Numerical mathematics is concerned with the development and analysis of efficient algorithms for the solution of mathematical problems, like 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 reallife processes, scientific computing combines numerical mathematics with mathematical modeling and adavanced computations.

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. Two research schools also focus on
interdisciplinary research in numerical mathematics and scientific computing – 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 (Mehrmann, Holtz) ranging from the solution of large sparse eigenvalue problems to compressed sensing provide a sound basis for almost all kinds of advanced numerical algorithms. In this respect, large systems of ordinary differential equations (ODEs) and differential-algebraic systems (DAEs) play an important role (Deuflhard, Mehrmann).

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, Deuflhard). This approach can also be applied to apparently completely different problems, for instance from data analysis.

Efficient numerical solvers for PDEs (Carstensen, Hintermüller, John, Kornhuber, Schneider, Yserentant) 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.

Numerical simulation of turbulent flows (John, Klein) remains one of the major challenges of scientific computing. Further problems concerning the interaction of multiple time and length scales and are treated in the DFG Priority Program Scale-selective modeling in fluid dynamics and meteorology (coordinator: Klein). The numerical solution of geometric PDEs benefits from strong collaboration of applied and numerical analysis (Ecker, Huisken, Kornhuber). As the goal in most applications is optimization rather than simulation, nonlinear optimization with PDEs and optimal control (Griewank, Hintermüller, Hömberg, Mehrmann, Tröltzsch) play an increasingly important role, as witnessed by the activities in MATHEON and in the DFG Priority Program Optimization with PDEs.

Last but not least, scientific computing on campus strongly benefits from close cooperation with visualization groups (Hege, Polthier) to better interpret computational results. There are also overlapping research themes regarding segmentation, shape statistics, and PDE numerics on manifold meshes.