We analyze the semi-implicit scheme of high-index saddle dynamics, which provides a powerful numerical method for finding the any-index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi-implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi-implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi-implicit scheme via, e.g. technical splittings and multi-variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi-implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi-implicit scheme in constructing solution landscapes of complex systems.

翻译：我们分析了高阶鞍点动力学的半隐式格式，该格式为寻找任意阶鞍点并构建解景观提供了强大的数值方法。与显式鞍点动力学格式相比，半隐式离散化放宽了步长限制并加速了收敛，但相应的数值分析相较于显式格式遇到了新的困难。具体而言，由于半隐式处理，每时间步特征向量的正交性质不能充分利用，且特征向量的计算与正交化过程耦合，进一步增加了数值分析的复杂性。我们通过技术性分裂和多变量循环归纳等方法解决了这些问题，证明了半隐式格式的误差估计。此外，我们进一步分析了求解半隐式系统的广义最小残差求解器的收敛速度。大量数值实验验证了半隐式格式在构建复杂系统解景观中的高效性和准确性。