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Uncoupling-Coupling Techniques for Metastable Dynamical Systems

  • Christof Schütte
  • Ralf Forster
  • Eike Meerbach
  • Alexander Fischer
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

We shortly review the uncoupling-coupling method, a Markov chain Monte Carlo based approach to compute statistical properties of systems like medium-sized biomolecules. This technique has recently been proposed for the efficient computation of biomolecular conformations. One crucial step of UC is the decomposition of reversible nearly uncoupled Markov chains into rapidly mixing subchains. We show how the underlying scheme of uncoupling-coupling can also be applied to stochastic differential equations where it can be translated into a domain decomposition technique for partial differential equations.

Keywords

Biomolecules Reversible Markov chains Stationary distribution Fokker-Planck operator Uncoupling-coupling Stochastic Differential Equations 

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References

  1. A. Brass, B. J. Pendleton, Y. Chen, and B. Robson. Hybrid Monte Carlo simulations theory and initial comparison with molecular dynamics. Biopolymers, 33:1307–1315, 1993.CrossRefGoogle Scholar
  2. P. Brémaud. Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York, 1999.Google Scholar
  3. G. E. Cho and C. D. Meyer. Aggregation/disaggregation methods for nearly uncoupled Markov chains. Technical Report NCSU #041600-0400, North Carolina State University, November 1999.Google Scholar
  4. P. Deuflhard, W. Huisinga, A. Fischer, and C. Schütte. Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Lin. Alg. Appl., 315:39–59, 2000.CrossRefGoogle Scholar
  5. P. Deuflhard and M. Weber. Robust Perron cluster analysis in conformation dynamics. ZIB-Report 03-19, Konrad-Zuse-Zentrum, Berlin, 2003.Google Scholar
  6. D. M. Ferguson, J. I. Siepmann, and D. G. Truhlar, editors. Monte Carlo Methods in Chemical Physics, volume 105 of Advances in Chemical Physics. Wiley, New York, 1999.Google Scholar
  7. A. Fischer. An Uncoupling-Coupling Method for Markov Chain Monte Carlo Simulations with an Application to Biomolecules. PhD thesis, Freie Universität Berlin, 2003.Google Scholar
  8. A. Fischer, C. Schütte, P. Deuflhard, and F. Cordes. Hierarchical uncoupling-coupling of metastable conformations. In T. Schlick and H. H. Gan, editors, Computational Methods for Macromolecules: Challenges and Applications, Proceedings of the 3rd International Workshop on Algorithms for Macromolecular Modeling, New York, Oct. 12–14, 2000, volume 24 of Lecture Notes in Computational Science and Engineering, Berlin, 2002. Springer.Google Scholar
  9. R. Forster. Ein Algorithmus zur Berechnung invarianter Dichten in metastabilen Systemen. Diploma thesis, Freie Universität Berlin, 2003.Google Scholar
  10. T. Friese, P. Deuflhard, and F. Schmidt. A multigrid method for the complex Helmholtz eigenvalue problem. In C.-H. Lai, P. E. Bjørstad, M. Cross, and O. B. Widlund, editors, Domain Decomposition Methods in Sciences and Engineering, DDM-org, pages 18–26, New York, 1999.Google Scholar
  11. W. Huisinga, S. Meyn, and C. Schütte. Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab., to appear 2004.Google Scholar
  12. J. S. Liu. Monte Carlo Strategies in Scientific Computing. Springer, New York, 2001.Google Scholar
  13. E. Meerbach, A. Fischer, and C. Schütte. Eigenvalue bounds on restrictions of reversible nearly uncoupled Markov chains. Preprint, 2003.Google Scholar
  14. C. D. Meyer. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev., 31:240–272, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  15. S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, Berlin, 1993.Google Scholar
  16. C. Schütte, A. Fischer, W. Huisinga, and P. Deuflhard. A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys., 151:146–168, 1999.MathSciNetCrossRefGoogle Scholar
  17. C. Schütte and W. Huisinga. Biomolecular conformations can be identified as metastable sets of molecular dynamics. In P. G. Ciaret and J.-L. Lions, editors, Handbook of Numerical Analysis, volume Computational Chemistry. North-Holland, 2003.Google Scholar
  18. C. Schütte, W. Huisinga, and P. Deuflhard. Transfer operator approach to conformational dynamics in biomolecular systems. In B. Fiedler, editor, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, 2001.Google Scholar
  19. W. J. Stewart and W. Wu. Numerical experiments with iteration and aggregation for Markov Chains. ORSA Journal on Computing, 4(3):336–350, 1992.Google Scholar
  20. L. Tierney. Markov chains for exploring posterior distributions (with discussion). Ann. Statist., 22:1701–1762, 1994.zbMATHMathSciNetGoogle Scholar
  21. M. Weber. Improved Perron cluster analysis. ZIB-Report 03-04, Konrad-Zuse-Zentrum, Berlin, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christof Schütte
    • 1
  • Ralf Forster
    • 1
  • Eike Meerbach
    • 1
  • Alexander Fischer
    • 1
  1. 1.Institute of Mathematics IIFree University BerlinBerlinGermany

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