Rational construction of stochastic numerical methods for molecular sampling

Applied Mathematics Research eXpress

Volumn 2013, Issue 1, 2013, Pages 34-56

  Leimkuhler, Benedict ^a   Matthews, Charles ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

CONSTANT TEMPERATURE; EULER-MARUYAMA METHOD; INVARIANT DISTRIBUTION; MOLECULAR DYNAMICS SIMULATIONS; POSITION COORDINATES; RATIONAL CONSTRUCTION; SAMPLING ACCURACIES; SIMPLE MODIFICATIONS;

BOLTZMANN EQUATION; MOLECULAR DYNAMICS;

BROWNIAN MOVEMENT;

EID: 84874883342     PISSN: 16871200     EISSN: 16871197     Source Type: Journal    
DOI: 10.1093/amrx/abs010     Document Type: Article

References (27)

1. Bou-Rabee, N. and H. Owhadi. "Long-run accuracy of variational integrators in the stochastic context." SIAM Journal on Numerical Analysis 48 (2010): 278-97.
2. Brünger, A., C. Brooks III, and M. Karplus. "Stochastic boundary-conditions for molecular-dynamics simulations of ST2 water." Chemical Physics Letters 105 (1982): 495-500.
3. Burrage, K. and G. Lythe. "Accurate stationary densities with partitioned numerical methods for stochastic differential equations." SIAM Journal on Numerical Analysis 47 (2009): 1601-18.
4. Davidchack, R. L. "Discretization errors in molecular dynamics simulations with deterministic and stochastic thermostats." Journal of Computational Physics 229 (2010): 9323-46.
5. Debussche, A. and E. Faou. "Weak backward error analysis for SDEs." Preprint (2011). ArXiv: 1105.0489 http://arxiv.org/abs/1105.0489.
6. Frenkel, D. and B. Smit. Understanding Molecular Simulation, 2nd ed. New York: Academic Press, 2001.
7. Hairer, E., C. Lubich, and G. Wanner. Geometric Numerical Integration, 2nd ed. Berlin: Springer, 2006.
8. Hörmander, L. "Hypoelliptic second order differential equations." Acta Mathematica 119 (1967): 147-71.
9. Joubaud, R. and G. Stoltz. "Nonequilibrium shear viscosity computations with Langevin dynamics." SIAM Multiscale Modeling and Simulation 10 (2012): 191-216.
10. Larini, L., R. Mannella, and D. Leporini. "Langevin stabilization of molecular-dynamics simulations of polymers by means of quasisymplectic algorithms." Journal of Chemical Physics 126 (2007): 104101.
11. Leimkuhler, B. and S. Reich. Simulating Hamiltonian Dynamics. Cambridge: Cambridge University Press, 2005.
12. Lelièvre, T., M. Rousset, and G. Stoltz. Free Energy Computations. A Mathematical Perspective. Imperial College Press, 2010.
13. Liu, J. Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Berlin: Springer, 2001.
14. Mattingly, J. C., A. M. Stuart and D. J. Higham. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise." Stochastic Processes and Their Applications 101 (2002): 185-232.
15. Mattingly, J. C., A. M. Stuart, and M. Tretyakov. "Convergence of numerical time-averaging and stationary measures via Poisson equations." SIAM Journal on Numerical Analysis 48 (2010): 552-77.
16. Melchionna, S. "Design of quasi-symplectic propagators for Langevin dynamics." Journal Chemical Physics 127 (2007): 044108.
17. Milstein, G. and M. Tretyakov. "Quasi-symplectic methods for Langevin-type equations." The IMA Journal of Numerical Analysis 23 (2003): 593-626.
18. Milstein, G. and M. Tretyakov. Stochastic Numerics for Mathematical Physics. Berlin: Springer, 2004.
19. Mishra, B. and S. Schlick. "The notion of error in Langevin dynamics. I. Linear analysis." Journal of Chemical Physics 105 (1996): 299-318.
20. Pavliotis, G. and Hairer, M. "From ballistic to diffusive behavior in periodic potentials." Journal of Statistical Physics 131 (2008): 175-202.
21. Shardlow, T. "Modified equations for stochastic differential equations." BIT Numerical Mathematics 46 (2006): 111-25.
22. Sivak, D., J. Chodera, and G. Crooks. "Driven Langevin dynamics: heat, work and pseudo-work." Preprint (2011). ArXiv:1107.2967v2. http://arxiv.org/abs/1107.2967v2.
23. Skeel, R. and J. Izaguirre. "An impulse integrator for Langevin dynamics." Molecular Physics 100 (2002): 3885-91.
24. Talay, D. "Simulation and numerical analysis of stochastic differential systems: a review." In Probabilistic Methods in Applied Physics, edited by P. Kree and W. Wedig, 63-106. Lecture Notes in Physics 451. Berlin: Springer, 1995.
25. Talay, D. "Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme." Markov Processes and Related Fields 8, no. 2 (2002): 163-98.
26. Villani, C. "Hypocoercivity." Memoirs of the American Mathematical Society 202 (2009); http://www.ams.org/journals/memo/2009-202-950/ S0065-9266-09-00567-5/.
27. Zygalakis, K. "On the existence and applications of modified equations for stochastic differential equations." SIAM Journal on Scientific Computing 33 (2011): 102-30.