Point cloud discretization of Fokker-Planck operators for committor functions. (English) Zbl 1431.65235
The paper develops a point cloud discretization method for computing committor functions of stochastic systems. The committor function provides useful information in understanding the transition of the stochastic system between disjoint regions in the phase space. Numerical experiments on model systems confirm that the method provides a promising tool to study stochastic systems in the framework of the transition path theory.
Reviewer: Zhiming Chen (Beijing)
MSC:
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
| 65D25 | Numerical differentiation |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 82C35 | Irreversible thermodynamics, including Onsager-Machlup theory |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 65D05 | Numerical interpolation |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Keywords:
point cloud discretization; Fokker-Planck operators; committor functions; transition path theoryReferences:
| [1] | M. Belkin, J. Sun, and Y. Wang, {\it Constructing Laplace operator from point clouds in \({R}^d\)}, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2009, pp. 1031-1040, . · Zbl 07051273 |
| [2] | P. G. Bolhuis, D. Chandler, C. Dellago, and P. L. Geissler, {\it Transition path sampling: Throwing ropes over rough mountain passes, in the dark}, Annu. Rev. Phys. Chem., 53 (2002), pp. 291-318. |
| [3] | M. Cameron and E. Vanden-Eijnden, {\it Flows in complex networks: Theory, algorithms, and application to Lennard-Jones cluster rearrangement}, J. Stat. Phys., 156 (2014), pp. 427-454. · Zbl 1301.82031 |
| [4] | R. R. Coifman, I. G. Kevrekidis, S. Lafon, M. Maggioni, and B. Nadler, {\it Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems}, Multiscale Model. Simul., 7 (2008), pp. 842-864. · Zbl 1175.60058 |
| [5] | R. R. Coifman and S. Lafon, {\it Diffusion maps}, Appl. Comput. Harmon. Anal., 21 (2006), pp. 5-30. · Zbl 1095.68094 |
| [6] | C. Dellago, P. G. Bolhuis, and P. L. Geissler, {\it Transition path sampling}, Adv. Chem. Phys., 123 (2002), pp. 1-78. |
| [7] | G. Dziuk, {\it Finite elements for the Beltrami operator on arbitrary surfaces}, in Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin, 1988, pp. 142-155. · Zbl 0663.65114 |
| [8] | W. E, W. Ren, and E. Vanden-Eijnden, {\it Finite temparture string method for the study of rare events}, J. Phys. Chem. B, 109 (2005), pp. 6688-6693. |
| [9] | W. E and E. Vanden-Eijnden, {\it Toward a theory of transition paths}, J. Stat. Phys., 123 (2006), pp. 503-523. · Zbl 1101.82016 |
| [10] | W. E and E. Vanden-Eijnden, {\it Transition path theory and path-finding algorithms for the study of rare events}, Annu. Rev. Phys. Chem., 61 (2010), pp. 391-420. |
| [11] | G. Hummer, {\it From transition paths to transition states and rate coefficients}, J. Chem. Phys., 120 (2004), pp. 516-523. |
| [12] | I. Jolliffe, {\it Principal Component Analysis}, Springer, New York, 2002. · Zbl 1011.62064 |
| [13] | S. Kirmizialtin and R. Elber, {\it Revisiting and computing reaction coordinates with directional milestoning}, J. Phys. Chem. A, 115 (2011), pp. 6137-6148. |
| [14] | R. Lai, J. Liang, and H.-K. Zhao, {\it A local mesh method for solving PDEs on point clouds}, Inverse Probl. Imaging, 7 (2013), pp. 737-755. · Zbl 1273.65034 |
| [15] | W. Lechner, J. Rogal, J. Juraszek, B. Ensing, and P. G. Bolhuis, {\it Nonlinear reaction coordinate analysis in the reweighted path ensemble}, J. Chem. Phys., 133 (2010), 174110. |
| [16] | J. Liang, R. Lai, T. Wong, and H. Zhao, {\it Geometric understanding of point clouds using Laplace-Beltrami operator}, Computer Vision and Pattern Recognition (CVPR), IEEE, Piscataway, NJ, 2012, pp. 214-221. |
| [17] | J. Liang and H. Zhao, {\it Solving partial differential equations on point clouds}, SIAM J. Sci. Comput., 35 (2013), pp. A1461-A1486. · Zbl 1275.65080 |
| [18] | A. V. Little, Y.-M. Jung, and M. Maggioni, {\it Multiscale estimation of intrinsic dimensionality of data sets}, in AAAI Fall Symposium: Manifold Learning and its Applications, Vol. 9, AAAI Press, Menlo Park, CA, 2009, pp. 26-33. |
| [19] | J. Lu and J. Nolen, {\it Reactive trajectories and the transition path process}, Probab. Theory Related Fields, 161 (2015), pp. 195-244. · Zbl 1343.60074 |
| [20] | A. Ma and A. R. Dinner, {\it Automatic method for identifying reaction coordinates in complex systems}, J. Phys. Chem. B, 109 (2005), pp. 6769-6779. |
| [21] | P. Majek and R. Elber, {\it Milestoning without a reaction coordinate}, J. Chem. Theory Comput., 6 (2010), pp. 1805-1817. |
| [22] | R. Metzler, G. Oshanin, and S. Redner, {\it First-Passage Phenomena and their Applications}, World Scientific, Singapore, 2014. · Zbl 1291.00067 |
| [23] | P. Metzner, C. Schütte, and E. Vanden-Eijnden, {\it Illustration of transition path theory on a collection of simple examples}, J. Chem. Phys., 125 (2006), 084110. |
| [24] | P. Metzner, C. Schütte, and E. Vanden-Eijnden, {\it Transition path theory for Markov jump processes}, Multiscale Model. Simul., 7 (2009), pp. 1192-1219. · Zbl 1185.60086 |
| [25] | B. Nadler, S. Lafon, R. R. Coifman, and I. G. Kevrekidis, {\it Diffusion maps, spectral clustering and reaction coordinates of dynamical systems}, Appl. Comput. Harmon. Anal., 21 (2006), pp. 113-127. · Zbl 1103.60069 |
| [26] | B. Peters, {\it Reaction coordinates and mechanicstic hypothesis tests}, Annu. Rev. Phys. Chem., 67 (2016), pp. 669-690. |
| [27] | B. Peters and B. L. Trout, {\it Obtaining reaction coordinates by likelihood maximization}, J. Chem. Phys., 125 (2006), 054108. |
| [28] | M. A. Rohrdanz, W. Zheng, M. Maggioni, and C. Clementi, {\it Determination of reaction coordinates via locally scaled diffusion map}, J. Chem. Phys., 134 (2011), 124116. |
| [29] | E. Vanden-Eijnden and M. Venturoli, {\it Revisiting the finite temperature string method for the calculation of reaction tubes and free energies}, J. Chem. Phys., 130 (2009), 194103. · Zbl 1198.82019 |
| [30] | W. Zheng, M. A. Rohrdanz, M. Maggioni, and C. Clementi, {\it Polymer reversal rate calculated via locally scaled diffusion map}, J. Chem. Phys., 134 (2011), 144108. |
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.
