Understanding high-index saddle dynamics via numerical analysis
Lei Zhang, Pingwen Zhang, Xiangcheng Zheng
TL;DR
This work analyzes discrete high-index saddle dynamics (HiSD) for finding index-$k$ saddle points and mapping solution landscapes. It shows that a common discretization that drops Lagrangian multiplier terms and uses Gram-Schmidt retraction recovers the same HiSD dynamics as the continuous system up to an $O(\tau)$ perturbation, revealing that Gram-Schmidt essentially encodes the multiplier mechanism. The authors extend the analysis to constrained HiSD on the unit sphere, demonstrating that manifold-preserving operations such as normalization, vector transport, and Gram-Schmidt underpin the preservation of key constraints. They provide rigorous error estimates and validate the theory with numerical experiments, confirming first-order convergence and practical equivalence with alternative schemes, while highlighting computational efficiency benefits. Overall, the results clarify the mechanism by which discrete HiSD preserves manifold properties and offer guidance for efficient implementations in high-dimensional settings.
Abstract
High-index saddle dynamics (HiSD) serves as a competitive instrument in searching the any-index saddle points and constructing the solution landscape of complex systems. The Lagrangian multiplier terms in HiSD ensure the Stiefel manifold constraint, which, however, are dropped in the commonly-used discrete HiSD scheme and are replaced by an additional Gram-Schmidt orthonormalization. Though this scheme has been successfully applied in various fields, it is still unclear why the above modification does not affect its effectiveness. We recover the same form as HiSD from this scheme, which not only leads to error estimates naturally, but indicates that the mechanism of Stiefel manifold preservation by Lagrangian multiplier terms in HiSD is nearly a Gram-Schmidt process (such that the above modification is appropriate). The developed methods are further extended to analyze the more complicated constrained HiSD on high-dimensional sphere, which reveals more mechanisms of the constrained HiSD in preserving several manifold properties.