A Parallel-in-Time Multigrid Preconditioner for Optimal Control
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.
Cite
@article{arxiv.2405.04808,
title = {A Parallel-in-Time Multigrid Preconditioner for Optimal Control},
author = {Radoslav Vuchkov and Eric C. Cyr and Aurya Javeed and Denis Ridzal},
journal= {arXiv preprint arXiv:2405.04808},
year = {2025}
}