Prof. Dr. Reinhold Schneider Prof. Dr. Christof Schütte Dr. Péter Koltai
transfer operator; Markov operator; Galerkin projection; reversible Markov chains
515 Analysis 519 Wahrscheinlichkeitstheorien, mathematische Statistik 518 Numerische Analysis
The focus of this doctoral thesis is the transfer operator, a tool that describes the propagation
of probability densities of an arbitrary dynamical system. This tool is usable for any moving
object that one wants to analyze, and thus has applications in subjects like population
statistics, the prediction of stock prices, and computational drug design.
The first part of this doctoral thesis is a purely theoretical investigation of the transfer
operator. Characterizations of transfer operators and adjoint transfer operators are revealed.
It is shown that Markov operators and transfer operators are equivalent. Further it is shown
that an adjoint operator of a transfer operator is equivalent to a generalized Koopman
operator, and that an adjoint operator of a transfer operator with an invariant measure is
equivalent to a Brown-Markov operator. All three characterizations are independent of a
transition kernel. The last characterization is disproving a claim made in 1966.
Diverse applications require a Galerkin projection of the transfer operator. Therefore, the
second part of this thesis reveals possible ways of improving the computation of a Galerkin
projection on an arbitrary function space. An exact formula of the error by the difference
in the L2 norm between the Galerkin entry and its approximation through a Monte Carlo
method is deduced for long and short-term trajectory approaches. The formula enables us
to approximate the Galerkin error itself by trajectories. It is shown that the error of the
Galerkin projection is dramatically reduced when using short-term trajectories instead of
a single long-term trajectory. Further, a characteristic of reversible processes is discovered,
which shows that reversible processes are more likely to return to set than to be there.
Next, a reweighting scheme is introduced that improves available techniques for obtaining a
Galerkin projection for a typical scenario that often appears in computational drug design. It
is shown that the Galerkin projections for multiple, similar ligands that bind to one receptor
can be computed using trajectories of just one single ligand and the corresponding weights.
Computation of the weights proves more advantageous than computing the trajectories
separately for each ligand.
The final result presented in this thesis shows how to correct the numerical error of a
Galerkin projection. This is useful for cases in which the numerical error might render the
frequently employed clustering method PCCA+ to be inapplicable. It is shown that one
can restore a particular property of a Galerkin projection that assures applicability of the
method PCCA+. More precisely, for almost any norm and any transition matrix a closest
reversible matrix exists, which can be computed by solving a strongly convex quadratic
problem. Further, a norm is introduced which heavily weights transition probabilities of
rare events. This norm has the property that the closest reversible matrix will preserve the
spectrum. Application of the method PCCA+ was until now restricted to reversible processes.
However, the correction scheme for the Galerkin projection opens the door to use of the
method PCCA+ for arbitrary systems.
In summary, this thesis reveals theoretical aspects of the transfer operator that are then
used to derive methods to optimize and correct the computation of the Galerkin projection.