A Parallel-in-Time Multigrid Preconditioner for Optimal Control

TL;DR

This paper introduces a parallel-in-time multigrid preconditioner for saddle-point systems in optimal control, significantly accelerating solution times when combined with SQP methods.

Contribution

It develops a novel parallel-in-time multigrid preconditioner using virtual interface variables and geometric multigrid, enhancing the efficiency of solving optimal control problems.

Findings

Preconditioner effectively accelerates convergence for scaled problems.

Parallelization in time simplifies computational complexity.

Demonstrated success on van der Pol oscillator and viscous Burgers' equation.

Abstract

We develop a parallel-in-time multigrid preconditioner for augmented systems. These saddle-point systems are foundational to numerical optimization. Our preconditioner, when paired with a suitable optimization method, accelerates the solution of optimal control problems. We construct the preconditioner by introducing virtual interface variables that enable time-domain decomposition. After permuting the resulting augmented system into block tridiagonal form, we develop a geometric multigrid scheme with a block Jacobi smoother, which parallelizes trivially in time. As the coarse grid solver we use GMRES preconditioned with a symmetric Gauss-Seidel iteration. We use the multigrid scheme to precondition a flexible GMRES [1] iteration for the solution of the augmented system. We combine our preconditioner with the matrix-free sequential quadratic programming (SQP) algorithm [2] to solve optimal control problems involving the van der Pol oscillator and the viscous Burgers' equation. We find that the preconditioner is remarkably effective when the problems are suitably scaled.