Multigrid-in-time for sensitivity analysis of chaotic dynamical systems

TL;DR

This paper explores the use of multigrid-in-time methods to improve the efficiency of sensitivity analysis in chaotic dynamical systems using Least Squares Shadowing, addressing computational challenges in large-scale KKT systems.

Contribution

It introduces multigrid-in-time schemes tailored for LSS, analyzing factors affecting convergence and demonstrating their effectiveness for chaotic systems.

Findings

Multigrid-in-time significantly accelerates LSS sensitivity computations.

Choice of smoother and grid coarsening impacts convergence rates.

Proper weighting of the least squares objective improves stability.

Abstract

The following paper discusses the application of a multigrid-in-time scheme to Least Squares Shadowing (LSS), a novel sensitivity analysis method for chaotic dynamical systems. While traditional sensitivity analysis methods break down for chaotic dynamical systems, LSS is able to compute accurate gradients. Multigrid is used because LSS requires solving a very large Karush-Kuhn-Tucker (KKT) system constructed from the solution of the dynamical system over the entire time interval of interest. Several different multigrid-in-time schemes are examined, and a number of factors were found to heavily influence the convergence rate of multigrid-in-time for LSS. These include the iterative method used for the smoother, how the coarse grid system is formed and how the least squares objective function at the center of LSS is weighted.