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Metastability and Dominant Eigenvalues of Transfer Operators

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New Algorithms for Macromolecular Simulation

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 49))

Abstract

Metastability is an important characteristic of molecular systems, e.g., when studying conformation dynamics, computing transition paths or speeding up Markov chain Monte Carlo sampling methods. In the context of Markovian (molecular) systems, metastability is closely linked to spectral properties of transfer operators associated with the dynamics. In this article, we prove upper and lower bounds for the metastability of a state-space decomposition for reversible Markov processes in terms of dominant eigenvalues and eigenvectors of the corresponding transfer operator. The bounds are explicitly computable, sharp, and do not rely on any asymptotic expansions in terms of some smallness parameter, but rather hold for arbitrary transfer operators satisfying a reasonable spectral condition.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 2-6, D-14195, Berlin, Germany

    Wilhelm Huisinga

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103, Leipzig, Germany

    Bernd Schmidt

Authors
  1. Wilhelm Huisinga
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  2. Bernd Schmidt
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Benedict Leimkuhler

  2. Institut nancéien de chimie moléculaire, Université Henri Poincaré - Nancy I, B.P. 239, 54506, Vandoeuvre-lès-Nancy, France

    Christophe Chipot

  3. Department of Computer Science, Cornell University, 4130 Upson Hall, Ithaca, NY, 14853-7501, USA

    Ron Elber

  4. Arrhenius Laboratory Division of Physical Chemistry, Stockholm University, 106 91, Stockholm, Sweden

    Aatto Laaksonen

  5. Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands

    Alan Mark

  6. Department of Chemistry, Courant Institute of Mathematical Sciences, New York University, Mercer Street 251, New York, NY, 10012, USA

    Tamar Schlick

  7. FB Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

    Christoph Schütte

  8. Department of Computer Science, Purdue University, N. University Street 250, West Lafayette, IN, 47907-2066, USA

    Robert Skeel

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Huisinga, W., Schmidt, B. (2006). Metastability and Dominant Eigenvalues of Transfer Operators. In: Leimkuhler, B., et al. New Algorithms for Macromolecular Simulation. Lecture Notes in Computational Science and Engineering, vol 49. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31618-3_11

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  • DOI: https://doi.org/10.1007/3-540-31618-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25542-0

  • Online ISBN: 978-3-540-31618-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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