一种在能量景观上定位指定指标临界点的二阶方法

中文摘要 AI 辅助

在高维能量景观上进行探索是许多应用中的一个具有挑战性的问题，在这些应用中，基态和过渡态都提供了关于底层物理系统的重要信息。为了促进计算探索，我们提出了一种三次正则化的二阶方法，用于定位指定指标的临界点，包括能量极小值（指标为$0$）和过渡态（正指标的鞍点）。在迭代的每一步中，所提出的方法涉及几个组成部分，例如根据指定指标更新Hessian矩阵的特征方向，沿选定的不稳定特征方向（如果有）反射梯度和Hessian矩阵，以及通过求解三次正则化子问题来构造下一步。在适当的假设下，我们证明，当特征空间和三次子问题被精确求解时，迭代局部二次收敛到指定指标的临界点。我们进一步建立了不精确设置下的收敛结果，其中特征空间计算和三次子问题的求解都近似执行。我们表明，所得迭代保持局部线性收敛，收敛速率取决于不精确计算的精度。此外，我们证明了算法的固定点恰好是指定指标的临界点。另外，我们引入了一种自适应策略来更新三次正则化参数，以提高所提出算法的鲁棒性和计算效率。数值实验证实了预测的收敛行为，展示了自适应策略的有效性，并说明了所提出方法在定位临界点和构建解景观方面的能力。

英文摘要

Exploring high dimensional energy landscape is a challenging problem in many applications where both ground states and transition states offer important information about the underlying physical systems. To facilitate the computational exploration, we propose a cubic-regularized second-order method for locating critical points of prescribed index, including both energy minima (with an index $0$) and transition states (saddle points with positive index). At each step of the iteration, the proposed method involves several ingredients, such as updating the eigen-directions of the Hessian according to the prescribed index, reflecting the gradient and the Hessian along selected unstable eigen-directions, if any, and constructing the next step by solving a cubic-regularized subproblem. Under suitable assumptions, we show that, when the eigenspace and the cubic subproblem are solved exactly, the iteration converges locally and quadratically to a critical point of the prescribed index. We further establish convergence results in the inexact setting, where both the eigenspace computation and the solution of the cubic subproblem are performed approximately. We show that the resulting iteration retains local linear convergence, with the convergence rate depending on the accuracy of the inexact computations. Moreover, we prove that the fixed points of the algorithm are precisely critical points of the prescribed index. In addition, we introduce an adaptive strategy for updating the cubic regularization parameter to improve robustness and computational efficiency of the proposed algorithm. Numerical experiments confirm the predicted convergence behavior, demonstrate the effectiveness of the adaptive strategy, and illustrate the capability of the proposed method for locating critical points and constructing solution landscapes.

发表机构

• Columbia University（哥伦比亚大学）

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