Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project A2

## Numerical analysis of high dimensional transfer operators

Principal Investigator(s) Other Investigators

## Summary:

The topic of the project is the numerical analysis of transfer operators that are generated by parabolic systems, in particular semilinear reaction diffusion equations. The spatial discretization of such systems via Galerkin or finite element methods leads to finite but high-dimensional dynamical systems for which we want to approximate global attractors and invariant measures. Due to the origin of the discrete equations it is assumed that attractors are imbedded into low-dimensional submanifolds of the phase space and invariant measures are supported by such low-dimensional manifolds.

The approach followed in the project combines in an adaptive way methods of dimension reduction (POD-modes, Proper Orthogonal Decomposition) with recent set-valued methods for attractors and invariant measures that have been developed by Dellnitz and co-workers. Several limit processes linked to this approach will be studied, such as the number of Galerkin modes tending to infinity, varying the number of POD-modes and increasing the refinement of the box collection covering the attractor. The first limit process has direct relations to Kolmogorov operators and their associated semigroups on infinite-dimensional spaces (cf. project B4, Röckner). Spectral structures of the transfer operators play an important role for the computation of invariant measures and measure-theoretic aspects are essential when investigating the relation to stochastic differential equations.

## Recent Preprints:

 09032 Computing Sacker-Sell spectra in discrete time dynamical systems PDF | PS.GZ 09027 A General Approach to Hyperbolicity for Set-Valued Maps PDF | PS.GZ 09008 Connecting orbits in perturbed systems PDF | PS.GZ 08118 Computing invariant measures with dimension reduction methods PDF | PS.GZ 08115 A $C^{\infty}$ density theorem for differential inclusions with Lipschitz continuous right hand sides PDF | PS.GZ 08114 Preservation of bifurcations under Runge-Kutta methods PDF | PS.GZ 08081 Numerical computation of dichotomy rates and projectors in discrete time PDF | PS.GZ 08079 On the approximation of integrated semigroups PDF | PS.GZ 08053 Shadowing and the Viability Kernel Algorithm PDF | PS.GZ 07082 Shadowing and inverse shadowing in set-valued dynamical systems. Hyperbolic case PDF | PS.GZ