This paper provides a provably quasi-optimal preconditioning strategy of the linear Schr\"odinger eigenvalue problem with periodic potentials for a possibly non-uniform spatial expansion of the domain. The quasi-optimality is achieved by having the iterative eigenvalue algorithms converge in a constant number of iterations for different domain sizes. In the analysis, we derive an analytic factorization of the spectrum and asymptotically describe it using concepts from the homogenization theory. This decomposition allows us to express the eigenpair as an easy-to-calculate cell problem solution combined with an asymptotically vanishing remainder. We then prove that the easy-to-calculate limit eigenvalue can be used in a shift-and-invert preconditioning strategy to bound the number of eigensolver iterations uniformly. Several numerical examples illustrate the effectiveness of this quasi-optimal preconditioning strategy.
翻译:本文为线性Schr\"dotinger egenvaly 提供了一种可以想象的近似最佳的先决条件策略, 其周期性潜力有可能使域内空间扩张不统一。 半最佳的实现方法是将迭代的 egenvaly 算法集中在不同域大小的恒定迭代数中。 在分析中, 我们得出了频谱的解析因子化, 并使用同质化理论的概念来描述它。 这种分解让我们能够将eigenpair 表达为容易到计算的细胞问题解决方案, 加上一个无线式消失的剩余部分。 然后我们证明, 容易到的计算限制 igenvality 可以在一个转变和反向的前提战略中使用, 以统一地将eigensoolver Iteration数捆绑绑在一起。 几个数字例子说明了这种准优劣的前提策略的有效性 。