@article{SchuetteNielsenWeber, author = {Sch{\"u}tte, Christof and Nielsen, Adam and Weber, Marcus}, title = {Markov State Models and Molecular Alchemy}, series = {Molecular Physics}, volume = {113}, journal = {Molecular Physics}, number = {1}, doi = {10.1080/00268976.2014.944597}, pages = {69 -- 78}, abstract = {In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.}, language = {en} } @misc{Lie, type = {Master Thesis}, author = {Lie, Han Cheng}, title = {A Meshless Method for Conformational Analysis}, language = {en} } @article{Nielsen, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, series = {Statistics \& Probability Letters}, volume = {118}, journal = {Statistics \& Probability Letters}, publisher = {Elsevier}, doi = {10.1016/j.spl.2016.06.011}, pages = {163 -- 170}, abstract = {In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the \$L^2\$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains.}, language = {en} } @misc{QuerLie, author = {Quer, Jannes and Lie, Han Cheng}, title = {Some connections between importance sampling and enhanced sampling methods in molecular dynamics}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64289}, abstract = {Enhanced sampling methods play an important role in molecular dynamics, because they enable the collection of better statistics of rare events that are important in many physical phenomena. We show that many enhanced sampling methods can be viewed as methods for performing importance sampling, by identifying important correspondences between the language of molecular dynamics and the language of probability theory. We illustrate these connections by highlighting the similarities between the rare event simulation method of Hartmann and Sch{\"u}tte (J. Stat. Mech. Theor. Exp., 2012), and the enhanced sampling method of Valsson and Parrinello (Phys. Rev. Lett. 113, 090601). We show that the idea of changing a probability measure is fundamental to both enhanced sampling and importance sampling.}, language = {en} } @article{KoltaiWuNoeetal., author = {Koltai, Peter and Wu, Hao and No{\´e}, Frank and Sch{\"u}tte, Christof}, title = {Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics}, series = {Computation}, volume = {6}, journal = {Computation}, number = {1}, publisher = {MDPI}, address = {Basel, Switzerland}, doi = {10.3390/computation6010022}, pages = {22}, language = {en} } @article{Weber, author = {Weber, Marcus}, title = {Implications of PCCA+ in Molecular Simulation}, series = {Computation}, volume = {6}, journal = {Computation}, number = {1}, doi = {10.3390/computation6010020}, pages = {20}, abstract = {Upon ligand binding or during chemical reactions the state of a molecular system changes in time. Usually we consider a finite set of (macro-) states of the system (e.g., 'bound' vs. 'unbound'), although the process itself takes place in a continuous space. In this context, the formula chi=XA connects the micro-dynamics of the molecular system to its macro-dynamics. Chi can be understood as a clustering of micro-states of a molecular system into a few macro-states. X is a basis of an invariant subspace of a transfer operator describing the micro-dynamics of the system. The formula claims that there is an unknown linear relation A between these two objects. With the aid of this formula we can understand rebinding effects, the electron flux in pericyclic reactions, and systematic changes of binding rates in kinetic ITC experiments. We can also analyze sequential spectroscopy experiments and rare event systems more easily. This article provides an explanation of the formula and an overview of some of its consequences.}, language = {en} } @misc{Weber, author = {Weber, Marcus}, title = {Supplementary: Implications of PCCA+ in Molecular Simulation}, abstract = {Matlab-software and data sets to recapitulate the presented results in M. Weber: Implications of PCCA+ in Molecular Simulation. Computation, 6(1):20, 2018.}, language = {en} } @misc{Weber, author = {Weber, Marcus}, title = {Eigenvalues of non-reversible Markov chains - A case study}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62191}, abstract = {Finite reversible Markov chains are characterized by a transition matrix P that has real eigenvalues and pi-orthogonal eigenvectors, where pi is the stationary distribution of P. This means, that a transition matrix with complex eigenvalues corresponds to a non-reversible Markov chain. This observation leads to the question, whether the imaginary part of that eigendecomposition corresponds to or indicates the "pattern" of the nonreversibility. This article shows that the direct relation between imaginary parts of eigendecompositions and the non-reversibility of a transition matrix is not given. It is proposed to apply the Schur decomposition of P instead of the eigendecomposition in order to characterize its nonreversibility.}, language = {en} } @article{KlusKoltaiSchuette, author = {Klus, Stefan and Koltai, Peter and Sch{\"u}tte, Christof}, title = {On the numerical approximation of the Perron-Frobenius and Koopman operator}, series = {Journal of Computational Dynamics}, volume = {3}, journal = {Journal of Computational Dynamics}, number = {1}, doi = {10.3934/jcd.2016003}, pages = {51 -- 77}, abstract = {Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di� fferent methods that have been developed over the last decades to compute � infinite-dimensional approximations of these in� finite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di� fferences between these approaches. The results will be illustrated using simple stochastic di� fferential equations and molecular dynamics examples.