Sharp a priori convergence estimates for Krylov subspace eigensolvers

椭圆型和自伴随微分算子的特征值问题在各种科学技术应用中都有出现。通过自适应有限元离散化和离散算子期望特征对的迭代计算，获得了数值解。子空间迭代是一种快速求解高维矩阵特征值问题的方法，比经典的对角化方法要有效得多。流行的子空间迭代在Krylov子空间中工作，可以理解为近70年历史的Lanczos方法的改进变体。主要的程序变体包括重启、块执行和预处理。相关的收敛理论未能跟上新的程序变体的积极发展。此外，许多收敛估计具有后验特征，即它们通过依赖于（待）计算的里兹值的复杂公式来限定收敛速率。本课题研究了实矩阵和对称矩阵特征值问题的Krylov子空间迭代收敛分析的新方法。首先分析了四种基本迭代方法：标准Krylov子空间迭代、重启Krylov子空间迭代、块-Krylov子空间迭代和重启块-Krylov子空间迭代。由此产生的估计将导致对这些子空间迭代的收敛行为的更好理解。计划对相关的预置迭代进行扩展。一个重点是先验估计，它可以在相当弱的假设和较不复杂的边界下推导出来。概率技术在推导真实的收敛率方面具有很高的潜力，并将与瑞利商水平集的几何解释和预处理相结合。新的块数和块大小的自适应控制技术表明了效率的提高，这应该在量子力学密度泛函理论中的自洽场迭代等应用问题中得到证明。

Eigenvalue problems of elliptic and self-adjoint differential operators occur in various scientific and technical applications. Their numerical solution succeeds by means of an adaptive finite element discretization and the iterative computation of the desired eigenpairs of the discretized operators. Subspace iterations are well known to be fast solution methods for the high-dimensional matrix eigenvalue problems and are considerably more efficient than classical diagonalization methods. Popular subspace iterations work in Krylov subspaces and can be understood as improved variants of the almost 70 years old Lanczos method. The main procedural variants include restarts, blockwise implementation and preconditioning. The associated convergence theory has not been able to keep pace with the active development of new procedural variants. In addition, many convergence estimates have an a posteriori character, i.e. they bound the rates of convergence by means of complicated formula that depend on the (to be) calculated Ritz values.The proposed project deals with new approaches for the convergence analysis of Krylov subspace iterations for real and symmetric matrix eigenvalue problems. First of all, four basic iteration methods are analyzed: standard Krylov subspace iterations, restarted Krylov subspace iterations, block-Krylov subspace iterations, and restarted block-Krylov subspace iterations. The resulting estimates should lead to an improved understanding of the convergence behavior of these subspace iterations. An extension to the related preconditioned iterations is planned. One focus is on a priori estimates, which can be derived under rather weak assumptions and work with less complex bounds. Probabilistic techniques have a high potential for deriving realistic convergence rates and will be combined with geometric interpretations of Rayleigh quotient level sets and preconditioning. New adaptive control techniques for the block numbers and block sizes suggest a gain in efficiency, which should be demonstrated for application problems such as the self-consistent field iterations in the quantum mechanical density functional theory.