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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 efficienttool for computing index-$k$ saddle points and constructing solutionlandscapes. Nevertheless, the conventional HiSD method often encounters slowconvergence rates on ill-conditioned problems. To address this challenge, wepropose an accelerated high-index saddle dynamics (A-HiSD) by incorporating theheavy ball method. We prove the linear stability theory of the continuousA-HiSD, and subsequently estimate the local convergence rate for the discreteA-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a fasterconvergence rate compared to the conventional HiSD method, especially whendealing with ill-conditioned problems. We also perform various numericalexperiments including the loss function of neural network to substantiate theeffectiveness and acceleration of the A-HiSD method.

Accelerated high-index saddle dynamics method for searching high-index saddle points