A Parallel-in-Time Multigrid Preconditioner for Optimal Control
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914180166057984 |
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| author | Vuchkov, Radoslav Cyr, Eric C. Javeed, Aurya Ridzal, Denis |
| author_facet | Vuchkov, Radoslav Cyr, Eric C. Javeed, Aurya Ridzal, Denis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Parallel-in-Time Multigrid Preconditioner for Optimal Control Vuchkov, Radoslav Cyr, Eric C. Javeed, Aurya Ridzal, Denis Optimization and Control 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. |
| title | A Parallel-in-Time Multigrid Preconditioner for Optimal Control |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2405.04808 |