Subspace curvature-scaling high-index saddle dynamics for accelerating ill-conditioned saddle point searches
For researchers in computational chemistry and physics needing efficient saddle point searches, this method addresses the bottleneck of ill-conditioned systems, offering a practical acceleration.
The paper proposes SCS-HiSD, a method that uses curvature information from unstable Hessian eigenvectors to accelerate saddle point searches, eliminating convergence dependence on small negative eigenvalues. Numerical experiments show substantial acceleration for ill-conditioned saddle points, especially in severe cases.
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.