Faculty of Mathematics

Analysis/Numerical Analysis/Optimization

The research in the fields of analysis/numerical analysis/optimization covers a wide range of mathematical topics that are usually related to the properties of solutions to nonlinear problems, often in the form of partial differential equations. These problems are often closely related to applications in the natural and engineering sciences, stimulating concrete research cooperation with scientists from other faculties. This concerns, for example, applications in solid mechanics, where we develop mathematically secured methods for the modelling and simulation of material behaviour  upon which our engineering colleagues build. The majority of research groups possess references to more than one of the three subareas of analysis, numerical analysis, and optimization. The resulting methodological scope leads to comprehensive handling of the problems and is enforced by the strong cooperation between individual work groups. Supported by the UA Ruhr Centre for Partial Differential Equations, ruhr.paD, this collaboration is also currently intensified towards the neighbouring universities in Bochum and Dortmund.

A specific topic which has recently moved into the centre of the mathematical research interest consists of variational inequalities. Constrained minimization problems, for instance, can be written in this form and therefore made more accessible to mathematical treatment. Here, the nonlinearity of the problem is a result of the fact that the admissible set is not given by a complete vector space.

A third possibility for analysing such questions consists of the formulation as a complementarity problem: Constraints are active on the part of the domain, implying that the conventional conditions for the presence of a minimum need to be satisfied on its complement.

This problem area is one of the topics that is addressed by the DFG priority programme SPP 1962 entitled “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” which started in autumn 2016 and was co-initiated by Prof. Arnd Rösch. Our Faculty participates in this priority programme with four research groups (Clason, Rösch, Starke and Yousept) with funded projects. This underlines the high standing of our Faculty in this field.

The joint project of Christian Clason and Arnd Rösch is concerned with the parameter identification for multiphase processes with applications in diverse fields such as climate research (cloud formation, glacier melting), material sciences (crystal growth, steel hardening) and solid mechanics (contact, damage).

In his project, Irwin Yousept works on an optimization problem associated with the Maxwell equations for the description of electromagnetic fields. It is concerned with the analytical and numerical treatment of a non-smooth hyperbolic PDE model characterizing the magnetization process of type II superconductors.

In a joint project associated with the frictional contact of elastic solids, Gerhard Starke’s research group collaborates with the Institute of Computational Science at the Università della Svizzera italiana in Lugano headed by Rolf Krause. In this context, novel stress-based finite element approaches are developed for the occurring quasi-variational inequalities.

The four groups participating in this priority programme contribute their expertise in different mathematical subareas. Supported by a regular seminar series, their spatial closeness and a variety of international contacts, this allows them,  to assist each other in their work.

Moreover, since 2015, the Prof. Clason’s research group “Inverse problems” has been working in collaboration with project partners from Würzburg and Klagenfurt within a D-A-CH cooperation project on the regularization and discretization of inverse problems for partial differential equations in Banach spaces. Particularly, this problem class contains parameter identification problems with structural a priori information (“sparsity”, integer constraints etc.).

In the Rösch research group, a joint MERCUR project with colleagues Christian Meyer (mathematics, TU Dortmund) and Klaus Hackl (engineering, RU Bochum) on the topic of “Optimal control of mechanical damage processes” was successfully finished.

The already existing scientific cooperation between Prof. Yousept’s research group and the mathematical department of the Chinese University of Hong Kong (CUHK) was strengthened by different activities, among them a successful proposal for a postdoctoral research fellowship from the Alexander-von-Humboldt foundation and the initiation of a joint project on the numerical analysis of the Maxwell equations.

In September 2016, a workshop on the topic of “Optimal control meets inverse problems” with 33 participants took place in Essen, organized jointly by Christian Clason, Arnd Rösch and Irwin Yousept.

In the Starke research group, a joint project with the Institute of Mechanics of the UDE has been carried out since 2014 within the DFG priority programme SPP 1748 “Reliable Simulation Techniques in Solid Mechanics”. In this project, special finite element discretisations for finite elasto-plastic deformations are developed. Methodically, its focus is also the construction of appropriate finite element spaces for the approximation of stresses. Due to the geometrical and material nonlinearity of the considered processes, both the analytical and the numerical treatment are significantly more difficult.

This problem area is also a central topic of the research in Prof. Patrizio Neff’s “Nonlinear analysis and modelling” research group. The main problem in isotropic nonlinear elasticity consists of finding a suitable form of the elastic energy to be minimized. In joint work, we have been able to discover an a priori differential geometric characterization of the strain energy as the geodesic distance of the deformation gradient to the group of rotations.

