@misc{Weber2009, author = {Weber, Marcus}, title = {A Subspace Approach to Molecular Markov State Models via an Infinitesimal Generator}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11432}, number = {09-27}, year = {2009}, abstract = {Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.}, language = {en} } @misc{FackeldeyRoeblitzScharkoietal.2011, author = {Fackeldey, Konstantin and R{\"o}blitz, Susanna and Scharkoi, Olga and Weber, Marcus}, title = {Soft Versus Hard Metastable Conformations in Molecular Simulations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13189}, number = {11-27}, year = {2011}, abstract = {Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.}, language = {en} } @phdthesis{Weber2011, author = {Weber, Marcus}, title = {A Subspace Approach to Molecular Markov State Models via a New Infinitesimal Generator}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14025}, school = {Zuse Institute Berlin (ZIB)}, year = {2011}, abstract = {Supercomputers can simulate complex molecular systems. However, there is a very large gap between the fastest oscillations of covalent bonds of a molecule and the time-scale of the dominant processes. In order to extract the dominant time-scales and to identify the dominant processes, a clustering of information is needed. This thesis shows that only the subspace-based Robust Perron Cluster Analysis (PCCA+) can solve this problem correctly by the construction of a Markov State Model. PCCA+ allows for time-extrapolation in molecular kinetics. This thesis shows the difference between molecular dynamics and molecular kinetics. Only in the molecular kinetics framework a definition of transition rates is possible. In this context, the existence of an infinitesimal generator of the dynamical processes is discussed. If the existence is assumed, the Theorem of Gauß can be applied in order to compute transition rates efficiently. Molecular dynamics, however, is not able to provide a suitable statistical basis for the determination of the transition pattern.}, language = {en} } @misc{Weber2012, author = {Weber, Marcus}, title = {The funnel trap paradox}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14765}, number = {12-12}, year = {2012}, abstract = {In this article, an illustrative example is given for the coarse-graining of a Markov process which leads to a shift in the statistical weights of a two-states-system. The example is based on a 2D-funnel trap. The funnel trap is constructed in such a way, that the area inside and outside of the trap is identical. However, observing the flight of the insect as a Markov process, the probability for being "in the trap" is higher. This example can be transferred to several kinds of processes (like receptor-ligandbinding processes in chemistry) and describes the influence of "re-entering events".}, language = {en} } @misc{Fackeldey2010, author = {Fackeldey, Konstantin}, title = {Coupling Meshbased and Meshfree Methods by a Transfer Operator Approach}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11755}, number = {10-12}, year = {2010}, abstract = {In contrast to the well known meshbased methods like the finite element method, meshfree methods do not rely on a mesh. However besides their great applicability, meshfree methods are rather time consuming. Thus, it seems favorable to combine both methods, by using meshfree methods only in a small part of the domain, where a mesh is disadvantageous, and a meshbased method for the rest of the domain. We motivate, that this coupling between the two simulation techniques can be considered as saddle point problem and show the stability of this coupling. Thereby a novel transfer operator is introduced, which interacts in the transition zone, where both methods coexist.}, language = {en} } @misc{TungaWeber2012, author = {Tunga, Burcu and Weber, Marcus}, title = {Free Energy Calculation Using Mayer Cluster Expansion and Fluctuation Free Integration}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16031}, year = {2012}, abstract = {This work aims to develop a new algorithm to calculate the free energy of water molecules by using a deterministic way. For this purpose, we assume a closed system confined to a physical volume, having water molecules in gas phase. To calculate the free energy of this sytem we utilized Mayer cluster expansion and the fluctuation free integration method.}, language = {en} } @misc{WeberFackeldey2013, author = {Weber, Marcus and Fackeldey, Konstantin}, title = {Computing the Minimal Rebinding Effect Included in a Given Kinetics}, issn = {1438-0064}, doi = {10.1137/13091124X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17796}, year = {2013}, abstract = {The rebinding effect is a phenomenon which occurs when observing a ligand-receptor binding process. On the macro scale this process comprises the Markov property. This Makovian view is spoiled when switching to the atomistic scale of a binding process. We therefore suggest a model which accurately describes the rebinding effect on the atomistic scale by allowing ''intermediate'' bound states. This allows us to define an indicator for the magnitude of rebinding and to formulate an optimization problem. The results form our examples show good agreement with data form laboratory.}, language = {en} } @misc{LieFackeldeyWeber2013, author = {Lie, Han Cheng and Fackeldey, Konstantin and Weber, Marcus}, title = {A square root approximation of transition rates for a Markov State Model}, issn = {1438-0064}, doi = {10.1137/120899959}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42195}, year = {2013}, abstract = {Trajectory- or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality - their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge.}, language = {en} } @misc{NielsenFackeldeyWeber2013, author = {Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {On a Generalized Transfer Operator}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-43162}, year = {2013}, abstract = {We introduce a generalized operator for arbitrary stochastic processes by using a pre-kernel, which is a generalization of the Markov kernel. For deterministic processes, such an operator is already known as the Frobenius-Perron operator, which is defined for a large class of measures. For Markov processes, there exists transfer operators being only well defined for stationary measures in \$L^2\$. Our novel generalized transfer operator is well defined for arbitrary stochastic processes, in particular also for deterministic ones. We can show that this operator is acting on \$L^1\$. For stationary measures, this operator is also an endomorphism of \$L^2\$ and, therefore, allows for a mathematical analysis in Hilbert spaces.