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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References
C. Bandle. Isoperimetric Inequalities and Applications. Pitman, Boston, London, Melbourne, 1980.
Anton Bovier, Michael Eckhoff, Véronique Gayrard, and Markus Klein. Metastability in stochastic dynamics of disordered mean-field models. Probab. Theor. Rel. Fields, 119:99–161, 2001.
Anton Bovier, Michael Eckhoff, Véronique Gayrard, and Markus Klein. Metastability and low lying spectra in reversible Markov chains. Com. Math. Phys., 228:219–255, 2002.
Anton Bovier, Michael Eckhoff, Véronique Gayrard, and Markus Klein. Metastability in reversible diffusion processes I: Sharp estimates for capacities and exit times. J. Eur. Math. Soc., 6:399–424, 2004.
Anton Bovier, Véronique Gayrard, and Markus Klein. Metastability in reversible diffusion processes II: Precise estimates for small eigenvalues. J. Eur. Math. Soc., 7:69–99, 2005.
Edward Brian Davies. Metastability and the Ising model. J. Statist. Phys, 27:657–675, 1982.
Edward Brian Davies. Metastable states of symmetric Markov semigroups I. Proc. London Math. Soc., 45(3):133–150, 1982.
Edward Brian Davies. Metastable states of symmetric Markov semigroups II. J. London Math. Soc., 26(2):541–556, 1982.
Peter Deuflhard, Wilhelm Huisinga, Alexander Fischer, and Christof Schütte. Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl., 315:39–59, 2000.
Michael Dellnitz and Oliver Junge. On the approximation of complicated dynamical behavior. SIAM J. Num. Anal., 36(2):491–515, 1999.
M. Dellnitz, O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg R. Preis, S.D. Ross, and B. Thiere. Transport in dynamical astronomy and multibody problems. International Journal of Bifurcation and Chaos, 2005. to appear.
J. L. Doob. Stochastic Processes. John Wiley & Sons, Inc., New York, 1953.
P. Deuflhard and Ch. Schütte. Molecular conformation dynamics and computational drug design. In J.M. Hill and R. Moore, editors, Applied Mathematics Entering the 21st Century, Proc. ICIAM 2003, pages 91–119, Sydney, Australia, 7-11 July 2004.
P. Deuflhard and M. Weber. Robust Perron cluster analysis in conformation dynamics. Lin. Alg. Appl.-Special issue on Matrices and Mathematical Biology. In Dellnitz, S. Kirkland, M. Neumann and C. Schütte (eds.), 398C:161–184, 2004.
Alexander Fischer. An Uncoupling-Coupling Method for Markov chain Monte Carlo simulations with an application to biomolecules. PhD thesis, Free University Berlin, 2003.
M.I. Freidlin and A.D. Wentzell. Random Perturbations of Dynamical Systems. Springer, New York, 1984. Series in Comprehensive Studies in Mathematics.
C. W. Gardiner. Handbook of Stochastic Methods. Springer, Berlin, 2nd enlarged edition edition, 1985.
Wilhelm Huisinga, Christoph Best, Rainer Roitzsch, Christof Schütte, and Frank Cordes. From simulation data to conformational ensembles: Structure and dynamic based methods. J. Comp. Chem., 20(16):1760–1774, 1999.
I. Horenko, E. Dittmer, A. Fischer, and Ch. Schütte. Automated model reduction for complex systems exhibiting metastability. Multiscale Modeling and Simulation, 2005. submitted.
R. Hassin and M. Haviv. Mean passage times and nearly uncoupled Markov chains. SIAM J. Disc. Math., 5(3):386–397, 1992.
Bruce Hendrickson and Robert Leland. An improved spectral graph partitioning algorithm for mapping parallel computations. SIAM J. Sci. Comput., 16(2):452–469, 1995.
D. J. Hartfiel and C. D. Meyer. On the structure of stochastic matrices with a subdominant eigenvalue near 1. Lin. Alg. Appl., 272:193–203, 1998.
Wilhelm Huisinga, Sean Meyn, and Christof Schütte. Phase transitions & metastability in Markovian and molecular systems. Ann. Appl. Probab., 14:419–458, 2004.
Wilhelm Huisinga, Christof Schütte, and Andrew M. Stuart. Extracting macroscopic stochastic dynamics: Model problems. Comm. Pure Appl. Math., 56:234–269, 2003.
Wilhelm Huisinga. Metastability of Markovian systems: A transfer operator approach in application to molecular dynamics. PhD thesis, Freie Universität Berlin, 2001.
P. Imkeller and A. Monahan. Conceptual stochastic climate models. Stochastics and Dynamics, 2(3):311–326, 2002.
Andrzej Lasota and Michael C. Mackey. Chaos, Fractals and Noise, volume 97 of Applied Mathematical Sciences. Springer, New York, 2nd edition, 1994.
C. D. Meyer. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev., 31:240–272, 1989.
S.P. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability. Springer, Berlin, 1993.
Edward Nelson. Dynamical Theories of Brownian Motion. Mathematical Notes. Princeton Uni. Press, 1967.
D. Revuz. Markov Chains. North-Holland, Amsterdam, Oxford, 1975.
Hannes Risken. The Fokker-Planck Equation. Springer, New York, 2nd edition, 1996.
Ch. Schütte. Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules. Habilitation Thesis, Fachbereich Mathematik und Informatik, Freie Universität Berlin, 1998.
Ch. Schütte, A. Fischer, W. Huisinga, and P. Deuflhard. A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys., Special Issue on Computational Biophysics, 151:146–168, 1999.
Christof Schütte and Wilhelm Huisinga. Biomolecular conformations can be identified as metastable sets of molecular dynamics. In P. G. Ciarlet, editor, Handbook of Numerical Analysis, volume Special Volume Computational Chemistry, pages 699–744. North-Holland, 2003.
Christof Schütte, Wilhelm Huisinga, and Peter Deuflhard. Transfer operator approach to conformational dynamics in biomolecular systems. In Bernold Fiedler, editor, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 191–223. Springer, 2001.
Gregory Singleton. Asymptotically exact estimates for metatstable Markov semigroups. Quart. J. Math. Oxford, 35(2):321–329, 1984.
Alistair Sinclair. Algorithms for Random Generation and Counting-A Markov Chain Approach. Progress in Theoretical Computer Science. Birkhäuser, 1993.
John Stachurski. Stochastic growth with increasing returns: Stability and path dependence. Studies in Nonlinear Dynamics & Econometrics, 7(2):1104–1104, 2003.
Christof Schütte, Jessika Walter, Carsten Hartmann, and Wilhelm Huisinga. An averaging principle for fast degrees of freedom exhibiting long-term correlations. SIAM Multiscale Modeling and Simulation, 2:501–526, 2004.
W. F. van Gunsteren, S. R. Billeter, A. A. Eising, P. H. Hünenberger, P. Krüger, A. E. Mark, W. R. P. Scott, and I. G. Tironi. Biomolecular Simulation: The GROMOS96 Manual and User Guide. vdf Hochschulverlag AG, ETH Zürich, 1996.
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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
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