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<p>The high-index saddle dynamics (HiSD) method (SIAM J Sci Comput 41:A3576–A3595, 2019) serves as an efficient tool for computing index-<i>k</i> 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.</p>

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Accelerated High-Index Saddle Dynamics Method for Searching High-Index Saddle Points

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Yue Luo,
  • Lei Zhang,
  • Xiangcheng Zheng

摘要

The high-index saddle dynamics (HiSD) method (SIAM J Sci Comput 41:A3576–A3595, 2019) 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.