A short introducton

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.
To illustrate, what are SPDEs are exactly, I would like to explain it by the following example.
Imagine a pond into which flows a chemical substance which reacts with water. This system can be described by a reaction diffusion equation. The pond however is not isolated as it is exposed to external conditions such as wind and rain which influences the behaviour of the system by e.g. advection and mixing. Both wind and rain are too complex to be described deterministically and, as every wind or rain has its own individual shape, it cannot be reproduced. One possible way to deal with the problem is to model them by means of stochastic processes, which show the same statistical properties as the wind and rain. The dynamics of the system can be described by a reaction diffusion equation with a Wiener process acting on the surface of the pond, which is a typical example of a so called nonlinear Stochastic Partial Differential Equation (SPDE).
Let us assume that a large number of deposits, each containing a chemical substance, are placed along the bank. Most of the deposits have a leak and some of the substance is dripping out in the water. This 'dripping out' can be modelled by a Poisson point process, where the waiting time between the drips is exponential distributed with parameter l depending on the magnitude of the deposit - the size of the drips corresponds to the size of the jumps of the Poisson point process. This model results in a stochastic reaction diffusion equation driven by a space time Poissonian noise, in particular, in a nonlinear SPDE driven by a Poisson random measure.
The presence of noise leads to new and important phenomena. E.g. Crauel [1] had shown that the presence of noise smears out bifurcations. Moreover, Crauel and Flandoli [2] presented an example in which the noise significantly changes the dynamic behaviour of the deterministic equation (even for arbitrarily small intensity of the noise). Also, there exist several examples, where, in the deterministic case, the invariant measure is e.g. 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.
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.

References

[1] H. Crauel. White noise eliminates instability. Arch. Math., 75:472–480, 2000. [2] H. Crauel and F. Flandoli. Additive noise destroys a pitchfork bifurcation. J. Dynam. Differential Equations, 10:259–274, 1998.

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 reaction–diffusion, population dynamic, environmental pollution and nonlinear filtering. Another source was an internal development of analysis and the theory of stochastic processes.
Here I want only to point out four recent examples, two from nanotechnology, one from photonic and one from mathematical finance. However, partially, the examples described here are related to hyperbolic problems, my main emphasis will be on parabolic problems.
Submicro-sized ferromagnetic elements are the main building blocks in magnetoelectronics, where they are widely used as information devices. As these elements get smaller and smaller, the effects of the thermal noise increase. For this reason, many researcher introduce noise in the systems. Here, it is important that the thermal noise eventually allows the magnetisation to overcome any energy barrier, and, thereby visit all possible configuration. For a detailed description we refer e.g. to [2].
Another example comes from thin films. Here, one is interested in the dynamics of complex dewetting of very thin layers. Also, if the layers get thinner and thinner, the effects of the thermal noise cannot be neglected any more. Gr¨un and his group modelled the thermal noise by Wiener noise (for detailed description we refer to [5, 1]).
Another class of complex chaotic systems that exhibit random behaviour arises in nonlinear optics, and especially in Photonics. The 1D Nonlinear Schr¨odinger Equation (NLSE) appears in optical waveguide propagation and in optical communication, see e.g. [4]. Practical implementations of optical communication systems lead to a variety of stochastic perturbations of NLSE such as additive noise and stochastic variation of group velocity dispersion. Multi-dimensional (2D and 3D) generalisations of NLSE appear as the paraxial approximation in nonlinear propagation of laser beams in many applications.

References

[1] J. Becker, G. Gr¨un, R. Seemann, H. Mantz, and K. Jacobs. Complex dewetting scenarios captured by thin-film models. Nature, January 2003:59–63, 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. Bj¨ork. On the geometry of interest rate models. In Paris-Princeton Lectures on Mathematical Finance 2003, volume 1847 of Lecture Notes in Math., pages 133–215. Springer. [4] G. Falkovich, I. Kolokolov, V. Lebedev, V. Mezentsev, and S. Turitsyn. Non-Gaussian error probability in optical soliton transmission. Physica D, 195:1–28, 2004. [5] G. Gr¨un, K.Mecke, andM. Rauscher. Thin-film flow influenced by thermal noise. J. Stat. Phys., 122:1261–1291, 2006.

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Groups Acting on Graphs

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.

Products of Graphs

This area is best described by 'The Product Graph Website'

Algorithms and the Structure of Graphs

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.

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.

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Our research in this direction is best described here

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