Scientific computing is the scientific discipline that deals with modeling and computer-based numerical simulations of processes as they are investigated for example within natural, engineering or economical sciences. The numerical simulation supplements the two classical pillars of knowledge acquisition within the respective applied sciences, namely the theoretical investigation and the laboratory experiment. Scientific computing with its central methods of modeling, numerical simulations and high-performance computing has developed to a key technology for the understanding and handling of scientific and technical challenges. Such different and complex problems like the design and simulation of the inlet port of a cylinder head in engines, the modeling and visualization of electrical excitation waves in the brain and heart muscle tissue or the design and control of nano and micro systems require the application of scientific computing. With scientific computing it is possible to discover new features of real systems as with complicated experiments in a physics or chemistry laboratory.

Multi-ITN STRIKE - Novel Methods in Computational Finance

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.

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.