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.
Mots-clés : accelerated dynamics, rare events, parallel replica
@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 }, pages = {1287--1314}, publisher = {EDP-Sciences}, volume = {47}, number = {5}, year = {2013}, doi = {10.1051/m2an/2013068}, mrnumber = {3100764}, zbl = {1298.65016}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2013068/} }
TY - JOUR AU - Simpson, Gideon AU - Luskin, Mitchell TI - Numerical analysis of parallel replica dynamics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1287 EP - 1314 VL - 47 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013068/ DO - 10.1051/m2an/2013068 LA - en ID - M2AN_2013__47_5_1287_0 ER -
%0 Journal Article %A Simpson, Gideon %A Luskin, Mitchell %T Numerical analysis of parallel replica dynamics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1287-1314 %V 47 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013068/ %R 10.1051/m2an/2013068 %G en %F M2AN_2013__47_5_1287_0
Simpson, Gideon; Luskin, Mitchell. Numerical analysis of parallel replica dynamics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1287-1314. doi : 10.1051/m2an/2013068. http://archive.numdam.org/articles/10.1051/m2an/2013068/
[1] Sobolev spaces. Academic Press 140 (2003). | MR | Zbl
and ,[2] Non-extinction of a Fleming-Viot particle model. Probab. Theory Relat. Fields (2011). | MR | Zbl
, and ,[3] Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Prob. 17 (2012). | MR | Zbl
, and ,[4] A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl. 18 (2012) 119-146. | MR | Zbl
, , and ,[5] Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37 (2009) 1926-1969. | MR | Zbl
, , , , and ,[6] Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction. J. Math. Biol. 60 (2010) 797-829. | MR | Zbl
and ,[7] Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23 (1995) 1300-1314. | MR | Zbl
, and ,[8] Spectral theory and differential operators. Cambridge University Press 42 (1996). | MR | Zbl
,[9] Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications, Springer (2011). | MR | Zbl
,[10] Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175-1207. | MR | Zbl
and ,[11] Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | MR | Zbl
and ,[12] Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | MR | Zbl
, and ,[13] Partial Differential Equations. Amer. Math. Soc. 2002. | JFM
,[14] Elliptic partial differential equations of second order. Springer Verlag 224 (2001). | MR | Zbl
and ,[15] Hydrodynamic limit for a Fleming-Viot type system. Stoch. Process. Their Appl. 110 (2004) 111-143. | MR | Zbl
and ,[16] Distributions, Sobolev spaces, elliptic equations. Europ. Math. Soc. (2008). | MR | Zbl
and ,[17] Computing the principal eigenvalue of the Laplace operator by a stochastic method. Math. Comput. Simul. 73 (2007) 351-363. | MR | Zbl
and ,[18] Computing the principal eigenelements of some linear operators using a branching Monte Carlo method. J. Comput. Phys. 227 (2008) 9794-9806. | MR | Zbl
and ,[19] Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (1994) 911-920. | MR | Zbl
and ,[20] Classification of killed one-dimensional diffusions. Ann. Probab. 32 (2004) 530-552. | MR | Zbl
and .[21] Implementation of Parallel Replica Dynamics, Personal Communication (2012).
,[22] Accelerated molecular dynamics methods: introduction and recent developments. Ann. Reports Comput. Chemistry 5 (2009) 79-98.
, , , and ,[23] On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844 (electronic). | MR | Zbl
,[24] Principles of Mathematical Analysis. McGraw-Hill (1976). | MR | Zbl
,[25] Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359 (2007) 1285-1324 (electronic). | MR | Zbl
and ,[26] Parallel replica method for dynamics of infrequent events. Phys. Rev. B 57 (1998) 13985-13988.
,[27] Extending the time scale in atomistic simulation of materials. Ann. Rev. Materials Sci. 32 (2002) 321-346.
, and ,Cité par Sources :