Searching the solution landscape by generalized high-index saddle dynamics
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https://doi.org/10.1007/s11425-020-1737-1
Searching the solution landscape by generalized high-index saddle dynamics Jianyuan Yin1 , Bing Yu1 & Lei Zhang2,∗ 1School 2Beijing
of Mathematical Sciences, Peking University, Beijing 100871, China; International Center for Mathematical Research, Center for Quantitative Biology, Peking University, Beijing 100871, China Email: [email protected], [email protected], [email protected] Received February 25, 2020; accepted July 8, 2020
Abstract
We introduce a generalized numerical algorithm to construct the solution landscape, which is a
pathway map consisting of all stationary points and their connections. Based on the high-index optimizationbased shrinking dimer (HiOSD) method for gradient systems, a generalized high-index saddle dynamics (GHiSD) is proposed to compute any-index saddles of dynamical systems. Linear stability of the index-k saddle point can be proved for the GHiSD system. A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape, which not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses, but also reveals the relationships between different solutions. Numerical examples, including a three-dimensional example and the phase field model, demonstrate the novel concept of the solution landscape by showing the connected pathway maps. Keywords MSC(2010)
saddle point, energy landscape, solution landscape, pathway map, dynamical system, phase field 37M05, 49K35, 37N30, 34K28, 65P99
Citation: Yin J Y, Yu B, Zhang L. Searching the solution landscape by generalized high-index saddle dynamics. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1737-1
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Introduction
The energy landscape, which is a mapping of all possible configurations of the system to their energy, exhibits a number of local minima separated by barriers. The energy landscape has been widely devoted to elucidating the structure and thermodynamics of the energy functions in a broad range of applications, such as protein folding [30, 32, 37, 42], catalysis [1, 29], Lennard-Jones clusters [3, 43], phase transitions [8, 11, 21] and artificial neural networks [9, 10, 13, 19]. The index-1 saddle point, often characterized as the transition state, is located at the point of the minimal energy barrier between two minima. The minimum energy path connects two minima and the transition state via a continuous curve on the energy landscape [14, 39]. Besides minima and transition states, the stationary points of the energy function also include high-index saddle points. To characterize the nature of a nondegenerate saddle point, the Morse index of a saddle point is the maximal dimension of a subspace on which its Hessian is negative definite [35]. An intriguing mathematical-physics problem is to efficiently search all stationary points of * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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