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：本文研究用于计算Gross-Pitaevskii特征向量问题（GPE）基态的广义反迭代方法。我们证明了该方法的显式线性收敛速率，该速率取决于加权线性特征值问题中最大特征值的模。进一步地，我们证明该特征值可被线性化Gross-Pitaevskii算子的第一谱隙所界定，从而恢复与线性特征向量问题相同的收敛速率。由此，我们建立了无阻尼GPE基本反迭代的首个局部收敛性结果。同时，我们展示了研究结果如何直接推广至扩展反迭代方法，包括[W. Bao, Q. Du, SIAM J. Sci. Comput., 25 (2004)]提出的梯度流离散归一化（GFDN）方法以及[P. Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]建议的阻尼反迭代方法。分析还揭示了GPE反迭代为何对谱平移不敏感——这一经验观察现可通过权重函数的爆破现象得到解释，该函数对收敛速率起关键作用。数值实验验证了我们的发现。