Preconditioned High-index Saddle Dynamics for Computing Saddle Points

TL;DR

This paper introduces a preconditioned high-index saddle dynamics method that improves convergence efficiency for computing saddle points in ill-conditioned problems by leveraging a Riemannian metric and theoretical analysis.

Contribution

It develops a preconditioned HiSD framework that enhances convergence rates and stability for saddle point computations in stiff and ill-conditioned systems.

Findings

Reduces iteration complexity from $O( ext{kappa}\log(1/))$ to $O( ext{kappa}_M\log(1/))$.

Demonstrates improved convergence and stability in stiff PDE discretizations.

Enables larger step sizes and resolves convergence failures in numerical experiments.

Abstract

High-index saddle dynamics (HiSD) is an effective approach for computing saddle points of a prescribed Morse index and constructing solution landscapes for complex nonlinear systems. However, for problems with ill-conditioned Hessians arising from fine discretizations or stiff potentials, the efficiency of standard HiSD deteriorates as its convergence rate worsens with the spectral condition number $\kappa$. To address this issue, we propose a preconditioned HiSD (p-HiSD) framework that reformulates the continuous dynamics within a Riemannian metric induced by a symmetric positive definite preconditioner $M$. By generalizing orthogonal reflections and unstable-subspace tracking to the $M$-inner product, the proposed scheme modifies the geometry of the saddle-search dynamics while remaining computationally efficient. Rigorous theoretical analysis confirms that the equilibria and their Morse indices are invariant under this metric. Furthermore, we establish the local exponential stability of the continuous dynamics and prove a discrete linear convergence rate governed by the preconditioned condition number $\kappa_M$. Consequently, the iteration complexity is sharply reduced from $O(\kappa\log(1/\epsilon))$ to $O(\kappa_M\log(1/\epsilon))$. Numerical experiments across stiff model problems and PDE discretizations demonstrate that p-HiSD resolves stiffness-induced convergence failures, permits substantially larger step sizes, and significantly reduces iteration counts.