论文标题
光谱间隙对非线性特征向量问题的反向迭代的收敛性的依赖性
The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem
论文作者
论文摘要
在本文中，我们考虑了用于计算GROSS-PITAEVSKII特征向量问题（GPE）的基础状态的广义反迭代。为此，我们证明了在加权线性特征值问题的大小中取决于最大特征值的显式线性收敛速率。此外，我们表明，该特征值可以通过线性化的Gross-Pitaevskii操作员的第一光谱差距来界定，从而恢复了与线性特征向量问题相同的速率。因此，我们为GPE的基本反向迭代而无需阻尼而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反迭代，例如[W.鲍，Q. du，Siam J.Sci。 Comput。，25（2004）]或[P. Henning，D。Peterseim，Siam J. Numer。肛门，53（2020）]。我们的分析还揭示了为什么GPE的反迭代对光谱变化的反应不佳。现在，可以通过对加权函数的爆炸来解释这种经验观察，这对收敛速度至关重要。我们的发现通过数值实验说明。
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.
