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Parallelizing the Circumcentered-Reflection Method

Pablo Barros, Roger Behling, Vincent Guigues, Luiz-Rafael Santos

TL;DR

The paper addresses the projection-based solution of the Convex Feasibility Problem over the intersection of affine subspaces by introducing a parallelized Circumcentered Reflection Method (P-CRM). It builds a general framework for Simultaneous Projection Methods (F-SPM) that includes Cimmino's method and proves linear convergence to the projection $s^*=P_S(x)$; it then develops P-CRM, proving linear convergence with a rate $r_P<1$ that is at least as good as the best F-SPM rate. The authors establish invariance and contraction properties through a geometric analysis of projections, reflections, and circumcenters, deriving explicit bounds and comparing rates to F-SPM. Numerical experiments show that P-CRM is competitive with CRM and benefits from parallelization, suggesting strong scalability for large-scale problems and potential GPU acceleration. Overall, the work provides a solid theoretical and practical acceleration framework for simultaneous projection methods in affine settings.

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.

Parallelizing the Circumcentered-Reflection Method

TL;DR

The paper addresses the projection-based solution of the Convex Feasibility Problem over the intersection of affine subspaces by introducing a parallelized Circumcentered Reflection Method (P-CRM). It builds a general framework for Simultaneous Projection Methods (F-SPM) that includes Cimmino's method and proves linear convergence to the projection ; it then develops P-CRM, proving linear convergence with a rate that is at least as good as the best F-SPM rate. The authors establish invariance and contraction properties through a geometric analysis of projections, reflections, and circumcenters, deriving explicit bounds and comparing rates to F-SPM. Numerical experiments show that P-CRM is competitive with CRM and benefits from parallelization, suggesting strong scalability for large-scale problems and potential GPU acceleration. Overall, the work provides a solid theoretical and practical acceleration framework for simultaneous projection methods in affine settings.

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.

Paper Structure

This paper contains 10 sections, 16 theorems, 73 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. F-SPM: a framework for SPM
  3. Preliminary results
  4. Convergence of F-SPM
  5. Convergence of the Parallelized Circumcentered Reflection Method method
  6. P-CRM method
  7. Preliminary results
  8. Convergence of P-CRM
  9. Numerical experiments
  10. Concluding Remarks

Key Result

Proposition 2.1

For a closed, convex set $C \neq \emptyset$ and any $x \in \mathds{R}^n,$ there exists a unique $z \in C$ such that Such $z$ is called the projection of $x$ onto $C$ and denoted $P_C(x).$ Moreover, for $z \in C,$

Figures (2)

  • Figure 1: P-CRM in $\mathds{R}^2$
  • Figure 2: P-CRM in $\mathds{R}^3$

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 26 more