Searching the solution landscape by generalized high-index saddle dynamics

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 optimization-based shrinking dimer (HiOSD) method for gradient systems, a generalized high-index saddle dynami...

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Bibliographic Details
Published in:Science China. Mathematics 2021-08, Vol.64 (8), p.1801-1816
Main Authors: Yin, Jianyuan, Yu, Bing, Zhang, Lei
Format: Article
Language:English
Subjects:
Algorithms
Applications of Mathematics
Dimers
Dynamic stability
Mathematics
Mathematics and Statistics
Numerical analysis
Optimization
Saddle points
Saddles
Search algorithms
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Summary: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 optimization-based 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.
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-020-1737-1