}, language = {en} } @misc{Helfmann2018, type = {Master Thesis}, author = {Helfmann, Luzie}, title = {Stochastic Modeling of Interacting Agent Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71887}, pages = {68}, year = {2018}, language = {en} } @misc{BujotzekSchuettNielsenetal.2013, author = {Bujotzek, Alexander and Sch{\"u}tt, Ole and Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {Efficient Conformational Analysis by Partition-of-Unity Coupling}, journal = {Math Chem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42570}, year = {2013}, abstract = {Obtaining a sufficient sampling of conformational space is a common problem in molecular simulation. We present the implementation of an umbrella-like adaptive sampling approach based on function-based meshless discretization of conformational space that is compatible with state of the art molecular dynamics code and that integrates an eigenvector-based clustering approach for conformational analysis and the computation of inter-conformational transition rates. The approach is applied to three example systems, namely n-pentane, alanine dipeptide, and a small synthetic host-guest system, the latter two including explicitly modeled solvent.}, language = {en} } @misc{Weber2017, 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}, year = {2017}, 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} } @misc{Berg2012, type = {Master Thesis}, author = {Berg, Mascha}, title = {Laufzeitoptimierung der Robusten Perron Cluster Analyse (PCCA+)}, year = {2012}, language = {de} } @misc{Nielsen2016, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59933}, year = {2016}, 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{Breitbach2018, type = {Master Thesis}, author = {Breitbach, Paula}, title = {Molecular Binding Kinetics of CYP P450 by Using the Infinitesimal Generator Approach}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69076}, year = {2018}, language = {en} } @misc{Bender2019, type = {Master Thesis}, author = {Bender, Wilhelm}, title = {On a Novel Approach for Global Optimization of Non-Convex Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-77074}, year = {2019}, abstract = {In this work new approaches for global optimization are developed. We move from theoretical connections in complexity theory to very practical results in protein-ligand docking. In regard to the efficiency of the suggested optimizer, continuity methods are motivated, that yield an improved optimization method.}, language = {en} } @misc{Wulkow2017, type = {Master Thesis}, author = {Wulkow, Niklas}, title = {Modelling the Spread of Innovations by a Markov Process in a Bayesian Framework}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66869}, year = {2017}, language = {en} } @misc{HelfmannDjurdjevacConradDjurdjevacetal.2019, author = {Helfmann, Luzie and Djurdjevac Conrad, Natasa and Djurdjevac, Ana and Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {From interacting agents to density-based modeling with stochastic PDEs}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-73456}, year = {2019}, abstract = {Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation of such agent-based models is often intractable when the system becomes too large. In this paper, starting from a stochastic spatio-temporal agent-based model (ABM), we present a reduced model in terms of stochastic PDEs that describes the evolution of agent number densities for large populations. We discuss the algorithmic details of both approaches; regarding the SPDE model, we apply Finite Element discretization in space which not only ensures efficient simulation but also serves as a regularization of the SPDE. Illustrative examples for the spreading of an innovation among agents are given and used for comparing ABM and SPDE models.}, language = {en} } @misc{WeberFackeldey2015, author = {Weber, Marcus and Fackeldey, Konstantin}, title = {G-PCCA: Spectral Clustering for Non-reversible Markov Chains}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55505}, year = {2015}, abstract = {Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We extend this spectral clustering method also to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension, non-reversible Markov chains can be analyzed. Furthermore, the chains need not have a positive stationary distribution. And additionally to metastabilities, dominant cycles and sinks can be identified, too.}, language = {en} } @misc{NielsenWeber2014, author = {Nielsen, Adam and Weber, Marcus}, title = {Computing the nearest reversible Markov chain}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53292}, year = {2014}, 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} } @misc{WeberQuer2015, author = {Weber, Marcus and Quer, Jannes}, title = {Estimating exit rates in rare event dynamical systems via extrapolation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-56622}, year = {2015}, abstract = {In this article we present a new idea for approximating exit rates for diffusion processes living in a craggy landscape. We are especially interested in the exit rates of a process living in a metastable regions. Due to the fact that Monte Carlo simulations perform quite poor and are very computational expensive in this setting we create several similar situations with a smoothed potential. For this we introduce a new parameter \$\lambda \in [0,1]\$ (\$\lambda = 1\$ very smoothed potential, \$\lambda=0\$ original potential) into the potential which controls the influence the smoothing. We then sample the exit rate for different parameters \$\lambda\$ the exit rate from a given region. Due to the fact that \$\lambda\$ is connected to the exit rate we can use this dependency to approximate the real exit rate. The method can be seen as something between hyperdynamics and temperature accelerated MC.}, language = {en} }