Numerical analysis of high dimensional transfer operators
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.