Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces

TL;DR

This paper analyzes the linear convergence rates of the averaged alternating modified reflections (AAMR) method for two subspaces, optimizing parameters for faster convergence and validating results with numerical experiments.

Contribution

It derives explicit convergence rates for AAMR in the two subspace case and optimizes parameters to improve convergence speed over existing methods.

Findings

Convergence rate depends on Friedrichs angle and parameters.

Optimized parameters yield faster convergence.

Numerical experiments confirm theoretical improvements.

Abstract

The averaged alternating modified reflections (AAMR) method is a projection algorithm for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method can be seen as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. In this paper we consider the particular case of two subspaces in a Euclidean space. We obtain the rate of linear convergence of the AAMR method in terms of the Friedrichs angle between the subspaces and the parameters defining the scheme, by studying the linear convergence rates of the powers of matrices. We further optimize the value of these parameters in order to get the minimal convergence rate, which turns out to be better than the one of other projection methods. Finally, we provide some numerical experiments that demonstrate the theoretical results.