Iterative Proximal-Minimization for Computing Saddle Points with Fixed Index

TL;DR

This paper introduces a robust iterative proximal minimization algorithm for computing saddle points with fixed Morse index, improving stability and robustness over existing methods without sacrificing convergence speed, demonstrated on various physical models.

Contribution

The paper proposes a novel IPM algorithm that enhances stability and robustness in saddle point computation by incorporating a proximal term, based on a differential game interpretation.

Findings

IPM outperforms previous methods in robustness.

The algorithm maintains convergence rate and computational efficiency.

Successful application to physical models like Allen-Cahn and Cahn-Hilliard equations.

Abstract

Computing saddle points with a prescribed Morse index on potential energy surfaces is crucial for characterizing transition states for nosie-induced rare transition events in physics and chemistry. Many numerical algorithms for this type of saddle points are based on the eigenvector-following idea and can be cast as an iterative minimization formulation (SINUM. Vol. 53, p.1786, 2015), but they may struggle with convergence issues and require good initial guesses. To address this challenge, we discuss the differential game interpretation of this iterative minimization formulation and investigate the relationship between this game's Nash equilibrium and saddle points on the potential energy surface. Our main contribution is that adding a proximal term, which grows faster than quadratic, to the game's cost function can enhance the stability and robustness. This approach produces a robust Iterative Proximal Minimization (IPM) algorithm for saddle point computing. We show that the IPM algorithm surpasses the preceding methods in robustness without compromising the convergence rate or increasing computational expense. The algorithm's efficacy and robustness are showcased through a two-dimensional test problem, and the Allen-Cahn, Cahn-Hilliard equation, underscoring its numerical robustness.