimulated research in the direction of matrix analysis.

Prof. Dirk Pauly in conjunction with Dr. Olli Mali from Jyväskylä (Finland) succeeded in developing functional a posteriori error estimates for elliptic boundary value problems in exterior domains and new estimates for the Maxwell constants. For some years now, he and Prof. Repin (St. Petersburg) have been organising a workshop on “Analysis and Advanced Numerical Methods for Partial Differential Equations”, which was held in 2014 in St. Petersburg, in 2013 in Strobl and in 2012 in Konnevesi/Jyväskylä.