Parallelizing the Circumcentered-Reflection Method

TL;DR

This paper presents P-CRM, a parallelized method for efficiently solving convex feasibility and approximation problems, with proven linear convergence and advantages in large-scale computational settings.

Contribution

It introduces P-CRM, a parallelized circumcentered reflection method, and the F-SPM framework, both achieving linear convergence and improving scalability for large problems.

Findings

P-CRM converges at a rate comparable or superior to F-SPM.

Numerical experiments demonstrate P-CRM's competitiveness and scalability.

The paper provides a simplified proof of Cimmino's method convergence.

Abstract

This paper introduces the Parallelized Circumcentered Reflection Method (P-CRM), a circumcentric approach that parallelizes the Circumcentered Reflection Method (CRM) for solving Convex Feasibility Problems in affine settings. Beyond feasibility, P-CRM solves the best approximation problem for any finite collection of affine subspaces; that is, it not only finds a feasible point but directly computes the projection of an initial point onto the intersection. Within a fully self-contained scheme, we also introduce the Framework for the Simultaneous Projection Method (F-SPM) which includes Cimmino's method as a special case. Theoretical results show that both P-CRM and F-SPM achieve linear convergence. Moreover, P-CRM converges at a rate that is at least as fast as, and potentially superior to, the best convergence rate of F-SPM. As a byproduct, this also yields a new and simplified convergence proof for Cimmino's method. Numerical experiments show that P-CRM is competitive compared to CRM and indicate that it offers a scalable and flexible alternative, particularly suited for large-scale problems and modern computing environments.