Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods

TL;DR

This paper introduces a comprehensive framework for analyzing angles between infinite-dimensional subspaces in Hilbert spaces, connecting these angles to spectral theory, perturbation bounds, and iterative methods like Rayleigh-Ritz and alternating projectors.

Contribution

It defines new angle concepts for infinite-dimensional subspaces, relates them to classical gaps, and applies these to eigenvalue approximation and accelerated iterative methods.

Findings

Angles are characterized via spectral theory and orthogonal projectors.

Bounds on perturbations of angles and Ritz values are established.

An accelerated alternating projectors method using conjugate gradients is proposed.

Abstract

We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators is used to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. The classical gaps and angles of Dixmier and Friedrichs are characterized in terms of the angles. We introduce principal invariant subspaces and prove that they are connected by an isometry that appears in the polar decomposition of the product of corresponding orthogonal projectors. Point angles are defined by analogy with the point operator spectrum. We bound the Hausdorff distance between the sets of the squared cosines of the angles corresponding to the original subspaces and their perturbations. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method, where the former subspace serves as a trial subspace and the orthogonal projector of the latter subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the trial subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. Finally, we consider the classical alternating projectors method and propose its ultimate acceleration, using the conjugate gradient approach. The corresponding convergence rate estimate is obtained in terms of the angles. We illustrate a possible acceleration for the domain decomposition method with a small overlap for the 1D diffusion equation.