In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. 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 and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput.25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal.58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably 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.

在本文中，我们考虑了用于计算 Gross–Pitaevskii 特征向量 (GPE) 问题基态的广义逆迭代。为此，我们证明了显式线性收敛率取决于加权线性特征值问题的最大特征值。此外，我们表明该特征值可以受线性化 Gross–Pitaevskii 算子的第一个谱间隙的限制，恢复与线性特征向量问题相同的速率。有了这个，我们为没有阻尼的 GPE 的基本逆迭代建立了第一个局部收敛结果。我们还展示了我们的发现如何直接推广到扩展的逆迭代，例如 [W. Bao 和 Q. Du，通过归一化梯度流计算玻色-爱因斯坦凝聚体的基态解，SIAM J. Sci。电脑。25 (2004) 1674–1697] 或 [P. Henning 和 D. Peterseim，Gross-Pitaevskii 特征值问题的 Sobolev 梯度流：全局收敛和计算效率，SIAM J. Numer。肛门。58 (2020) 1744–1772]。我们的分析还揭示了为什么 GPE 的逆迭代对光谱偏移没有有利的反应。这种经验观察现在可以用对收敛速度至关重要的加权函数的放大来解释。数值实验说明了我们的发现。