On the linear convergence of the circumcentered-reflection method

Abstract

In order to accelerate the Douglas-Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.

The block-wise circumcentered–reflection method

“…The resulting algorithm significantly outperforms DRM and MAP numerically as presented in [11]. This numerical performance of CRM, together with the deficiency of DRM in dealing with more than two sets (see [1,Example 2.1] and some modifications [15,16] for DRM), motivated our theoretical study in [12]. The circumcenter schemes we came up with are already in the attention of specialists of the field (see Bauschke et al [9,10], Lindstrom and Sims [24] and Ouyang [25]) and questions on the possibility of successful behavior in more general and more important settings are arising.…”

“…The main purpose of this paper is applying the recently developed circumcentered-reflection method (CRM) [12] to solve problem (1) by taking advantage of a block-wise structure. This idea may be beneficial in certain problems coming from the discretization of partial differential equations as we describe and illustrate in our numerical section.…”

“…Nonetheless, Definition 2 allows for non-affine mappings. Later we will see and use the fact that the circumcenter operator defined in [12] is a best approximation mapping, even though it is usually non-affine.…”

“…Probably the two most famous and standard among these methods are the Douglas-Rachford method (DRM), or averaged alternating reflection method (see, e.g., [2]); and the method of alternating projections (MAP) which is also known as von Neumann's or Kaczmarz' algorithm (see, e.g., [4,27]). Upon our ideas presented in [11,12], we devote this work to study a circumcenter type method related to both DRM and MAP.…”

“…The linear convergence of the circumcentered-reflection method (CRM) was established in [12] for solving problem (1) with m ≥ 2 affine subspaces. Since the computation of a circumcenter requires the resolution of a suitable m × m linear system, this might not be of negligible computational cost for large m. To avoid this drawback for problems where the computation of z C RM is simply too demanding, we propose in the present work the Block-wise Circumcentered-Reflection Method (Bw-CRM) by using an arbitrary ordered partition of the indexes {1, 2, .…”