Mathematical and Numerical Analysis to ShrinkingDimer Saddle Dynamics with Local Lipschitz Conditions

Abstract

We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinkingdimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some space-time PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.

Keywords

Saddle dynamics / solution landscape / saddle points / local Lipschitz condition / error estimate / Richardson extrapolation

References

[1] E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, SIAM, 2003.

[2] C. Chen and Z. Xie, Search extension method for multiple solutions of a nonlinear problem, Comput. Math. Appl., 47:327-343, 2004.

[3] V. Comincioli, Analisi Numerica Metodi Modelli Applicazioni,McGraw-Hill Libri Italia, 1995.

[4] J. Doye and D. Wales, Saddle points and dynamics of Lennard-Jones clusters, solids, and supercooled liquids, J. Chem. Phys., 116:3777-3788, 2002.

[5] W. E, E. Vanden-Eijnden, Transition-path theory and path-finding algorithms for the study of rare events, Annu. Rev. Phys. Chem., 61:391-420, 2010.

[6] W. E and X. Zhou, The gentlest ascent dynamics, Nonlinearity, 24:1831-1842, 2011.

[7] P. E. Farrell, Á. Birkisson, and S. W. Funke, Deflation techniques for finding distinct solutions of nonlinear partial differential equations, SIAM J. Sci. Comput., 37:A2026-A 2045, 2015.

[8] W. Gao, J. Leng, and X. Zhou, An iterative minimization formulation for saddle point search, SIAM J. Numer. Anal., 53:1786- 1805, 2015.

[9] N. Gould, C. Ortner, and D. Packwood, A dimer-type saddle search algorithm with preconditioning and linesearch, Math. Comp., 85:2939-2966, 2016.

[10] W. Grantham, Gradient transformation trajectory following algorithms for determining stationary min-max saddle points, in: Advances in Dynamic Game Theory, Ann. Internat. Soc. Dynam. Games, 9: 639-657, 2007.

[11] Y. Han, Y. Hu, P. Zhang, and L. Zhang, Transition pathways between defect patterns in confined nematic liquid crystals, J. Comput. Phys., 396:1-11, 2019.

[12] Y. Han, Z. Xu, A. Shi, and L. Zhang, Pathways connecting two opposed bilayers with a fusion pore: a molecularly-informed phase field approach, Soft Matter., 16:366-374, 2020.

[13] Y. Han, J. Yin, Y. Hu, A. Majumdar, and L. Zhang,Solution landscapes of the simplified EricksenLeslie model and its comparison with the reduced Landau-de Gennes model, Proc. Math. Phys. Eng. Sci., 477: 2021.0458, 2021.

[14] Y. Han, J. Yin, P. Zhang, A. Majumdar, and L. Zhang, Solution landscape of a reduced Landau-de Gennes model on a hexagon, Nonlinearity, 34:2048-2069, 2021.

[15] D. Heidrich and W. Quapp, Saddle points of index 2 on potential energy surfaces and their role in theoretical reactivity investigations, Theor. Chim. Acta, 70: 89-98, 1986.

[16] G. Henkelman, H. Jónsson, A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys., 111:7010-7022, 1999.

[17] A. Levitt and C. Ortner, Convergence and cycling in Walker-type saddle search algorithms, SIAM J. Numer. Anal., 55:2204-2227, 2017.

[18] Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput., 23:840-865, 2001.

[19] D. Mehta, Finding all the stationary points of a potential-energy landscape via numerical polyno-mial-homotopy-continuation method, Phys. Rev. E, 84:025702, 2011.

[20] J. W. Milnor, Morse Theory, Princeton University Press, 1963.

[21] Q. Nie, L. Qiao, Y. Qiu, L. Zhang, and W. Zhao, Noise control and utility: from regulatory network to spatial patterning, Sci. China Math., 63:425-440, 2020.

[22] A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics. in: Texts in Applied Mathematics 37, 2007.

[23] W. Wang, L. Zhang, P. Zhang, Modelling and computation of liquid crystals, Acta Numer., 30:765-851, 2021.

[24] Z. Xu, Y. Han, J. Yin, B. Yu, Y. Nishiura, L. Zhang, Solution landscapes of the diblock copolymerhomopolymer model under two-dimensional confinement, Phys. Rev. E, 104:014505, 2021.

[25] J. Yin, K. Jiang, A.-C. Shi, P. Zhang, and L. Zhang,Transition pathways connecting crystals and quasicrystals, Proc. Natl. Acad. Sci., 118:e2106230118, 2021.

[26] J. Yin, Y. Wang, J. Chen, P. Zhang, and L. Zhang, Construction of a pathway map on a complicated energy landscape, Phys. Rev. Lett., 124:090601, 2020.

[27] J. Yin, B. Yu, and L. Zhang, Searching the solution landscape by generalized high-index saddle dynamics, Sci. China Math., 64: 1801, 2021.

[28] J. Yin, L. Zhang, and P. Zhang, High-index optimization-based shrinking dimer method for finding high-index saddle points, SIAM J. Sci. Comput., 41:A3576-A3595, 2019.

[29] J. Yin, L. Zhang, and P. Zhang, Solution landscape of the Onsager model identifies non-axisymmetric critical points, Physica D: Nonlinear Phenomena, 430:133081, 2022.

[30] B. Yu, X. Zheng, P. Zhang, and L. Zhang, Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm, J. Comput. Phys., 468:111513, 2022.

[31] J. Zhang and Q. Du, Shrinking dimer dynamics and its applications to saddle point search, SIAM J. Numer. Anal., 50:1899-1921, 2012.

[32] L. Zhang, L. Chen, and Q. Du, Morphology of critical nuclei in solid-state phase transformations, Phys. Rev. Lett., 98:265703, 2007.

[33] L. Zhang, L. Chen, and Q. Du, Simultaneous prediction of morphologies of a critical nucleus and an equilibrium precipitate in solids, Commun. Comput. Phys., 7:674-682, 2010.

[34] L. Zhang, Q. Du, and Z. Zheng, Optimization-based shrinking dimer method for finding transition states, SIAM J. Sci. Comput., 38:A528-A544, 2016.

[35] L. Zhang, W. Ren, A. Samanta, and Q. Du, Recent developments in computational modelling of nucleation in phase transformations, npj Comput. Mater., 2:16003, 2016.

[36] L. Zhang, P. Zhang, and X. Zheng, Error estimates of Euler discretization to high-index saddle dynamics, SIAM J. Numer. Anal., 60:2925-2944, 2022.