用于搜索高指数鞍点的加速高指数鞍点动力学方法
Accelerated high-index saddle dynamics method for searching high-index saddle points
AI总结：
针对传统高指数鞍点动力学方法在病态问题上收敛慢的问题，本文结合重球法提出加速方法 A-HiSD，证明了其稳定性并证实了其在病态问题上的更快收敛速度。
摘要：
高指数鞍点动力学（HiSD）方法是计算 index-$k$ 鞍点并构建解景观的有效工具。然而，传统的 HiSD 方法在病态问题上常面临收敛速度慢的问题。为应对此挑战，我们通过引入重球法提出了加速高指数鞍点动力学（A-HiSD）。我们证明了连续 A-HiSD 的线性稳定性理论，随后估计了离散 A-HiSD 的局部收敛速度。分析表明，与传统 HiSD 方法相比，A-HiSD 方法展现出更快的收敛速度，特别是在处理病态问题时。我们还进行了包括神经网络损失函数在内的多种数值实验，以证实 A-HiSD 方法的有效性与加速效果。
英文摘要：
The high-index saddle dynamics (HiSD) method [J. Yin, L. Zhang, and P. Zhang, {\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595] serves as an efficient tool for computing index-$k$ saddle points and constructing solution landscapes. Nevertheless, the conventional HiSD method often encounters slow convergence rates on ill-conditioned problems. To address this challenge, we propose an accelerated high-index saddle dynamics (A-HiSD) by incorporating the heavy ball method. We prove the linear stability theory of the continuous A-HiSD, and subsequently estimate the local convergence rate for the discrete A-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a faster convergence rate compared to the conventional HiSD method, especially when dealing with ill-conditioned problems. We also perform various numerical experiments including the loss function of neural network to substantiate the effectiveness and acceleration of the A-HiSD method.
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