Randomized Method of Subspace Corrections

TL;DR

This paper analyzes a randomized iterative subspace correction method, providing theoretical convergence rate estimates and demonstrating its fault-tolerance with high probability, enhancing robustness in numerical solutions.

Contribution

It introduces a probabilistic analysis of the convergence and fault-tolerance of randomized subspace correction methods, offering sharp error reduction estimates.

Findings

Expected convergence rates are derived and sharp estimates are provided.

The method converges with probability one even under fault conditions.

Fault-tolerance is achieved by rejecting corrections upon errors, maintaining convergence.

Abstract

In this paper, we consider the iterative method of subspace corrections with random ordering. We prove identities for the expected convergence rate, which can provide sharp estimates for the error reduction per iteration. We also study the fault-tolerant feature of the randomized successive subspace correction method by simply rejecting all the corrections when error occurs and show that the results iterative method converges with probability one. Moreover, we also provide sharp estimates on the expected convergence rate for the fault-tolerant, randomized, subspace correction method.