Convergence of Adaptive, Randomized, Iterative Linear Solvers

TL;DR

This paper establishes general conditions ensuring convergence of adaptive, randomized, iterative linear solvers, providing theoretical guarantees and convergence rates to aid practical solver design.

Contribution

It introduces a unified framework with assumptions that guarantee convergence of adapted linear solvers, both randomized and deterministic, with proven rates.

Findings

Provided a set of assumptions guaranteeing convergence with probability one

Derived worst-case convergence rates for adapted solvers

Guided the design of reliable, high-performance linear solvers

Abstract

Deterministic and randomized, row-action and column-action linear solvers have become increasingly popular owing to their simplicity, low computational and memory complexities, and ease of composition with other techniques. Moreover, in order to achieve high-performance, such solvers must often be adapted to the given problem structure and to the hardware platform on which the problem will be solved. Unfortunately, determining whether such adapted solvers will converge to a solution has required equally unique analyses. As a result, adapted, reliable solvers are slow to be developed and deployed. In this work, we provide a general set of assumptions under which such adapted solvers are guaranteed to converge with probability one, and provide worst case rates of convergence. As a result, we can provide practitioners with guidance on how to design highly adapted, randomized or deterministic, row-action or column-action linear solvers that are also guaranteed to converge.