Computation Schemes for Transfer Operators

by Adam Nielsen

Institution: Freie Universität Berlin
Year: 2016
Posted: 02/05/2017
Record ID: 2104735
Full text PDF: http://edocs.fu-berlin.de/diss/receive/FUDISS_thesis_000000101687


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…