@misc{KubeWeber2005, author = {Kube, Susanna and Weber, Marcus}, title = {Identification of Metastabilities in Monomolecular Conformation Kinetics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8956}, number = {06-01}, year = {2005}, abstract = {The identification of metastable conformations of molecules plays an important role in computational drug design. One main difficulty is the fact that the underlying dynamic processes take place in high dimensional spaces. Although the restriction of degrees of freedom to a few dihedral angles significantly reduces the complexity of the problem, the existing algorithms are time-consuming. They are based on the approximation of transition probabilities by an extensive sampling of states according to the Boltzmann distribution. We present a method which can identify metastable conformations without sampling the complete distribution. Our algorithm is based on local transition rates and uses only pointwise information about the potential energy surface. In order to apply the cluster algorithm PCCA+, we compute a few eigenvectors of the rate matrix by the Jacobi-Davidson method. Interpolation techniques are applied to approximate the thermodynamical weights of the clusters. The concluding example illustrates our approach for epigallocatechine, a molecule which can be described by seven dihedral angles.}, 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{Zha1989, author = {Zha, Hongyuan}, title = {A Numerical Algorithm for Computing the Restricted Singular Value Decomposition of Matrix Triplets.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-194}, number = {SC-89-01}, year = {1989}, abstract = {This paper presents a numerical algorithm for computing the restricted singular value decomposition of matrix triplets (RSVD). It is shown that one can use unitary transformations to separate the regular part from a general matrix triplet. After preprocessing on the regular part, one obtains a matrix triplet consisting of three upper triangular matrices of the same dimensions. The RSVD of this special matrix triplet is computed using the implicit Kogbetliantz technique. The algorithm is well suited for parallel computation. {\bf Keywords:} Restricted singular values, matrix triplets, unitary transformations, implicit Kogbetliantz technique.}, language = {en} } @misc{Deuflhard2002, author = {Deuflhard, Peter}, title = {From Molecular Dynamics to Conformational Dynamics in Drug Design}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6878}, number = {02-20}, year = {2002}, abstract = {Computational drug design studies molecular recognition in the {\em virtual lab}. The arising Hamiltonian dynamics is known to be chaotic and ill-conditioned already after picoseconds, whereas times are \$msec\$ up to \$min\$. Classical molecular dynamics with long term trajectory computation gives, at best, information about time and statistical ensemble averages. The present paper surveys a recent new modeling approach called {\em conformational dynamics}, which is due to the author and Ch. Sch{\"u}tte. This approach achieves information about the dy time scales by telescoping a short term deterministic model with a statistical model. Examples of small biomolecules are included.}, language = {en} } @misc{DeuflhardWeber2003, author = {Deuflhard, Peter and Weber, Marcus}, title = {Robust Perron Cluster Analysis in Conformation Dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7415}, number = {03-19}, year = {2003}, abstract = {The key to molecular conformation dynamics is the direct identification of metastable conformations, which are almost invariant sets of molecular dynamical systems. Once some reversible Markov operator has been discretized, a generalized symmetric stochastic matrix arises. This matrix can be treated by Perron cluster analysis, a rather recent method involving a Perron cluster eigenproblem. The paper presents an improved Perron cluster analysis algorithm, which is more robust than earlier suggestions. Numerical examples are included.}, language = {en} } @misc{Deuflhard1999, author = {Deuflhard, Peter}, title = {Differential Equations in Technology and Medicine. Computational Concepts, Adaptive Algorithms, and Virtual Labs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4223}, number = {SC-99-34}, year = {1999}, abstract = {This series of lectures has been given to a class of mathematics postdocs at a European summer school on Computational Mathematics Driven by Industrial Applications in Martina Franca, Italy (organized by CIME). It deals with a variety of challenging real life problems selected from clinical cancer therapy, communication technology, polymer production, and pharmaceutical drug design. All of these problems from rather diverse application areas share two common features: (a) they have been modelled by various differential equations -- elliptic, parabolic, or Schr{\"o}dinger--type partial differential equations, countable ordinary diffential equations, or Hamiltonian systems, (b) their numerical solution has turned out to be real challenge to computational mathematics.}, language = {en} } @misc{HelmbergKiwiel1999, author = {Helmberg, Christoph and Kiwiel, K.C.}, title = {A Spectral Bundle Method with Bounds}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4256}, number = {SC-99-37}, year = {1999}, abstract = {Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. Encouraging preliminary computational results are reported.}, language = {en} } @misc{KubeWeber2006, author = {Kube, Susanna and Weber, Marcus}, title = {Coarse Grained Molecular Kinetics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9299}, number = {06-35}, year = {2006}, abstract = {The dynamic behavior of molecules can often be described by Markov processes. From computational molecular simulations one can derive transition rates or transition probabilities between subsets of the discretized conformational space. On the basis of this dynamic information, the spatial subsets are combined into a small number of so-called metastable molecular conformations. This is done by clustering methods like the Robust Perron Cluster Analysis (PCCA+). Up to now it is an open question how this coarse graining in space can be transformed to a coarse graining of the Markov chain while preserving the essential dynamic information. In the following article we aim at a consistent coarse graining of transition probabilities or rates on the basis of metastable conformations such that important physical and mathematical relations are preserved. This approach is new because PCCA+ computes molecular conformations as linear combinations of the dominant eigenvectors of the transition matrix which does not hold for other clustering methods.}, language = {en} }