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arXiv · 2405.04808

A Parallel-in-Time Multigrid Preconditioner for Optimal Control

Radoslav Vuchkov·Eric C. Cyr·Aurya Javeed·Denis Ridzal
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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.

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Radoslav Vuchkov, Eric C. Cyr, Aurya Javeed, Denis Ridzal. 2024-05-08. A Parallel-in-Time Multigrid Preconditioner for Optimal Control. https://arxiv.org/abs/2405.04808

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