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Numerical analysis of parallel replica dynamics

Published online by Cambridge University Press:  09 July 2013

Gideon Simpson  and
Mitchell Luskin
Gideon Simpson
Affiliation:
School of Mathematics, University of Minnesota, 206 Church St SE, 127 Vincent Hall, Minneapolis MN 55455, USA.. gsimpson@umn.edu; luskin@umn.edu
Mitchell Luskin
Affiliation:
School of Mathematics, University of Minnesota, 206 Church St SE, 127 Vincent Hall, Minneapolis MN 55455, USA.. gsimpson@umn.edu; luskin@umn.edu

Abstract

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Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit distribution from a given well for a single process can be approximated by the distribution of the first exit of N independent identical processes, each run for only 1 / N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in [C. Le Bris, T. Lelièvre, M. Luskin and D. Perez, Monte Carlo Methods Appl. 18 (2012) 119–146], we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.

Keywords

Accelerated dynamicsrare eventsparallel replica

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1287 - 1314
Copyright
© EDP Sciences, SMAI, 2013

References

R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press 140 (2003).
M. Bieniek, K. Burdzy and S. Finch, Non-extinction of a Fleming–Viot particle model. Probab. Theory Relat. Fields (2011).
M. Bieniek, K. Burdzy and S. Pal, Extinction of Fleming–Viot-type particle systems with strong drift. Electron. J. Prob. 17 (2012).
Le Bris, C., Lelièvre, T., Luskin, M. and Perez, D., A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl. 18 (2012) 119–146. Google Scholar
Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and San Martín, J., Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37 (2009) 1926–1969. Google Scholar
Cattiaux, P. and Méléard, S., Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction. J. Math. Biol. 60 (2010) 797–829. Google ScholarPubMed
Collet, P., Martínez, S. and San Martín, J., Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23 (1995) 1300–1314. Google Scholar
E.B. Davies, Spectral theory and differential operators. Cambridge University Press 42 (1996).
P. Del Moral, Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications, Springer (2011).
Del Moral, P. and Doucet, A., Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175–1207. Google Scholar
Del Moral, P. and Miclo, L., Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups. ESAIM: PS 7 (2003) 171–208. Google Scholar
El Makrini, M., Jourdain, B. and Lelievre, T., Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189–213. Google Scholar
L.C. Evans, Partial Differential Equations. Amer. Math. Soc. 2002.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer Verlag 224 (2001).
Grigorescu, I. and Kang, M., Hydrodynamic limit for a Fleming-Viot type system. Stoch. Process. Their Appl. 110 (2004) 111–143. Google Scholar
D. Haroske and H. Triebel, Distributions, Sobolev spaces, elliptic equations. Europ. Math. Soc. (2008).
Lejay, A. and Maire, S., Computing the principal eigenvalue of the Laplace operator by a stochastic method. Math. Comput. Simul. 73 (2007) 351–363. Google Scholar
Lejay, A. and Maire, S., Computing the principal eigenelements of some linear operators using a branching Monte Carlo method. J. Comput. Phys. 227 (2008) 9794–9806. Google Scholar
Martínez, S. and San Martín, J., Quasi–stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (1994) 911–920. Google Scholar
Martínez, S. and San Martín, J.. Classification of killed one-dimensional diffusions. Ann. Probab. 32 (2004) 530–552. Google Scholar
D. Perez, Implementation of Parallel Replica Dynamics, Personal Communication (2012).
Perez, D., Uberuaga, B.P., Shim, Y., Amar, J.G. and Voter, A.F., Accelerated molecular dynamics methods: introduction and recent developments. Ann. Reports Comput. Chemistry 5 (2009) 79–98. Google Scholar
Rousset, M., On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824–844 (electronic). Google Scholar
W. Rudin, Principles of Mathematical Analysis. McGraw-Hill (1976).
Steinsaltz, D. and Evans, S.N., Quasistationary distributions for one–dimensional diffusions with killing. Trans. Amer. Math. Soc. 359 (2007) 1285–1324 (electronic). Google Scholar
Voter, A.F., Parallel replica method for dynamics of infrequent events. Phys. Rev. B 57 (1998) 13985–13988. Google Scholar
Voter, A.F., Montalenti, F. and Germann, T.C., Extending the time scale in atomistic simulation of materials. Ann. Rev. Materials Sci. 32 (2002) 321–346. Google Scholar