Efficient parallel inversion of ParaOpt preconditioners

TL;DR

This paper introduces a fast parallel inversion method for ParaOpt preconditioners, enhancing efficiency in solving systems within the time-parallel optimal control framework, especially for nonlinear problems.

Contribution

It generalizes the ParaOpt preconditioner to nonlinear cases and proposes a novel, efficient inversion technique for the associated smaller systems.

Findings

Significantly reduces inversion time for ParaOpt preconditioners.

Improves scalability and efficiency in nonlinear optimal control problems.

Avoids issues with boundary conditions in existing methods.

Abstract

Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.