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.
Supported by the DFG research center “Mathematics for key technologies” (FZT 86) in Berlin.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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.
P. Brémaud. Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York, 1999.
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.
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.
P. Deuflhard and M. Weber. Robust Perron cluster analysis in conformation dynamics. ZIB-Report 03-19, Konrad-Zuse-Zentrum, Berlin, 2003.
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.
A. Fischer. An Uncoupling-Coupling Method for Markov Chain Monte Carlo Simulations with an Application to Biomolecules. PhD thesis, Freie Universität Berlin, 2003.
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.
R. Forster. Ein Algorithmus zur Berechnung invarianter Dichten in metastabilen Systemen. Diploma thesis, Freie Universität Berlin, 2003.
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.
W. Huisinga, S. Meyn, and C. Schütte. Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab., to appear 2004.
J. S. Liu. Monte Carlo Strategies in Scientific Computing. Springer, New York, 2001.
E. Meerbach, A. Fischer, and C. Schütte. Eigenvalue bounds on restrictions of reversible nearly uncoupled Markov chains. Preprint, 2003.
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.
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.
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.
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.
W. J. Stewart and W. Wu. Numerical experiments with iteration and aggregation for Markov Chains. ORSA Journal on Computing, 4(3):336–350, 1992.
L. Tierney. Markov chains for exploring posterior distributions (with discussion). Ann. Statist., 22:1701–1762, 1994.
M. Weber. Improved Perron cluster analysis. ZIB-Report 03-04, Konrad-Zuse-Zentrum, Berlin, 2003.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schütte, C., Forster, R., Meerbach, E., Fischer, A. (2005). Uncoupling-Coupling Techniques for Metastable Dynamical Systems. In: Barth, T.J., et al. Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26825-1_8
Download citation
DOI: https://doi.org/10.1007/3-540-26825-1_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22523-2
Online ISBN: 978-3-540-26825-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)