×

Synchronous parallel kinetic Monte Carlo for continuum diffusion-reaction systems. (English) Zbl 1139.65003

Summary: A novel parallel kinetic Monte Carlo (kMC) algorithm formulated on the basis of perfect time synchronicity is presented. The algorithm is intended as a generalization of the standard \(n\)-fold kMC method, and is trivially implemented in parallel architectures. In its present form, the algorithm is not rigorous in the sense that boundary conflicts are ignored. We demonstrate, however, that, in their absence, or if they were correctly accounted for, our algorithm solves the same master equation as the serial method.
We test the validity and parallel performance of the method by solving several pure diffusion problems (i.e. with no particle interactions) with known analytical solution. We also study diffusion-reaction systems with known asymptotic behavior and find that, for large systems with interaction radii smaller than the typical diffusion length, boundary conflicts are negligible and do not affect the global kinetic evolution, which is seen to agree with the expected analytical behavior.
Our method is a controlled approximation in the sense that the error incurred by ignoring boundary conflicts can be quantified intrinsically, during the course of a simulation, and decreased arbitrarily (controlled) by modifying a few problem-dependent simulation parameters.

MSC:

65C05 Monte Carlo methods
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Kalos, M. H.; Whitlock, P. A., Monte Carlo Methods (1986), John Wiley & Sons: John Wiley & Sons New York · Zbl 0655.65004
[2] Landau, D. P.; Binder, K., Monte Carlo Simulations in Statistical Physics (2000), Cambridge University Press · Zbl 0998.82504
[3] Friedberg, R.; Cameron, J. E., J. Chem. Phys., 52, 6049 (1970)
[4] Swendsen, R. H.; Wang, J. S., Phys. Rev. Lett., 57, 2607 (1986)
[5] Lubachevsky, B. D., Complex Sys., 1, 1099 (1987)
[6] Lubachevsky, B. D., J. Comput. Phys., 75, 103 (1988)
[7] Korniss, G.; Toroczkai, Z.; Novotny, M. A.; Rikvold, P. A., Phys. Rev. Lett., 84, 1351 (2000)
[8] Korniss, G.; Novotny, M. A.; Guclu, H.; Toroczkai, Z.; Rikvold, P. A., Science, 299, 677 (2003)
[9] Shim, Y.; Amar, J. G., J. Comput. Phys., 212, 305 (2006)
[10] Jefferson, D. R., Virtual time, ACM Trans. Prog. Lang. Syst., 7, 404 (1985)
[11] Eick, S. G.; Greenberg, A. G.; Lubachevsky, B. D.; Weiss, A., ACM Trans. Model. Comp. Simul., 3, 287 (1993)
[12] B.D. Lubachevsky, A. Weiss, in: Proceedings of the 15th Workshop on Parallel and Distributed Simulations, Lake Arrowhead, CA, 2001, p. 185.; B.D. Lubachevsky, A. Weiss, in: Proceedings of the 15th Workshop on Parallel and Distributed Simulations, Lake Arrowhead, CA, 2001, p. 185.
[13] Shim, Y.; Amar, J. G., Phys. Rev. B, 71, 115436 (2005)
[14] Merrick, M.; Fichthorn, K. A., Phys. Rev. E, 75, 011606 (2007)
[15] Shim, Y.; Amar, J. G., Phys. Rev. B, 71, 125432 (2005)
[16] Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L., J. Comput. Phys., 17, 10 (1975)
[17] Hanusse, P.; Blanché, A., J. Chem. Phys., 74, 6148 (1981)
[18] ben-Avraham, D., J. Chem. Phys., 88, 941 (1987)
[19] Beichl, I.; Sullivan, F., IEEE Comput. Sci. Eng., 4, 91 (1997)
[20] Amar, J. G., Comput. Sci. Eng., 8, 9 (2006)
[21] Berger, M. J.; Bokhari, S. H., IEEE Trans. Comput., 36, 570 (1987)
[22] i.e. The formal solution to Eq. \((10) \operatorname{\Phi;}_m - \operatorname{\nabla;}^2 [ - \operatorname{\nabla;}^2 + \frac{ \operatorname{\partial;}}{ \operatorname{\partial;} t} ] G ( x , x_0 \text{;} t ) = \sum;_m \operatorname{e}^{- \operatorname{\Lambda;}_m^2 t} \operatorname{\Phi;}_m ( \mathbf{x} ) \operatorname{\Phi;}_m ( \mathbf{x}_0 )\); i.e. The formal solution to Eq. \((10) \operatorname{\Phi;}_m - \operatorname{\nabla;}^2 [ - \operatorname{\nabla;}^2 + \frac{ \operatorname{\partial;}}{ \operatorname{\partial;} t} ] G ( x , x_0 \text{;} t ) = \sum;_m \operatorname{e}^{- \operatorname{\Lambda;}_m^2 t} \operatorname{\Phi;}_m ( \mathbf{x} ) \operatorname{\Phi;}_m ( \mathbf{x}_0 )\)
[23] Kang, K.; Redner, S., Phys. Rev. A, 32, 435 (1985)
[24] Leyvraz, F.; Redner, S., Phys. Rev. Lett., 66, 216 (1991)
[25] Oppelstrup, T.; Bulatov, V. V.; Gilmer, G. H.; Kalos, M. H.; Sadigh, B., Phys. Rev. Lett., 97, 230602 (2006)
[26] Caturla, M. J.; Soneda, N.; Alonso, E.; Wirth, B. D.; Díaz de la Rubia, T.; Perlado, J. M., J. Nucl. Mater., 276, 13 (2001)
[27] The LLNL “Zeus” cluster (http://www.llnl.gov/computing/tutorials/lc_resources/#IntelSystems; The LLNL “Zeus” cluster (http://www.llnl.gov/computing/tutorials/lc_resources/#IntelSystems
[28] D.J. Kerbyson, H.J. Alme, A. Hoisie, F. Petrini, H.J. Wasserman, M. Gittings, in: Proceedings of the ACM/IEEE SC2001 Conference, 2001.; D.J. Kerbyson, H.J. Alme, A. Hoisie, F. Petrini, H.J. Wasserman, M. Gittings, in: Proceedings of the ACM/IEEE SC2001 Conference, 2001.
[29] Goedecker, S.; Hoisie, A., Performance Optimization of Numerically Intensive Codes (2001), SIAM: SIAM Philadelphia, PA · Zbl 0985.68006
[30] G. Amdahl, in: AFIPS Conference Proceedings, vol. 30, 1967, pp. 483-485.; G. Amdahl, in: AFIPS Conference Proceedings, vol. 30, 1967, pp. 483-485.
[31] Wong, C. K.; Easton, M. C., SIAM J. Comput., 9, 111 (1980)
[32] Knuth, D. E., The Art of Computer Programming: Sorting and Searching, vol. 3 (1998), Addison-Wesley Professional · Zbl 0918.68079
[33] Beichl, I.; Sullivan, F., IEEE Comput. Sci. Eng., 3, 13 (1996)
[34] Culler, D. E.; Karp, R.; Patterson, D. A.; Sahay, A.; Schauser, K. E.; Santos, E.; Subramonian, R.; von Eicken, T., Commun. ACM, 39, 78 (1996)
[35] Cameron, K. W.; Ge, R.; Sun, X. H., IEEE Trans. Comput., 56, 314 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.