Sharp a priori convergence estimates for Krylov subspace eigensolvers

Krylov 子空间特征求解器的尖锐先验收敛估计

• 批准号: 463329614
• 负责人: Professor Dr. Klaus Neymeyr
• 金额: --
• 依托单位: Lehrstuhl für Numerische Mathematik und Approximationstheorie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/463329614?language=en
• 关键词: Sharp; priori; convergence; estimates; Krylov

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.

椭圆和自伴微分算子的特征值问题出现在各种科学和技术应用中。他们的数值解通过自适应有限元离散化和离散算子所需特征对的迭代计算而成功。众所周知，子空间迭代是高维矩阵特征值问题的快速求解方法，并且比经典的对角化方法更有效。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新启动、分块实施和预处理。相关的收敛理论未能跟上新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 Ritz 值的复杂公式来限制收敛速率。所提出的项目涉及用于实数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。由此产生的估计应该有助于更好地理解这些子空间迭代的收敛行为。计划对相关的预处理迭代进行扩展。其中一个重点是先验估计，它可以在相当弱的假设下得出，并且可以在不太复杂的范围内工作。概率技术在推导实际收敛率方面具有很高的潜力，并将与瑞利商水平集和预处理的几何解释相结合。针对块数量和块大小的新自适应控制技术表明效率的提高，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。