A randomized operator splitting scheme inspired by stochastic optimization methods

TL;DR

This paper introduces a randomized operator splitting scheme inspired by stochastic optimization, which reduces computational costs and achieves convergence orders comparable to traditional methods, with potential improvements.

Contribution

It combines operator splitting with stochastic methods to create a randomized scheme that decreases computational costs while maintaining convergence properties.

Findings

Convergence order is at least 1/2, with potential to reach order 1.

Numerical experiments confirm theoretical convergence rates.

Randomization can improve accuracy without full operator application.

Abstract

In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.