Subspace Curvature-Scaling High-Index Saddle Dynamics for Accelerating Ill-Conditioned Saddle Point Searches

Jul 16, 2026

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A subspace curvature-scaling high-index saddle dynamics (SCS-HiSD) method is proposed to accelerate high-index saddle dynamics (HiSD) for locating ill-conditioned saddle points. HiSD already yields unstable Hessian eigenvector approximations, which are reused to build an inverse-Hessian approximation on the unstable subspace with negligible extra cost. Subspace curvature information adaptively rescales dynamics along each unstable direction, removing dependence on the smallest-magnitude negative eigenvalues and greatly improving convergence. Linear stability and local discrete convergence are proven, and experiments confirm strong gains on benchmark and liquid-crystal problems, especially for severe ill-conditioning.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

Introduction

Energy landscapes and saddle points

Numerical methods for locating saddle points

High-index saddle dynamics and conditioning challenges

arXiv :2607 .03030v1 [math .NA] 3 Jul 2026 SUBSPACE CURVATURE-SCALING HIGH-INDEX SADDLE DYNAMICS FOR ACCELERATING ILL-CONDITIONED SADDLE POINT SEARCHES∗ JIANYUAN YIN†, LEI ZHANG‡, AND ZHIYI ZHANG§ Abstract. We propose a subspace curvature-scaling high-index saddle dynamics (SCS-HiSD) method to accelerate high-index saddle dynamics (HiSD) for locating ill-conditioned saddle points. The key observation is that HiSD already computes approximations of the unstable Hessian eigenvectors during iteration, which can be used at negligible additional cost to construct an inverse-Hessian approximation on the unstable subspace. This subspace curvature information is incorporated to adaptively scale the dynamics along each unstable direction, eliminating the dependence of the convergence rate on the smallest-magnitude negative eigenvalues and thereby substantially accelerating the convergence for ill-conditioned saddle points. We establish the linear stability of the continuous SCS-HiSD system and provide a local convergence analysis for the discrete iterative scheme. This method extends naturally to address slow convergence caused by small positive eigenvalues. Numerical experiments on benchmark problems and a liquid-crystal model demonstrate that SCS-HiSD substantially accelerates the computation of ill-conditioned saddle points, particularly in severely ill-conditioned cases. Key words. saddle point, high-index saddle dynamics, ill-conditioned problem, linear stability, local convergence, solution landscape AMS subject classifications. 65K10, 65L20, 37C10 1. Introduction. Exploring complex energy landscapes is a fundamental problem in physics, chemistry, and materials science [35, 30 , 22 , 29] . Local minima correspond to (meta)stable states, whereas the transitions between these states are characterized by saddle points [16, 7] . Primary attention has been devoted to index-1 saddle points, also known as transition states, which lie on the minimum energy path connecting two adjacent local minima, and are closely related to the corresponding transition pathways. Developing efficient algorithms for locating saddle points is essential for a comprehensive understanding of complex energy landscapes. Existing numerical methods for locating saddle points can be classified mainly into two categories, path-finding methods and surface-walking methods. Representative path-finding methods include the nudged elastic band method [18] and the string method [6], which iteratively compute a minimum energy path connecting two local minima. Surface-walking methods directly search for saddle points by using local gradient and Hessian information, or their approximations. Typical examples include gentlest ascent dynamics [8, 15], the eigenvector-following method [4], the activationrelaxation method [3], the iterative minimization formulation [10], and dimer-type methods [17, 42 , 43] . Some surface-walking methods for saddle points have also been generalized to data-driven cases [14, 2 , 11 , 13] . Beyond transition states, high-index saddle points can provide important struc- ∗ Submitted to. Funding: This work was supported by the National Natural Science Foundation of China (No. 12225102, T2321001, 12288101) . †School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing 100875, China ([jyyin@bnu.edu.cn](jyyin@bnu.edu.cn)). ‡School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Machine Learning Research, Center for Quantitative Biology, Peking University, Beijing 100871, China ([zhangl@math.pku.edu.cn](zhangl@math.pku.edu.cn)). § School of Mathematical Sciences, Peking University, Beijing 100871, China ([zzy2323@pku.edu.cn](zzy2323@pku.edu.cn)). 2 YIN, ZHANG, AND ZHANG tural information about energy landscapes [9, 36 , 19] . Several surface-walking methods designed for index-1 saddle points have been e

FAQs

What is the main idea behind SCS-HiSD?

SCS-HiSD reuses unstable Hessian eigenvector approximations computed during HiSD iterations to construct an inverse-Hessian approximation on the unstable subspace. It then uses subspace curvature to adaptively scale the dynamics along each unstable direction.

How does the method improve convergence for ill-conditioned saddle points?

By adaptively rescales dynamics along unstable directions, SCS-HiSD eliminates dependence of the convergence rate on the smallest-magnitude negative eigenvalues. This avoids severe slowdown caused by nearly flat unstable directions.

What theoretical results are provided for SCS-HiSD?

The work establishes linear stability of the continuous SCS-HiSD system and provides local convergence analysis for the discrete iterative scheme, with additional discussion of addressing slow convergence from small positive eigenvalues.

Jul 16, 2026

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