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Preconditioned High-index Saddle Dynamics for Computing Saddle Points

Bingzhang Huang, Hua Su, Lei Zhang, Jin Zhao

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 $κ$. 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 $κ_M$. Consequently, the iteration complexity is sharply reduced from $O(κ\log(1/ε))$ to $O(κ_M\log(1/ε))$. 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.

Preconditioned High-index Saddle Dynamics for Computing Saddle Points

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 . 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 . By generalizing orthogonal reflections and unstable-subspace tracking to the -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 . Consequently, the iteration complexity is sharply reduced from to . 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.

Paper Structure

This paper contains 17 sections, 6 theorems, 54 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preconditioned High-Index Saddle Dynamics
  4. A metric reformulation of HiSD
  5. Invariance of the target saddle structure
  6. Local Exponential Stability
  7. Discrete Convergence Analysis
  8. Preconditioner Design
  9. Spectral-type preconditioners
  10. Exploiting Algebraic Structure
  11. Practical Selection Guidelines
  12. Numerical Experiments
  13. Quadratic model: rate verification
  14. Butterfly landscape: improved geometric robustness
  15. Stiff coupled bistable chain
  16. ...and 2 more sections

Key Result

Theorem 2.1

\newlabelthm:hisd-stability-original0 Let $x^*$ be a nondegenerate index-$k$ saddle with Hessian eigenvalues $\lambda_1 \le \cdots \le \lambda_k < 0 < \lambda_{k+1} \le \cdots \le \lambda_n$, all distinct. Let $\{u_i\}_{i=1}^k$ be eigenvectors corresponding to the negative eigenvalues. Then $(x^*,

Figures (4)

  • Figure 1: Verification of convergence rate on a quadratic model. The observed rates closely match the theoretical prediction $q = (\kappa_M-1)/(\kappa_M+1)$, showing that preconditioning accelerates convergence by reducing the effective condition number.
  • Figure 2: Butterfly function ($c=1$). Starting near a local minimizer, standard HiSD (cyan) fails to reach the target saddle point (green), whereas p-HiSD with the spectral preconditioner (magenta) and the subspace-inertial preconditioner (orange) successfully converges to it.
  • Figure 3: Convergence histories of $\|\nabla E(\mathbf{x})\|$ for the coupled bistable chain. Preconditioning mitigates the severe stiffness ($K/\delta = 10^4$), allowing p-HiSD variants to operate stably at $\eta = 0.5$ and converge rapidly. By contrast, standard HiSD stagnates under its stringent stability limit $\eta = 5\times 10^{-5}$.
  • Figure 4: Allen--Cahn: convergence histories of the discrete gradient norm $\|\nabla E(u_m)\|$. Standard HiSD decays extremely slowly, whereas the two-stage p-HiSD strategy reaches the prescribed tolerance in 177 iterations, with the preconditioner switched at iteration 171.

Theorems & Definitions (11)

  • Theorem 2.1: Local stability Yin2019HiSD
  • Theorem 2.2: Discrete convergence Luo2022Convergence
  • Proposition 3.1
  • Proof 1
  • Proposition 4.1: Equilibrium characterization
  • Proof 2
  • Theorem 4.2: Local exponential stability
  • Proof 3
  • Remark 4.3
  • Theorem 5.2: Local linear convergence
  • ...and 1 more