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

TL;DR

This paper introduces a novel constrained saddle dynamics method on Riemannian manifolds for efficiently finding saddle points of energy functionals with equality constraints, demonstrated on physical problems.

Contribution

The paper develops the CHiSD method using Riemannian geometry tools, providing stability analysis and a numerical framework for solution landscape construction under constraints.

Findings

Successfully applied to Thomson problem and Bose-Einstein condensation

Proved linear stability of the method at saddle points

Demonstrated efficiency in constructing solution landscapes

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.