Numerical analysis of parallel replica dynamics

Simpson, Gideon; Luskin, Mitchell

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1287-1314.

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.

Classification : 60H35, 65C20, 65C30, 70K70, 74S60
Mots clés : accelerated dynamics, rare events, parallel replica

[1] R.A. Adams and J.J.F. Fournier, Sobolev spaces. Academic Press 140 (2003). | MR | Zbl

[2] M. Bieniek, K. Burdzy and S. Finch, Non-extinction of a Fleming-Viot particle model. Probab. Theory Relat. Fields (2011). | MR | Zbl

[3] M. Bieniek, K. Burdzy and S. Pal, Extinction of Fleming-Viot-type particle systems with strong drift. Electron. J. Prob. 17 (2012). | MR | Zbl

[4] C. Le Bris, T. Lelièvre, M. Luskin and D. Perez, A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl. 18 (2012) 119-146. | MR | Zbl

[5] P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard and J. San Martín, Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37 (2009) 1926-1969. | MR | Zbl

[6] P. Cattiaux and S. Méléard, Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned on non-extinction. J. Math. Biol. 60 (2010) 797-829. | MR | Zbl

[7] P. Collet, S. Martínez and J. San Martín, Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23 (1995) 1300-1314. | MR | Zbl

[8] E.B. Davies, Spectral theory and differential operators. Cambridge University Press 42 (1996). | MR | Zbl

[9] P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications, Springer (2011). | MR | Zbl

[10] P. Del Moral and A. Doucet, Particle motions in absorbing medium with hard and soft obstacles. Stoch. Anal. Appl. 22 (2004) 1175-1207. | MR | Zbl

[11] P. Del Moral and L. Miclo, Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: PS 7 (2003) 171-208. | Numdam | MR | Zbl

[12] M. El Makrini, B. Jourdain and T. Lelievre, Diffusion Monte Carlo method: Numerical analysis in a simple case. ESAIM: M2AN 41 (2007) 189-213. | Numdam | MR | Zbl

[13] L.C. Evans, Partial Differential Equations. Amer. Math. Soc. 2002. | JFM

[14] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer Verlag 224 (2001). | MR | Zbl

[15] I. Grigorescu and M. Kang, Hydrodynamic limit for a Fleming-Viot type system. Stoch. Process. Their Appl. 110 (2004) 111-143. | MR | Zbl

[16] D. Haroske and H. Triebel, Distributions, Sobolev spaces, elliptic equations. Europ. Math. Soc. (2008). | MR | Zbl

[17] A. Lejay and S. Maire, Computing the principal eigenvalue of the Laplace operator by a stochastic method. Math. Comput. Simul. 73 (2007) 351-363. | MR | Zbl

[18] A. Lejay and S. Maire, Computing the principal eigenelements of some linear operators using a branching Monte Carlo method. J. Comput. Phys. 227 (2008) 9794-9806. | MR | Zbl

[19] S. Martínez and J. San Martín, Quasi-stationary distributions for a Brownian motion with drift and associated limit laws. J. Appl. Probab. 31 (1994) 911-920. | MR | Zbl

[20] S. Martínez and J. San Martín. Classification of killed one-dimensional diffusions. Ann. Probab. 32 (2004) 530-552. | MR | Zbl

[21] D. Perez, Implementation of Parallel Replica Dynamics, Personal Communication (2012).

[22] D. Perez, B.P. Uberuaga, Y. Shim, J.G. Amar and A.F. Voter, Accelerated molecular dynamics methods: introduction and recent developments. Ann. Reports Comput. Chemistry 5 (2009) 79-98.

[23] M. Rousset, On the control of an interacting particle estimation of Schrödinger ground states. SIAM J. Math. Anal. 38 (2006) 824-844 (electronic). | MR | Zbl

[24] W. Rudin, Principles of Mathematical Analysis. McGraw-Hill (1976). | MR | Zbl

[25] D. Steinsaltz and S.N. Evans, Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359 (2007) 1285-1324 (electronic). | MR | Zbl

[26] A.F. Voter, Parallel replica method for dynamics of infrequent events. Phys. Rev. B 57 (1998) 13985-13988.

[27] A.F. Voter, F. Montalenti and T.C. Germann, Extending the time scale in atomistic simulation of materials. Ann. Rev. Materials Sci. 32 (2002) 321-346.