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

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.

References in corpus (15)

• String Method for the Study of Rare Events
• Mathematical theory and numerical methods for Bose-Einstein condensation
• Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation
• The Gentlest Ascent Dynamics
• Some improvements of the ART method for finding transition pathways on potential energy surfaces
• Stable Vortex-Bright Soliton Structures in Two-Component Bose Einstein Condensates
• Construction of a pathway map on a complicated energy landscape
• A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
• Searching the solution landscape by generalized high-index saddle dynamics
• Duplication, collapse and escape of magnetic skyrmions revealed using a systematic saddle point search method
• Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation
• Bifurcation analysis of stationary solutions of two-dimensional coupled Gross-Pitaevskii equations using deflated continuation
• Kinetic Transition Networks for the Thomson Problem and Smale's 7th Problem
• Deflation-based Identification of Nonlinear Excitations of the 3D Gross--Pitaevskii equation
• Dynamic and Energetic Stabilization of Persistent Currents in Bose-Einstein Condensates

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• Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs
• Gentlest ascent dynamics on manifolds defined by adaptively sampled point-clouds