Computation of invariant measures with dimension reduction methods

In recent years, Dellnitz, Junge and co-workers developed a subdivision algorithm for the approximation of invariant measures in discrete dynamical systems based on the so-called Ulam's approach. In high dimensions, this adaptive invariant measure (AIM) algorithm suffers from the ``curse of dimension'' even when the support of the system's invariant measure is known to be low-dimensional. In our thesis we develop algorithms facing this problem by combining the subdivision technique with proper orthogonal decomposition (POD) as a model reduction method. We derive explicit error bounds concerning the long-time behavior of POD solutions, propose a discrete version of the Prohorov metric as a proper distance notion for discrete measures computed by the algorithms, and analytically compare the approximation processes of the AIM algorithm and the POD-based algorithms. A marginal-like representation of discrete measures is proposed in order to visualize the numerical experiments. The algorithms are applied to finite element discretizations of the Chafee-Infante problem in order to show the power of our approach.