Numerical analysis of high-dimensional
transfer operators
Description
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.
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.
Members
Preprints
