WIAS Research Highlights

A general optimal control framework that incorporates physical processes that are enriched through data-driven components is introduced, and its feasibility in two key applications is shown. This idea combines the power of both traditional mathematical modeling with machine learning methods, and is able to deliver more accurate physical models.

The microscope, from its invention in the seventeenth century to modern transmission electron microscopy (TEM), has revolutionized the fields of science and technology. We explore the structure and composition of materials such as semiconductor quantum dots and discover how mathematical theory plays a critical role in solving the reconstruction problem and automated processing of TEM images.

EDP-convergence provides a thermodynamical consistent way to derive effective gradient systems for multi-scale problems. The derived evolutionary systems satisfy the desired physical principles. Previously unknown physical structures have been uncovered. They provide theoretical insights in chemical reaction systems and, in this way, a better understanding of everyday biochemical processes.

We develop and analyze physics preserving numerical techniques for new semiconductor techniques, materials and devices that innovate established technologies. Among them are low-cost perovskites for next-generation solar cells, resource-efficient nanowires as well as accurate lasers for self-driving cars.

The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. It allows to estimate the integration error when approximating spherical integrals. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates. We provide a fully explicit, easy to implement formula for the spherical cap discrepancy.

Perovskite solar cells outperform classical silicon solar cells, presenting an efficient green energy solution. Unfortunately, they degrade too fast. So how can we further improve efficiency while preventing degradation? We answer such a question in four stages: modeling, discretization, analysis, and simulation.

One of the greatest unsolved problems in contemporary mathematical physics is a mathematical understanding of the famous Bose-Einstein condensation (BEC) phase transition. Two alternative approaches to achieve progress are pursued: large deviation analysis of the free energy, respectively reflection-positivity techniques.

Pressure-robustness and quality meshes are two important ingredients for accurate discretizations in computational fluid dynamics. Polyhedral meshes have the potential to simplify quality mesh generation and can be easier adapted to the problem at hand. We summarize recent research on pressure-robustness and showcase a pressure-robust (virtual element) method on polyhedral meshes.

The explosion of interconnected devices requires radical advancements in the design of communication networks including for example peer-to-peer data transmission. Probabilistic methods can help to analyze the potential benefits but also challenges of such systems.

Quasi-variational inequalities (QVIs) are powerful mathematical objects that can be used to describe real-world phenomena as varied as thermoforming, or fluid flow in the heart. The inherently complex structure of QVIs makes their analysis a delicate matter. The control of QVIs is also important from the point of view of theoretical understanding as well as for applications.

When operated at high power, the emission quality of high-power broad-area edge-emitting semiconductor lasers suffers from undesired nonlinear processes induced by heating, multimode dynamics, and optical filamentation. We model these effects at different levels of complexity, analyze the hierarchy of models, create and exploit efficient numerical algorithms, and simulate the models in collaboration with physicists and engineers.

To foster communication between different groups of mathematicians and to develop new ideas and research directions, the Research Center MATH+ runs Thematic Einstein Semesters. Within the scope of the issue reviewed here, research on novel materials, scaling limits, bulk-interface coupling, and structure-preserving discretizations have been identified as particularly relevant topics.