Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constrained high-index saddle dynamics for the solution landscape with equality constraints

Jianyuan Yin, Zhen Huang, Lei Zhang

TL;DR

The paper introduces Constrained High-Index Saddle Dynamics (CHiSD), a Riemannian minimax approach to locate index-$k$ saddles of an energy functional subject to equality constraints. By formulating CHiSD with constrained gradients and Hessians on a manifold, and enforcing manifold compatibility via retractions and vector transport, it provides a linear-stability theory and a practical numerical scheme. The method enables construction of a constrained solution landscape through downward and upward search protocols and is demonstrated on the Thomson problem and Bose–Einstein condensation, illustrating its efficiency in identifying multiple stationary points and their connections under constraints. The work offers a systematic framework for exploring constrained energy landscapes with potential applications to molecular systems, electronic structure, and nonlinear eigenvalue problems, while highlighting avenues for computational improvements and extensions to rotating quantum systems.

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.

Constrained high-index saddle dynamics for the solution landscape with equality constraints

TL;DR

The paper introduces Constrained High-Index Saddle Dynamics (CHiSD), a Riemannian minimax approach to locate index- saddles of an energy functional subject to equality constraints. By formulating CHiSD with constrained gradients and Hessians on a manifold, and enforcing manifold compatibility via retractions and vector transport, it provides a linear-stability theory and a practical numerical scheme. The method enables construction of a constrained solution landscape through downward and upward search protocols and is demonstrated on the Thomson problem and Bose–Einstein condensation, illustrating its efficiency in identifying multiple stationary points and their connections under constraints. The work offers a systematic framework for exploring constrained energy landscapes with potential applications to molecular systems, electronic structure, and nonlinear eigenvalue problems, while highlighting avenues for computational improvements and extensions to rotating quantum systems.

Abstract

We propose a constrained high-index saddle dynamics (CHiSD) method to search for index- 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- 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.

Paper Structure

This paper contains 14 sections, 1 theorem, 64 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Constrained high-index saddle dynamics
  3. Constrained saddle points
  4. Review of the HiSD method
  5. Formulation of CHiSD
  6. Linear stability
  7. Numerical implementation
  8. Retractions and vector transport
  9. Numerical algorithms of CHiSD
  10. Construction of the solution landscape
  11. Numerical examples
  12. Thomson problem
  13. Bose--Einstein condensation
  14. Conclusions and discussions

Key Result

theorem 1

Assume that $E(\bm x)$ is a $\mathcal{C}^3$ functional, $\bm x^\ast \in \mathcal{M}$, $\left\{\bm v_i^\ast\right\}_{i=1}^k \subset T(\bm x^*)$ satisfies $\|\bm v_i^\ast\|=1$, $\mu>0$, and $\operatorname{Hess}E(\bm x^\ast)$ is nondegenerate, whose eigenvalues are $\lambda^\ast_1<\cdots<\lambda^\ast_k

Figures (3)

  • Figure 1: Illustration of (a) an energy landscape of the function \ref{['eqn:toy']} and (b) the solution landscape starting from the maximum, $A$, down to minima $C_1$ and $C_2$. Two saddle points $B_1$ and $B_2$ are connected to $A$. The indices labeled in (b) are those according to the Morse definition.
  • Figure 2: The solution landscape of the Thomson problem with particle numbers (a) $N=5$. (b) $7$, and (c) $9$. The configurations of stationary points are presented in the solution landscape. Each red ball represents a particle on the sphere, and the red line segments are drawn to show the relative positions. The height of each configuration approximately corresponds to its relative energy. The index of each stationary point is labelled on the left side, and some important stationary points are further labelled with their configurations. Each arrow from a higher-index stationary point to a lower-index stationary point corresponds to a CHiSD pathway by the downward search.
  • Figure 3: An upward pathway sequence from the ground state to a 10-saddle of BEC. Each dashed arrow represents an upward search to an excited state, and three transient states on some upward dynamical pathways are presented. We show the probability density $|\phi|^2$ for each state, and present the energy of each stationary solution.

Theorems & Definitions (6)

  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1
  • proof
  • remark thmcounterremark
  • remark thmcounterremark