Construction of Solution Landscape on the Energy Landscape by Saddle Dynamics
Time: 2021-10-26
Published By: He Liu
Speaker(s): Bing Yu (PKU)
Time: 15:15-17:00 November 3, 2021
Venue: Room 29, Quan Zhai, BICMR
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. An intriguing mathematical-physics problem is to efficiently search all stationary points of a multivariable energy function, including both minima and saddle points. Most existing methods focus on the computing of the minima and transition states (index-1 saddle point), and depend on a mass of initial guesses.
In this talk, we introduce a novel and more informative concept of the solution landscape based on the energy landscape, which is a pathway map consisting of all stationary points and their connections. In particular, a high-index saddle dynamics method is proposed to compute any-index saddles of the energy landscape. A combination of the downward search algorithm and the upward search algorithm is applied to systematically construct the solution landscape. The solution landscape not only provides a powerful and efficient way to compute multiple solutions without tuning initial guesses, but also reveals the relationships between different solutions. Furthermore, we generalize the saddle dynamics method to non-gradient systems and obtain similar results, indicating the concept of solution landscape could be applied in dynamical systems. Numerical examples, including the phase field model and its modified non-gradient models, are presented to show the wide applications of the solution landscape. Joint work with Jianyuan Yin, Lei Zhang, Xiangcheng Zheng.