}, language = {en} } @article{RuedrichSarichSchuette, author = {R{\"u}drich, S. and Sarich, Marco and Sch{\"u}tte, Christof}, title = {Utilizing hitting times for finding metastable sets in non-reversible Markov chains}, series = {Journal of Comp. Dynamics}, journal = {Journal of Comp. Dynamics}, language = {en} } @article{KlusNueskeKoltaietal., author = {Klus, Stefan and N{\"u}ske, Feliks and Koltai, Peter and Wu, Hao and Kevrekidis, Ioannis and Sch{\"u}tte, Christof and No{\´e}, Frank}, title = {Data-driven model reduction and transfer operator approximation}, series = {Journal of Nonlinear Science}, volume = {28}, journal = {Journal of Nonlinear Science}, number = {3}, doi = {10.1007/s00332-017-9437-7}, pages = {985 -- 1010}, language = {en} } @misc{Lange, type = {Master Thesis}, author = {Lange, Christoph}, title = {Approximating and Clustering Markov Chains via Kullback-Leibler-Divergence}, language = {en} } @misc{Zwingel, type = {Master Thesis}, author = {Zwingel, Oliver}, title = {Die Cross-Entropy-Methode f{\"u}r stochastische Optimierungsprobleme}, language = {de} } @article{NielsenWeber, author = {Nielsen, Adam and Weber, Marcus}, title = {Computing the nearest reversible Markov chain}, series = {Numerical Linear Algebra with Applications}, volume = {22}, journal = {Numerical Linear Algebra with Applications}, number = {3}, doi = {10.1002/nla.1967}, pages = {483 -- 499}, abstract = {Reversible Markov chains are the basis of many applications. However, computing transition probabilities by a finite sampling of a Markov chain can lead to truncation errors. Even if the original Markov chain is reversible, the approximated Markov chain might be non-reversible and will lose important properties, like the real valued spectrum. In this paper, we show how to find the closest reversible Markov chain to a given transition matrix. It turns out that this matrix can be computed by solving a convex minimization problem.}, language = {en} } @book{OPUS4-5339, title = {Molecular Dynamics Simulation}, series = {Entropy (special issue)}, journal = {Entropy (special issue)}, editor = {Ciccotti, Giovanni and Ferrario, Mauro and Sch{\"u}tte, Christof}, publisher = {Multidisciplinary Digital Publishing Institute (MDPI)}, language = {en} } @misc{SarichSchuette, author = {Sarich, Marco and Sch{\"u}tte, Christof}, title = {Markov Model Theory}, series = {An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation}, volume = {797}, journal = {An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation}, editor = {Bowman, Gregory R. and Pande, Vijay S. and No{\´e}, Frank}, publisher = {Springer}, doi = {10.1007/978-94-007-7606-7}, pages = {23 -- 44}, abstract = {This section reviews the relation between the continuous dynamics of a molecular system in thermal equilibrium and the kinetics given by a Markov State Model (MSM). We will introduce the dynamical propagator, an error-less, alternative description of the continuous dynamics, and show how MSMs result from its discretization. This allows for an precise understanding of the approximation quality of MSMs in comparison to the continuous dynamics. The results on the approximation quality are key for the design of good MSMs. While this section is important for understanding the theory of discretization and related systematic errors, practitioners wishing only to learn how to construct MSMs may skip directly to the discussion of Markov model estimation.}, language = {en} } @article{SikorskiWeberSchuette, author = {Sikorski, Alexander and Weber, Marcus and Sch{\"u}tte, Christof}, title = {The Augmented Jump Chain}, series = {Advanced Theory and Simulations}, volume = {4}, journal = {Advanced Theory and Simulations}, number = {4}, publisher = {Wiley-VCH}, doi = {10.1002/adts.202000274}, abstract = {Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. A representation of non-autonomous Markov jump processes is presented as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so-called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.}, language = {en} } @article{LelievreZhang2018, author = {Leli{\`e}vre, Tony and Zhang, Wei}, title = {Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates}, series = {Multiscale Modeling and Simulation}, journal = {Multiscale Modeling and Simulation}, number = {17}, doi = {10.1137/18M1186034}, pages = {1019 -- 1051}, year = {2018}, abstract = {Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Leli{\`e}vre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Leli{\`e}vre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.}, language = {en} } @article{Zhang2018, author = {Zhang, Wei}, title = {Ergodic SDEs on submanifolds and related numerical sampling schemes}, series = {ESAIM: Mathematical Modelling and Numerical Analysis}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, year = {2018}, abstract = {In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure \$\mu\$ on the level set of a smooth function ξ:R^d→R^k, 1≤k