Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints
We propose a constrained high-index saddle dynamics (CHiSD) method to search for index- k saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index- k saddle point is proved...
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| Published in | Journal of scientific computing Vol. 91; no. 2; p. 62 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.05.2022
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| Abstract | We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-
k
saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-
k
saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose–Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method. |
|---|---|
| AbstractList | We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-
k
saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-
k
saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose–Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method. We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-k saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-k saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose–Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method. |
| ArticleNumber | 62 |
| Author | Yin, Jianyuan Zhang, Lei Huang, Zhen |
| Author_xml | – sequence: 1 givenname: Jianyuan surname: Yin fullname: Yin, Jianyuan organization: School of Mathematical Sciences, Peking University – sequence: 2 givenname: Zhen surname: Huang fullname: Huang, Zhen organization: School of Mathematical Sciences, Peking University – sequence: 3 givenname: Lei orcidid: 0000-0001-9972-2051 surname: Zhang fullname: Zhang, Lei email: zhangl@math.pku.edu.cn organization: Beijing International Center for Mathematical Research, Center for Quantitative Biology, Peking University |
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| Keywords | 65P99 Bose–Einstein condensation Solution landscape Thomson problem 37M05 Manifold optimization Energy landscape Saddle point 34K21 49K35 37N30 |
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| Snippet | We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-
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saddle points of an energy functional subject to equality... We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-k saddle points of an energy functional subject to equality constraints.... |
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| SubjectTerms | Algorithms Computational Mathematics and Numerical Analysis Constraints Eigenvalues Eigenvectors Energy Euclidean space Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Minimax technique Numerical analysis Riemann manifold Saddle points Search algorithms Theoretical |
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| Title | Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints |
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