The main areas of our scientific work are: Stochastic Partial Differential Equations, Graph Theory, Numerical Analysis and Backward Stochastic Differential Equations for Lévy Processes Information about Scientific Collaborations. -
Stochastic Partial Differential Equations
Partial Differential Equations (PDEs) play an essential role for mathematical modelling of many physical
phenomena, and the literature devoted to their theory and applications is enormous. SPDEs are quite a young
research area, the first articles appeared in the mid 60's. The presence of noise leads to new and important
phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing,
where in the deterministic case, the invariant measure is not unique, and, in the stochastic case the system is
uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads
often to a more realistic description of real systems than their deterministic counterpart.
[1] H. Crauel. White noise eliminates instability. Arch. Math., 75:472â€“480, 2000. [2] H. Crauel and F. Flandoli. Additive noise destroys a pitchfork bifurcation. J. Dynam. Differential Equations, 10:259â€“274, 1998. Impact for other Branches of other Science
Stochastic Partial Differential Equations were motivated by the need to describe random phenomena studied
in the natural sciences such as control theory, physics, chemistry and biology. They are used, for example,
in neurophysiology, mathematical finance, chemical reactionâ€“diffusion, population dynamic, environmental
pollution and nonlinear filtering. Another source was an internal development of analysis and the theory of
stochastic processes.
[1] J. Becker, G. GrÂ¨un, R. Seemann, H. Mantz, and K. Jacobs. Complex dewetting scenarios captured by thin-film models. Nature, January 2003:59â€“63, 2006. [2] R. Kohn, M. Reznikoff and E. Vanden-Eijnden; Magnetic Elements at Finite Temperature and Large Deviation Theory; Journal of Nonlinear Science, 15:223-253, 2005. [3] T. BjÂ¨ork. On the geometry of interest rate models. In Paris-Princeton Lectures on Mathematical Finance 2003, volume 1847 of Lecture Notes in Math., pages 133â€“215. Springer. [4] G. Falkovich, I. Kolokolov, V. Lebedev, V. Mezentsev, and S. Turitsyn. Non-Gaussian error probability in optical soliton transmission. Physica D, 195:1â€“28, 2004. [5] G. GrÂ¨un, K.Mecke, andM. Rauscher. Thin-film flow influenced by thermal noise. J. Stat. Phys., 122:1261â€“1291, 2006. Graph Theory
We study automorphism groups of transitive and almost transitive graphs in connection with the end structure and growth properties of the underlying graph. At the core of many of these investigations is Gromov's charcterization of groups of polynomial growth and its generalization to groups acting transitively on graphs. Topics persued up to now and still being investigated by our group include automorphism groups of graphs with polynomial growth, groups and graphs with linear growth, s-transitivity, covering graphs, groups acting on trees and groups of products of graphs. Furthermore, many of these concepts have been successfully applied to the investigation of the subgroup structure of free and virtually free groups. Although the primary goal of these activities are infinite graphs and groups, many important applications pertain to finite structures, e.g. the construction of graphs with large girth and contractors or expanders. Of particular interest in this respect are counting methods for subgroups of given index in free groups and related groups. In addition we started to investigate transitive directed graphs, in particular highly arc-transitive digraphs. This is a quite young field of interest which has close connections to topology. Besides structural properties of those graphs we are mainly interested in their automorphism groups.
This area is best described by 'The Product Graph Website'
The enormous interest in good algorithms for the solution of large systems of linear equations, both by sequential and parallel methods, has increased the importance of structural investigations of associated networks. Thus, the research interests of our group pertaining to products of graphs, isometric embeddings of graphs into Cartesian products, efficient sequential and parallel algorithms for the decomposition of graphs into Cartesian products, realizations of metrics by graphs, eigenvalue methods for the decomposition of graphs and other problems have gained new dimensions. zum Seitenanfang Numerical Analysis
Numerical simulation of hydrocarbon flow in porous media and turbulent flow in combustion engines initiated our interest in iterative solvers for large, sparse systems of linear equations. We investigate multilevel incomplete factorizations of matrices arising from finite-difference discretizations. Our interest lies in hierarchical ordering strategies and estimates for resulting condition numbers. These methods are closely related to algebraic multigrid methods. Promising coarsening strategies based on minimum spanning trees in the grid are considered. We also use eigenvalue methods for recursive spectral decomposition of graphs. These methods are implemented for domain decomposition on distributed-memory parallel computers. Here we deal with additional constraints like load-balance conditions or edge sets that must not be cut. The modeling of flow and transport, and discretication of the resulting partial differential equations is another area of interest. For example, discretization strategies in regions where a moving grid glides along stationary grid cells were developed. The practical application behind this task was to simulate air flow in rotating fans to optimize the performance of laundry dryers. Current activities also include the drying of porous refractory bricks and thermal monitoring of steel slabs. Backward Stochastic Differential Equations for Lévy Processes
Our research in this direction is best described here |