Improved high-index saddle dynamics for finding saddle points and solution landscape

TL;DR

The paper tackles the challenge of locating saddle points and assembling a complete solution landscape for complex energy functions. It introduces iHiSD, a crossover dynamics that combines gradient flow with high-index saddle dynamics and switches to the HiSD near saddle points, governed by time-dependent weights $\beta_1(t)$, $\beta_2(t)$ and $\alpha(t)$. Through a rigorous analysis of the reflection manifold $\mathcal{R}_k$, stability results, and a transitivity-based nonlocal convergence theorem, the authors establish that iHiSD can reach target saddles from outside their regions of attraction and connect arbitrary saddles via gradient-flow paths when available. A discretized scheme is developed by representing $R$ through low-rank eigen-subspaces, and the method is proven to converge similarly to the continuous system. Using Morse theory, the authors prove that any two stationary points can be connected by a sequence of iHiSD trajectories, enabling construction of a complete solution landscape, which is demonstrated with 2D and Morse-potential clustering examples showing improved exploration over traditional HiSD methods.

Abstract

We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD, and then prove its stable and nonlocal convergence from outside of the region of attraction to the saddle point, which resolves the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD that inherits these convergence properties. Furthermore, based on the Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Theoretically, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.

Figures (5)

• Figure 6.1: Comparison of different dynamics for searching saddle points starting from a local minimum for $c=1$. The GAD method could not cross the boundary of the region of attraction, while the gradient ascent method diverges rapidly due to numerical instability. In contrast, the iHiSD method could traverse this boundary to locate the saddle point under different initial values of $\alpha$.
• Figure 6.2: Construction of solution landscape with iHiSD starting from a minimum ($c=1.5$) (left) and a maximum ($c=2$) (right).
• Figure 6.3: Stationary patterns of Morse potential for different parameter $a$. The supercritical pitchfork bifurcation occurs at $a\approx 1.74$.
• Figure 6.4: Solution landscape for the particle clustering with $a=1.5$.
• Figure 6.5: Solution landscape for the particle clustering with $a=6$.

Theorems & Definitions (17)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3: transitive lemma
• proof
• Theorem 3.4: nonlocal convergence
• Remark 1
• proof
• Remark 2