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A Minimum Action Method with Optimal Linear Time Scaling

Part of: Stochastic analysis Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  23 November 2015

Xiaoliang Wan
Xiaoliang Wan*
Affiliation:
Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge 70803, USA.
*
*Corresponding author. Email address: [email protected](X. Wan)
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Abstract

In this work, we develop a minimum action method (MAM) with optimal linear time scaling, called tMAM for short. The main idea is to relax the integration time as a functional of the transition path through optimal linear time scaling such that a direct optimization of the integration time is not required. The Feidlin-Wentzell action functional is discretized by finite elements, based on which h-type adaptivity is introduced to tMAM. The adaptive tMAM does not require reparametrization for the transition path. It can be applied to deal with quasi-potential: 1) When the minimal action path is subject to an infinite integration time due to critical points, tMAM with a uniform mesh converges algebraically at a lower rate than the optimal one. However, the adaptive tMAM can recover the optimal convergence rate. 2) When the minimal action path is subject to a finite integration time, tMAM with a uniform mesh converges at the optimal rate since the problem is not singular, and the optimal integration time can be obtained directly from the minimal action path. Numerical experiments have been implemented for both SODE and SPDE examples.

Keywords

Large deviation principlesmall random perturbationsminimum action methodrare eventsuncertainty quantification.

MSC classification

Secondary: 49M25: Discrete approximations 60H30: Applications of stochastic analysis (to PDE, etc.) 65K10: Optimization and variational techniques
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1352 - 1379
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Cerjan, C. and Miller, W., On finding transition states, J. Chem. Phys. 75(6) (1981), 2800–2806.Google Scholar
[2]Cerrai, S. and Freidlin, M., Approximation of quasi-potential and exit problems for multidimensional RDE's with noise, Trans. Amer. Math. Soc., 363(7) (2011), 3853–3892.Google Scholar
[3]Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, UK, 1992.Google Scholar
[4], W. E, Ren, W. and Vanden-Eijnden, E., String method for the study of rare events, Phys. Rev. B, 66 (2002), 052301.Google Scholar
[5], W. E, Ren, W. and Vanden-Eijnden, E., Minimum action method for the study of rare events, Commun. Pure Appl. Math., 57 (2004), 637–565.Google Scholar
[6], W. E, Ren, W. and Vanden-Eijnden, E., Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys., 126 (2007), 164103.Google Scholar
[7]W. E, and Zhou, X., The gentlest ascent dynamics, Nonlinearity, 24(6) (2011), 1831.Google Scholar
[8]W. E, , Zhou, X. and Cheng, X., Subcritical bifurcation in spatially extended systems, Nonlinearity, 25(3) (2012), 761.Google Scholar
[9]Faris, W. and Jona-Lasinio, G., Large fluctuations for a nonlinear hear equation with noise, J. Phys. A: Math. Gen. 15 (1982), 3025–3055.Google Scholar
[10]Freidlin, M., Random perturbations of reaction-diffusion equations: the quasideterministic approximation, Trans. Amer. Math. Soc., 305 (1988), 665–697.Google Scholar
[11]Freidlin, M. and Wentzell, A., Random Perturbations of Dynamical Systems, second ed., Springer-Verlag, New York, 1998.CrossRefGoogle Scholar
[12]Hairer, M. and Weber, H., Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions, preprint, 2014.Google Scholar
[13]Heymann, M. and Vanden-Eijnden, E., The geometric minimum action method: A least action principle on the space of curves, Commun. Pure Appl. Math., 61 (2008), 1052–1117.Google Scholar
[14]Henkelman, G. and Jónsson, H., A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys., 111(15) (1999), 7010–7022.Google Scholar
[15]Jònsson, H., Mills, G. and Jacobsen, K., Nudged elastic band method for finding minimum energy paths of transitions, Classical and Quantum Dynamics in Condensed Phase Simulations, Ed. Berne, B., Ciccotti, G. and Coker, D., 1998.Google Scholar
[16]van Kampen, N., Stochastic Processes in Physics and Chemistry, North-Holland, 1981.Google Scholar
[17]Karniadakis, G. and Sherwin, S., Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, 2nd Edition, 2005.Google Scholar
[18]Nocedal, J. and Wright, S., Numerical Optimization, Springer Series in Operations Research, Springer, New York, 1999.CrossRefGoogle Scholar
[19]Onsager, L. and Machlup, S., Fluctuations and irreversible processes, Phys. Rev., 91 (1953), 1505–1512.Google Scholar
[20]Zhou, X., Ren, W. and W. E, , Adaptive minimum action method for the study of rare events, J. Chem. Phys., 128(2008), 104111.Google Scholar
[21]Zhou, X. and W. E, , Study of noise-induced transitions in the Lorenz system using the minimum action method, Comm. Math. Sci., 8(2) (2010), 341–355.Google Scholar
[22]Wan, X., Zhou, X. and W.E, , Study of the noise-induced transition and the exploration of the configuration space for the Kuramoto-Sivashinsky equation using the minimum action method, Nonlin-earity, 23 (2010), 475–493.Google Scholar
[23]Wan, X., An adaptive high-order minimum action method, J. Compt. Phys., 230 (2011), 8669–8682.Google Scholar
[24]Wan, X., A minimum action method for small random perturbations of two-dimensional parallel shear flows, J. Compt. Phys., 235 (2013), 497–514.Google Scholar
[25]Wan, X. and Lin, G., Hybrid parallel computing of minimum action method, Parallel Computing, 39 (2013), 638–651.Google Scholar
[26]Wan, X., Yu, H. and W. E, , Model the nonlinear instability of wall-bouned shear flows as a rare event: A study on two-dimensional Poiseuille flows, Nonlinearity, in press.Google Scholar
[27]Zhang, J. and Du, Q., Shrinking dimer dynamics and its applications to saddle point search, SIAM Journal on Numerical Analysis, 50(4) (2012), 1899–1921.CrossRefGoogle Scholar