Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints

• Published in: Journal of scientific computing Vol. 91; no. 2; p. 62
• Main Authors: Yin, Jianyuan, Huang, Zhen, Zhang, Lei
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2022; Springer Nature B.V
• Subjects: 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; 65P99; Bose–Einstein condensation; Solution landscape; Thomson problem; 37M05; Manifold optimization; Energy landscape; Saddle point; 34K21; 49K35; 37N30
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01838-3

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.