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.

翻译：在势能面上计算具有指定莫尔斯指数的鞍点，对于刻画物理和化学中噪声诱导的稀有跃迁事件的过渡态至关重要。针对此类鞍点的许多数值算法基于特征向量跟踪思想，并可表述为迭代最小化形式（SINUM. Vol. 53, p.1786, 2015），但这些方法可能面临收敛性问题且需要良好的初始猜测。为应对这一挑战，我们探讨了该迭代最小化形式的微分博弈解释，并研究了该博弈的纳什均衡与势能面上鞍点之间的关系。我们的主要贡献在于：通过在博弈成本函数中添加一个增长速度快于二次项的近端项，能够增强算法的稳定性和鲁棒性。该方法产生了一种用于鞍点计算的鲁棒迭代近端最小化（IPM）算法。我们证明，IPM算法在保持收敛速度不降低且不增加计算成本的前提下，其鲁棒性超越了现有方法。通过二维测试问题、Allen-Cahn方程和Cahn-Hilliard方程的数值实验，展示了该算法的有效性和鲁棒性，突显了其数值稳健性。