@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{SchuetteNielsenWeber, author = {Sch{\"u}tte, Christof and Nielsen, Adam and Weber, Marcus}, title = {Markov State Models and Molecular Alchemy}, issn = {1438-0064}, doi = {10.1080/00268976.2014.944597}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-46718}, 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 ex- periments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multi- valent 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 proper- ties 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 en- ergy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.}, language = {en} } @misc{Nielsen, type = {Master Thesis}, author = {Nielsen, Adam}, title = {Von Femtosekunden zu Minuten}, pages = {107}, language = {de} } @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} } @phdthesis{Nielsen, author = {Nielsen, Adam}, title = {Computation Schemes for Transfer Operators}, language = {en} } @misc{NielsenWeber, 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}, 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} } @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} } @misc{Nielsen, 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}, 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{BujotzekSchuettNielsenetal., 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}, series = {Math Chem}, journal = {Math Chem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42570}, 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} } @article{BujotzekSchuettNielsenetal., author = {Bujotzek, Alexander and Sch{\"u}tt, Ole and Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {ZIBgridfree: Efficient Conformational Analysis by Partition-of-Unity Coupling}, series = {Journal of Mathematical Chemistry}, volume = {52}, journal = {Journal of Mathematical Chemistry}, number = {3}, doi = {10.1007/s10910-013-0265-1}, pages = {781 -- 804}, language = {de} } @misc{NielsenFackeldeyWeber, 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}, 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} }