@misc{SarichSchuette2014, author = {Sarich, Marco and Sch{\"u}tte, Christof}, title = {Utilizing hitting times for finding metastable sets in non-reversible Markov chains}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-51209}, year = {2014}, abstract = {Techniques for finding metastable or almost invariant sets have been investigated, e.g., for deterministic dynamical systems in set-oriented numerics, for stochastic processes in molecular dynamics, and for random walks on complex networks. Most prominent algorithms are based on spectral apporaches and identify metastable sets via the doimant eigenvalues of the transfer operator associated with the dynamical system under consideration. These algorithms require the dominant eigenvalues to be real-valued. However, for many types of dynamics, e.g. for non-reversible Markov chains, this condition is not met. In this paper we utilize the hitting time apporach to metastable sets and demonstrate how the wellknown statements about optimal metastable decompositions of reversible chains can be reformulated for non-reversible chains if one switches from a spectral approach to an exit time approach. The performance of the resulting algorithm is illustrated by numerical experiments on random walks on complex networks.}, language = {en} } @misc{KoltaiSchuette2017, author = {Koltai, P{\´e}ter and Sch{\"u}tte, Christof}, title = {A multi scale perturbation expansion approach for Markov state modeling of non-stationary molecular dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64868}, year = {2017}, abstract = {We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one.}, language = {en} } @misc{KoltaiCiccottiSchuette2016, author = {Koltai, Peter and Ciccotti, Giovanni and Sch{\"u}tte, Christof}, title = {On metastability and Markov state models for non-stationary molecular dynamics}, volume = {174103}, journal = {The Journal of Chemical Physics}, edition = {145}, issn = {1438-0064}, doi = {10.1063/1.4966157}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57869}, year = {2016}, abstract = {We utilize the theory of coherent sets to build Markov state models for non- equilibrium molecular dynamical systems. Unlike for systems in equilibrium, "meta- stable" sets in the non-equilibrium case may move as time evolves. We formalize this concept by relying on the theory of coherent sets, based on this we derive finite-time non-stationary Markov state models, and illustrate the concept and its main differences to equilibrium Markov state modeling on simple, one-dimensional examples.}, language = {en} }