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:在本文中,我们考虑了Gross-Pitaevskii egenvictor 问题(GPE)在计算地面状态方面的普遍反向循环。 因为,我们证明,在加权线性电子值问题的规模上,取决于最大二次值的明显线性趋同率。 此外,我们表明,这一二次值可以被线性Gross-Pitaevskii操作员的第一个光谱差距所束缚,恢复与线性电子素问题相同的速度。我们由此建立了第一个本地趋同结果,使GPE基本反向相趋同而没有阻断。我们还表明,我们的调查结果如何直接概括地扩大反向的趋同率,例如[W. Bao, Q. Du, SIAM J. Sci. comput., 25(2004) 或[P. Henning, D. Peterseim, SIAM J. Num. Anal., 53 (2020) 。我们的分析还表明,我们目前对GPE值的反比重值的反比重率的反比值反应,说明我们如何解释。</s>