Alongside an experienced postdoc student from the US, Dr. Smit Vega Garcia, the research group on the analysis of partial differential equations, led by Prof. Georg Weiss and funded by the DFG Individual Grants Programme, investigated singularities of Electrohydrodynamic Equations by using analytic methods. Moreover, Prof. Weiss works in collaboration with Prof. Sagun Chanillo of our partner university Rutgers on the analysis of the free surface of neutron stars.

Prof. Christoph Scheven began an international collaboration with two scientists from the Università di Napoli “Federico II” and used existing contacts with the Seoul National University for a successful cooperation. Both research projects led to publications on regularity questions for nonlinear parabolic differential equations. A further research emphasis was obstacle problems with linear growth. In particular, an existence result for obstacle problems associated with total variation flow was achieved together with cooperation partners from Salzburg and Erlangen. The regularity theory of nonlinear differential equations is also a current research topic of Prof. Andreas Gastel’s group. Elliptic systems with critical nonlinearity were investigated in this regard within the scope of a DFG-funded project.

Prof. Petra Wittbold’s work group continued the successful scientific collaboration with colleagues from the universities in Pau and Marseille in the field of nonlinear partial differential equations with stochastic perturbation. Furthermore, two postdoctoral researchers from Brazil visited the group during the reporting period with research projects on “Asymptotics of Solutions for Nonlinear Partial Differential Inclusions.”

Geometrical and analytical aspects of variational problems and nonlinear differential equations motivated by physical or geometrical phenomena are investigated in the group led by Prof. Ulrich Dierkes. This constitutes, for example, questions associated with uniqueness and the regularity of generalized surfaces with prescribed curvature. As part of a doctoral project, finished in 2016, optimal regularity results for a class of singular differential equations were derived.

Geometrical partial differential equations are also the focus of Paola Pozzi’s research group, where numerical topics are considered in addition to the analytical aspects. In international collaborations, results were obtained on the long-time existence of the flow associated with elastic curves under different boundary conditions and on the analysis of a numerical model for elastic embedded curves. Emphasis should be given to the results achieved in cooperation with Björn Stinner of the University of Warwick.

These are associated with original error estimates for discretisations of coupled systems arising, for instance, in biology. In this context, the motion of a closed curve (idealized cell) is coupled to the solution of a parabolic partial differential equation to be solved on the curve, which by itself is influenced by the evolution of the curve. Systems of this kind are relevant for applications; and a rigorous numerical analysis is usually associated with severe technical complications.

In 2016, the Kraus research group succeeded in developing extremely efficient solvers for stable discretisations of the Darcy and Brinkman equations that are employed for the simulation of flow in porous media. Of interest in this context is the construction and convergence analysis of optimal and robust multigrid methods with convergence properties that are independent of jumps in the permeability. A research project based at the Johann Radon Institute in Linz/Austria (and still headed by Prof. Johannes Kraus) was mainly concerned with a posteriori error estimations for the (nonlinear) Poisson-Boltzmann equation. Applications are associated with the representation of the electrostatic potential of biological macromolecules.

In many areas of economic activity it is almost never possible to make decisions based on complete knowledge of all relevant input parameters. There are almost always decisions that extend well into the future where they must fulfil conditions, the detailed form of which emerges only then. The optimization of such activities is the subject of stochastic optimization, the main research area of the group led by Prof. Rüdiger Schultz. The work group is permanently established in the project-based research. This includes the participation in the GRK 1855 and TRR 154 programmes coordinated by the DFG, and a project of the Hadamard foundation. Furthermore, a cooperation project with the Fraunhofer institute Dortmund and the TU Dortmund on logistics, funded by the Mercator foundation, was successfully completed in 2015. In the context of these research projects, three members of the AG received their doctorate in 2015 and 2016.

Cooperation work at the level of research groups which led to first publications provide connections to Prof. Volker Krätschmer and the Belomestny research group from our Faculty, as well as PD Dr. René Henrion from Weierstraß Institute Berlin, Prof. Martin Gugat, University of Erlangen, Profs. Sergio Conti und Martin Rumpf, both from University Bonn, with Dr. Ward Romeijnders from the University Groningen and Prof. David Woodruff, University of California at Davis. Of special importance is the broad engagement in the DFG Transregio ransregional collaborative research field TRR 154 “Mathematical Modelling, Simulation and Optimization for Gas Networks”, which began in autumn 2014.