Numerical mathematics is concerned with the development and analysis of efficient algorithms for the solution of mathematical problems such as those arising in natural sciences and engineering, and has thus always played an important role in applications. The larger field of scientific computing has emerged from the mutual interplay of numerical mathematicians and external users. Aiming at the simulation and optimization of real-life processes, scientific computing combines numerical mathematics with mathematical modeling and advanced computations.
Berlin Research Groups
Berlin has joined this worldwide process, starting in 1986 with the foundation of ZIB and continuing over the years by a well-coordinated hiring policy at all three universities. A major success has been the DFG Research Center MATHEON, which – since its founding in 2002 – has established a close network of research groups in scientific computing at the three universities as well as WIAS and ZIB. The Einstein Center for Mathematics (ECMath) operates with a wider focus, providing a foundation of various research activities in the rich Berlin scientific landscape, but shaping research by projects in particular in the area of modeling, simulation, and optimization. Three research schools also focus on interdisciplinary research in numerical mathematics and scientific computing – the Berlin International Graduate School in Model- and Simulation based Research (BIMoS), the IMPRS Computational Biology and Scientific Computing (2004-) and the Helmholtz research school GeoSim for Explorative Simulation in Earth Sciences (2010-).
All important research areas of modern numerical mathematics and scientific computing are strongly represented in Berlin, most by leading scientists on an international scale.
Fundamental techniques of numerical linear algebra (Liesen, Mehl, Mehrmann, Nabben) ranging from the solution of large sparse eigenvalue problems to fast matrix factorization provide a sound basis for almost all types of advanced numerical algorithms. Numerical methods for the simulation and control of dynamical systems require the treatment of large scale systems of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) which is another important topic of fundamental research (Mehrmann, Tischendorf) with applications in many engineering areas..
Compressed sensing (Kutyniok) is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It is based on the novel paradigm in mathematical data science that data typically allows a sparse approximation/representation by a suitable basis, and provides efficient methodologies for problems such as data acquisition, feature extraction, or recovery of missing data. Applications to problem instances in electrical engineering such as communications are treated in the DFG Priority Program Compressed Sensing in information processing (coordinator: Kutyniok).
New algorithms for large highly oscillatory systems of ODEs arising in biomolecules are obtained by merging concepts from geometric numerical integration and applied stochastics (Schütte). This approach can also be applied to apparently completely different problems, for instance from data analysis.
Imaging Science (Hintermüller, Kutyniok) aims to develop methods for image compression, denoising, deblurring, and inpainting, to name a few, as well as to provide a precise analysis of such. The utilized techniques to solve the associated typically highly ill-posed inverse problems range from applied harmonic analysis and sparse regularization over partial differential equations and variational methods to probabilistic and statistical methodologies.
Efficient numerical solvers for PDEs (Carstensen, Hintermüller, John, Kornhuber, Kutyniok, Liesen, Mehrmann, Nabben, Schneider) are based on a posteriori error estimates and fast multilevel and domain decomposition methods, which benefit from structural properties of the underlying continuous problems. High-dimensional PDE systems like the Fokker–Planck or Schrödinger equations additionally require advanced techniques from wavelet compression, sparse grids, and tensor product approximation.Recently, also model reduction techniques become essential for many application areas (Mehrmann).
Numerical simulation of turbulent flows (John, Klein) remains one of the major challenges of scientific computing. The numerical solution of geometric PDEs benefits from strong collaboration of applied and numerical analysis (Ecker, Kornhuber). The new DFG collaborative research centre on Scaling cascades in complex systems (speaker: Klein) focusses in the area on methodological developments for the modeling and simulation of complex processes involving cascades of scales derived from prototypical challenges in the natural sciences (König, Kornhuber, Mielke, Noe, Schneider, Schütte).
As the goal in most applications is optimization rather than simulation, nonlinear optimization with PDEs and optimal control (Hintermüller, Hömberg, Mehrmann, Tröltzsch) play an increasingly important role, as witnessed by the activities in MATHEON. Several mathematicians in this area are also involved in the DFG collaborative research centre on control of self-organizing nonlinear systems: Theoretical methods and concepts of application (Emmrich, Mehrmann, Miehlke, Stannat, Tröltzsch). The recently initiated DFG collaborative research centre on mathematical modelling, simulation and optimization using the example of gas networks (Hintermüller, Klimt, Koch, Mehrmann, Skutella, Tischendorf) is a beautiful example, which combines different expertises in this regime, but also across disciplines, to simulate and control the flow of gas in complex gas networks.
Last but not least, scientific computing on campus strongly benefits from close cooperation with groups working in geometry processing (Polthier) and visual computing (Hege) They help to cover pre- and post-processing (image processing, incl. segmentation, statistical shape models, mesh generation, data visualization and visual analytics). There are also overlapping research themes concerning mathematical image processing (Hintermüller, Kutyniok) and PDE numerics on manifold meshes.
The two Basic Courses provide a rigorous introduction to the most important strategies and concepts of modern numerical mathematics, and are independent of each other. The first course focuses mainly on numerical methods for ordinary differential equations, but also on deepening knowledge in numerical linear algebra, especially regarding iterative methods for large systems. The second semester gives an introduction to partial differential equations from fundamental theory to modern numerical concepts.
Prerequisites: Non-linear systems of equations, best approximation, linear regression, hermite interpolation, numerical quadrature and initial value problems for ODEs.
Outline of the contents:
Numerical methods for ODEs and numerical linear algebra
◦ Stiff initial value problems and stability
◦ Implicit Runge-Kutta and multistep methods
◦ Numerical methods for DAEs
◦ Iterative solution of linear equations and eigenvalue problems
Numerical methods for PDEs
◦ Modeling, variational problems and PDEs
◦ Classification and characterization
◦ Classical solutions and finite differences
◦ Weak solutions and finite elements
◦ Preconditioning, multigrid methods, and adaptivity
◦ Numerical methods for parabolic PDEs
Spectrum of Advanced Courses
Topics for advanced courses include Model Reduction, Optimal Control of PDEs, Finite Element Methods, Computational Photonics, Uncertainty Quantification, Control Theory, Multilevel Methods, Energy Based Modeling, Theory and Numerics of Non-Smooth Optimization, Numerical Analysis of PDAEs, Variational Inequalities, Inverse Problems.