bezeichnet die eigenständige wissenschaftliche Disziplin, die sich mit der rechnergestützten numerischen Simulation von Prozessen beschäftigt, wie sie etwa in den Natur-, Ingenieurwesen oder Wirtschaftswissenschaften untersucht werden. Die numerische Simulation tritt dabei neben die beiden klassischen Säulen des Erkenntniserwerbs in den betreffenden Anwendungswissenschaften, nämlich die theoretische Untersuchung und das Experiment. Das wissenschaftliche Rechnen mit seinen zentralen Methodenbereichen Modellierung, Numerische Simulation und Hochleistungsrechnen hat sich zu einer Schlüsseltechnologie für das Verständnis und die Bewältigung wissenschaftlich-technischer Herausforderungen entwickelt. So unterschiedliche komplexe Probleme wie die Gestaltung und Simulation des Einlasskanals eines Zylinderkopfes in Motoren, die Modellierung und Visualisierung von elektrischen Erregungswellen im Gehirn und Herzmuskelgewebe, oder das Design und die Steuerung von Nano- und Mikrosystemen erfordern den Einsatz von wissenschaftlichem Rechnen. Durch wissenschaftliches Rechnen erhält man oft genauso neue Erkenntnisse, wie es bei komplizierten Experimenten in einem Physik- oder Chemielabor passieren kann. Also, man kann mit dem Unerwarteten rechnen.
The computational complexity of mathematical models employed in financial mathematics has witnessed a tremendous growth that requires the development of advanced numerical techniques appropriate for the most present-day applications in financial industry. Besides a series of internationally recognized researchers from academics, leading quantitative analysts from the financial industry also participate in this network. The challenge lies in the necessity of combining transferable techniques and skills such as mathematical analysis, sophisticated numerical methods and stochastic simulation methods with deep qualitative and quantitative understanding of mathematical models arising from financial markets.
Control of quantum phenomena is becoming central in a variety of systems with present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures. In most cases, quantum control is aiming at quantum devices and molecular systems where there is the need to manipulate quantum states with highest possible precision. The successful application of optimal control theory in this field together with the enormous effort towards nanosciences explains the large growing interest in quantum control problems. Our focus is the development and investigation of fast and robust solvers of finite- and infinite-dimensional quantum control problems within an interdisciplinary scientific environment including biology, chemistry, mathematics, and physics.
Multigrid methods represent one of the most important breakthrough in the development of fast solvers of large-scale algebraic problems. In particular, multigrid principles allow to define iterative schemes with optimal computational complexity for solving discretized partial differential equations (PDEs). More recently, multigrid schemes have been applied successfully to solve PDE optimization problems and inverse problems. Our purpose is the development and analysis of multigrid methods for simulation and optimization with PDE models and the formulation of multigrid optimization schemes for unconstrained optimization.
Transient excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. This effect is described by a low-frequency variant of Maxwell's equations: the eddy current equations. These partial differential equations are of a mixed (parabolic and elliptic) type, which makes their study particularly challenging, both from a theoretical and a practical point of view. In this project (funded by the DFG grant HA 6158/1-1) we study the mathematical theory behind eddy current problems. Our particular aim is on inverse problems where a conducting object (or a flaw in a conducting object) is to be detected from electromagnetic measurements. We develop the neccessary theoretical tools to determine what measurements are needed to uniquely identify the shape of a conductor, and apply these tools to design rigorously justified reconstruction algorithms.
Protein structure prediction (PSP) is one of the most challenging problems in bio-mathematics, -informatics, and microbiology. It is the prediction of the three-dimensional structure of proteins from its amino acid sequence. The applications of PSP range from the design of new pharmaceutical drugs and enzymes to the investigation of how protein fold and interact with other molecules. We investigate fast multi-scale global optimization schemes having the objective to find the globally best (minimum) potential energy configuration corresponding to the tertiary structure of the protein.
The purpose of PDE optimization is to define ways of how optimally change, influence, or estimate features of systems modeled by partial differential equations. An important class of problems in PDE optimization results from optimal control applications. Other important classes of optimization problems are shape design, topology, and parameter optimization. Optimization is also an essential tool for solving many inverse problems. Our research work aims at the formulation, approximation, and efficient solution of PDE optimization problems also in the presence of data and model uncertainty.
The development of tomographic and other non-invasive imaging devices represent an exceptional contribution to medicine, material sciences, and arts. This development has led to a fast growth in the interdisciplinary field of imaging science where mathematics plays a central role providing a rigorous basis for imaging analysis. We investigate mathematical methods and fast algorithms for image processing and computer vision. The main focus is on techniques using partial differential equations and optimization.