The rate of convergence in the method of alternating projections

Abstract

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

Subspace arrangements and property T

“…This explains why the definition of the angle give in the introduction is equivalent to the Definition 3.20, see [1] for a detailed proof.…”

“…It is not clear what the exact geometric meaning of the angle defined below is; the definition is equivalent to the one described in the Introduction, see Remark 3.25. The same notion is studied in [1], where the authors define several ways to measure the "angle" between several subspaces. Our definition of angle is the same as the Friedrichs number used in [1].…”

“…The same notion is studied in [1], where the authors define several ways to measure the "angle" between several subspaces. Our definition of angle is the same as the Friedrichs number used in [1]. where .V 1 ; : : : ; V n / denotes the co-distance between V i as defined in [13].…”

An EM scattering algorithm for all materials

“…For methods involving alternating projections convergence is governed by the particular properties of the projections themselves [16], and may or may not occur. In [1] Chantaveerod demonstrates for perfect conductors that his iterative method does properly converge when the three dimensional Cauchy kernel is used.…”

Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Projections

“…In view of these applications it is important to understand the rate at which the convergence in (1.1) takes place; see for instance [1,2,6,7] for indepth investigations. Recall that the Friedrichs number c(L 1 , L 2 ) between the two subspaces L 1 , L 2 of X is defined as c(L 1 , L 2 ) = sup |(x 1 , x 2 )| : x k ∈ L k ∩ L ⊥ and x k ≤ 1 for k = 1, 2 , where L = L 1 ∩ L 2 .…”