Asynchronous Sequential Inertial Iterations for Common Fixed Points Problems with an Application to Linear Systems

TL;DR

This paper introduces an asynchronous inertial iterative framework for finding common fixed points of nonexpansive operators, enabling faster convergence and parallel processing without synchronization, with applications to linear systems.

Contribution

It extends the ARock algorithm to common fixed point problems with inertial terms, allowing asynchronous updates with bounded delays and improving convergence speed.

Findings

Proven convergence without delay distribution assumptions

Enhanced performance demonstrated with linear system example

Framework supports parallel, asynchronous processing

Abstract

The common fixed points problem requires finding a point in the intersection of fixed points sets of a finite collection of operators. Quickly solving problems of this sort is of great practical importance for engineering and scientific tasks (e.g., for computed tomography). Iterative methods for solving these problems often employ a Krasnosel'ski\u{\i}-Mann type iteration. We present an Asynchronous Sequential Inertial (ASI) algorithmic framework in a Hilbert space to solve common fixed points problems with a collection of nonexpansive operators. Our scheme allows use of out-of-date iterates when generating updates, thereby enabling processing nodes to work simultaneously and without synchronization. This method also includes inertial type extrapolation terms to increase the speed of convergence. In particular, we extend the application of the recent ARock algorithm" [Peng, Z. et al, SIAM J. on Scientific Computing 38, A2851-2879, (2016)] in the context of convex feasibility problems. Convergence of the ASI algorithm is proven with no assumption on the distribution of delays, except that they be uniformly bounded. Discussion is provided along with a computational example showing the performance of the ASI algorithm applied in conjunction with a diagonally relaxed orthogonal projections (DROP) algorithm for estimating solutions to large linear systems.