Numerical mathematics deals with the construction and analysis of efficient algorithms for the solution of mathematical problems. Already since the 1950’s, this discipline has played an important role in the simulation and optimization of problems that arise from science and engineering. Under this influence, numerical analysts were drawn more and more into mathematical and scientific modeling. As a consequence, the discipline numerical analysis widened into the larger field of scientific computing.

Berlin has joined this worldwide process, starting in 1986 with the foundation of ZIB. Today, there is a close scientific computing network of research groups at the universities and the external research institutes PIK, WIAS, and ZIB. Among their joint research activities, the most visible ones are the Berlin Center for Genome-based Bioinformatics (BCB, since 2001) and the DFG Research Center MATHEON (since 2002). These are complemented by joint PhD programs like the IMPRS Computational Biology and Scientific Computing (since 2004) and the RTG Analysis, Numerics, and Optimization of Multiphase Problems (since 2004).

Widely recognized work on numerical linear algebra (Mehrmann), like the solution of large sparse eigenvalue problems, forms a sound basis for a variety of numerical computations. In applications, large sets of ordinary differential equations and differential-algebraic systems play a crucial role (Deuflhard, Mehrmann). The numerical treatment of highly oscillatory differential equations describing, e.g., large biomolecules, requires completely new concepts partly imported from stochastics (Deuflhard, Schütte). The open problem of efficient numerical treatment of partial differential equations in high dimensions (e.g., Fokker–Planck equations and quantum calculations for real-world molecular systems) results in fascinating new approaches (Schütte, Yserentant).

In finite element methods for partial differential equations, a posteriori error estimates (Carstensen, Kornhuber) play a crucial role. Exceedingly challenging questions concern the interaction of small parameters, like, e.g., low Mach numbers in the Euler equations, with time steps or mesh sizes, studied by asymptotic numerical analysis (Klein). For the arising discretized large algebraic systems, multilevel methods and domain decomposition (Kornhuber, Yserentant) exploit structural properties of the underlying continuous problems for the construction of fast solvers. In most branches of technology, there is a growing demand for nonlinear optimization as well as for robust and optimal control (Griewank, Mehrmann, Tröltzsch). Activities range from linear and combinatorial optimization to nonlinear and PDE optimization, thus bridging even to discrete mathematics.

Comparably strong ties have evolved towards visualization by the joint development of virtual labs and by overlapping research themes regarding complex hierarchical meshes, shape statistics, and PDE numerics, backed by the close oncampus cooperation of scientific computing (Deuflhard, Klein, Kornhuber, Schütte) and visualization (Hege, Polthier). Last but not least, there are the traditionally close connections between numerical analysis and applied analysis, mainly regarding PDEs and dynamical systems extending also into the geosciences and meteorology. As a recent outcome, we mention the newly established DFG Priority Program Scale-selective modeling in fluid dynamics and meteorology.