Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.

翻译：预条件特征值求解器为求解大规模特征值问题提供了融入预条件子的可能性，这类问题常见于偏微分方程离散化。此类方法的收敛性分析较为复杂。即便对于相对简单的预条件逆迭代（PINVIT）——该方法以对称正定矩阵的最小特征值为目标——Neymeyr的经典分析也极为复杂，且仅当初始向量相当接近目标特征向量时才能保证收敛。本工作中，我们针对PINVIT的一种变体证明了新的非渐近收敛结果。我们的证明通过分析等价的黎曼最速下降法并利用类凸性性质来实现。我们展示的收敛速率几乎与PINVIT的收敛速率相匹配。其主要优势在于，我们对初始向量所需的条件往往更为宽松。这种改进的全局收敛特性通过理论界和一系列数值实验，在两类预条件子中得到了验证。