Research Group "Numerical Methods for high-dimensional Systems"

Karlsruhe Institute of Technology (KIT)

Department of Mathematics

Many important applications in financial mathematics (e.g. Black-Scholes equation), systems biology (e.g. chemical master equation), chemistry (e.g. Schrödinger equation), or physics require the solution of high-dimensional partial differential equations. Such problems are particularly challenging because they cannot be solved with traditional methods. The main reason is that the number of unknowns grows exponentially with the dimension such that the computational workload exceeds the capacity of most computers. For example, an equidistant discretization of the unit interval [0, 1] by mesh points with distance 0.1 has only 11 points, but a similar discretization of the unit cube requires 11³ = 1331 mesh points, and a corresponding mesh on the 10-dimensional hypercube contains 11¹⁰ = 25, 937, 424, 601 mesh points. This exponential growth of the size of the problem is known as the curse of dimensionality.

We are working on the development of different methods to solve such high dimensional problems in an adequate time on standard computers. We are developing direct numerical solvers as well as efficient sampling algorithms to simulate the time evolution of complex systems.