论文标题
索引的强大计算 - $ K $鞍点：迭代近端最小化和差异游戏模型
Towards Robust Calculation of index-$k$ Saddle Point: Iterative Proximal-Minimization and Differential Game Model
论文作者
论文摘要
在势能表面上带有给定摩尔斯指数的鞍点是与物理和化学中能量景观有关的重要对象。基于迭代最小化公式的有效数值方法已以最小化子问题或连续动力学的序列的形式提出。我们在这里提出了对该公式的差异性游戏解释，并理论上研究了拟议游戏的NASH平衡和势能表面上的原始鞍点。为了定义这种差异游戏，将新的近端功能提高到游戏中的成本函数，并派生出强大的迭代近端最小化算法（IPM）来计算鞍点。我们证明，游戏的NASH平衡正是关注的鞍点，并表明新算法比没有邻近的先前迭代最小化算法更强大，而相同的收敛速度和计算成本仍然保持。测试了二维问题和Cahn-Hillard问题，以证明这一数值优势。
Saddle point with a given Morse index on a potential energy surface is an important object related to energy landscape in physics and chemistry. Efficient numerical methods based on iterative minimization formulation have been proposed in the forms of the sequence of minimization subproblems or the continuous dynamics. We here present a differential game interpretation of this formulation and theoretically investigate the Nash equilibrium of the proposed game and the original saddle point on potential energy surface. To define this differential game, a new proximal function growing faster than quadratic is introduced to the cost function in the game and a robust Iterative Proximal-Minimization algorithm (IPM) is then derived to compute the saddle points. We prove that the Nash equilibrium of the game is exactly the saddle point in concern and show that the new algorithm is more robust than the previous iterative minimization algorithm without proximity, while the same convergence rate and the computational cost still hold. A two dimensional problem and the Cahn-Hillard problem are tested to demonstrate this numerical advantage.
