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.
@article{M2AN_2013__47_5_1287_0,
author = {Simpson, Gideon and Luskin, Mitchell},
title = {Numerical analysis of parallel replica dynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1287-1314},
doi = {10.1051/m2an/2013068},
mrnumber = {3100764},
zbl = {1298.65016},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_5_1287_0}
}
Simpson, Gideon; Luskin, Mitchell. Numerical analysis of parallel replica dynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1287-1314. doi : 10.1051/m2an/2013068. http://gdmltest.u-ga.fr/item/M2AN_2013__47_5_1287_0/
[1] and , Sobolev spaces. Academic Press 140 (2003). | MR 2424078 | Zbl 1098.46001
[2] , and , Non-extinction of a Fleming-Viot particle model. Probab. Theory Relat. Fields (2011). | MR 2925576 | Zbl 1253.60089
[3] , and , Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Prob. 17 (2012). | MR 2878790 | Zbl 1258.60031
[4] , , and , A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl. 18 (2012) 119-146. | MR 2926765 | Zbl 1243.82045
[5] , , , , and , Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37 (2009) 1926-1969. | MR 2561437 | Zbl 1176.92041
[6] and , Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction. J. Math. Biol. 60 (2010) 797-829. | MR 2606515 | Zbl 1202.92082
[7] , and , Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23 (1995) 1300-1314. | MR 1349173 | Zbl 0867.60046
[8] , Spectral theory and differential operators. Cambridge University Press 42 (1996). | MR 1349825 | Zbl 0893.47004
[9] , Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications, Springer (2011). | MR 2044973 | Zbl 1130.60003
[10] and , Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175-1207. | MR 2089064 | Zbl 1071.60100
[11] and , Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | MR 1956078 | Zbl 1040.81009
[12] , and , Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | MR 2339625 | Zbl 1135.81379
[13] , Partial Differential Equations. Amer. Math. Soc. 2002. | JFM 42.0398.04
[14] and , Elliptic partial differential equations of second order. Springer Verlag 224 (2001). | MR 1814364 | Zbl 0562.35001
[15] and , Hydrodynamic limit for a Fleming-Viot type system. Stoch. Process. Their Appl. 110 (2004) 111-143. | MR 2052139 | Zbl 1075.60124
[16] and , Distributions, Sobolev spaces, elliptic equations. Europ. Math. Soc. (2008). | MR 2375667 | Zbl 1133.46001
[17] and , Computing the principal eigenvalue of the Laplace operator by a stochastic method. Math. Comput. Simul. 73 (2007) 351-363. | MR 2289255 | Zbl 1110.65105
[18] and , Computing the principal eigenelements of some linear operators using a branching Monte Carlo method. J. Comput. Phys. 227 (2008) 9794-9806. | MR 2469034 | Zbl 1157.65303
[19] and , Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (1994) 911-920. | MR 1303922 | Zbl 0818.60071
[20] and . Classification of killed one-dimensional diffusions. Ann. Probab. 32 (2004) 530-552. | MR 2040791 | Zbl 1045.60083
[21] , Implementation of Parallel Replica Dynamics, Personal Communication (2012).
[22] , , , and , Accelerated molecular dynamics methods: introduction and recent developments. Ann. Reports Comput. Chemistry 5 (2009) 79-98.
[23] , On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844 (electronic). | MR 2262944 | Zbl 1174.60045
[24] , Principles of Mathematical Analysis. McGraw-Hill (1976). | MR 385023 | Zbl 0346.26002
[25] and , Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359 (2007) 1285-1324 (electronic). | MR 2262851 | Zbl 1107.60048
[26] , Parallel replica method for dynamics of infrequent events. Phys. Rev. B 57 (1998) 13985-13988.
[27] , and , Extending the time scale in atomistic simulation of materials. Ann. Rev. Materials Sci. 32 (2002) 321-346.