# Scientific Computing

Phenomena in Science and Engineering are frequently described by systems of mathematical equations that can be used to carry out simulations. The area of applied mathematics that deals with this kind of task is called Scientific Computing.

Mathematical models are frequently expressed as systems of differential or integral equations. Since solving these systems "by hand" is only possible in special cases, numerical algorithms are employed to obtain approximations of the solution. Typically the first step is to discretize the system, i.e., to approximate it by a finite number of variables. The second step then is to solve this finite-dimensional system. A third step may be to optimize certain parameters of the problem, e.g., to fit experimental data.

Our group currently focuses on the construction of efficient solvers based on hierarchical matrices, particularly H^{2}-matrices. These matrices provide us with an efficient way of representing matrices that appear in numerical algorithms for differential and integral equations, and this allows us to significantly reduce the computational work. An example is a solution algorithm for large systems of linear equations arising from an elliptic partial differential equation: classical direct solvers require O(n^{2}) operations to find the solution, while H^{2}-matrix solvers can handle the same task in O(n log(n)) operations.