In this paper we consider the generalized inverse iteration for computing ground states of the Gross-Pitaevskii eigenvector problem (GPE). For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross-Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis also reveals why the inverse iteration for the GPE does not react favourably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

翻译：本文考虑用于计算Gross-Pitaevskii特征值问题（GPE）基态的广义逆迭代方法。我们证明了显式的线性收敛速率，该速率依赖于加权线性特征值问题的最大模特征值。进一步表明，该特征值可通过线性化Gross-Pitaevskii算子的第一谱间隙进行界定，从而恢复与线性特征值问题相同的收敛速率。由此，我们建立了无阻尼GPE基本逆迭代的首个局部收敛性结果。同时展示了我们的发现如何直接推广至扩展逆迭代，例如[W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)]提出的梯度流离散归一化（GFDN）方法，或[P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]建议的阻尼逆迭代。我们的分析还揭示了为何GPE的逆迭代对谱平移不敏感——这一经验性现象现可归因于加权函数的爆破效应，该效应对收敛速率起关键作用。数值实验验证了我们的理论发现。