jk10 Jens Kemper.
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.
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| Publisher: | Logos Verlag, Berlin (2010) |
| Pages: | 158 |
| ISBN-10: | 3-8325-2452-5 |
| ISBN-13: | 978-3-8325-2452-4 |
