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A Parallel-in-Time Multigrid Preconditioner for Optimal Control

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

TL;DR

This work introduces a parallel-in-time multigrid preconditioner for augmented saddle-point systems arising in optimal control. By adding virtual interface variables and permuting to a time-major block-tridiagonal form, it enables a trivially parallelizable block Jacobi smoother, with a GMRES coarse-grid solve preconditioned by symmetric Gauss-Seidel inside a flexible GMRES outer loop. The method is integrated with a matrix-free SQP framework and validated on the van der Pol oscillator and viscous Burgers’ equation, showing strong performance when problem scaling is appropriate. Findings indicate significant reductions in iteration counts and robust scalability, including extensions to invariant hyperbolic problems. The approach preserves the full augmented-system structure and aligns closely with multigrid principles, offering a path toward scalable, time-parallel optimization workflows.

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.

A Parallel-in-Time Multigrid Preconditioner for Optimal Control

TL;DR

This work introduces a parallel-in-time multigrid preconditioner for augmented saddle-point systems arising in optimal control. By adding virtual interface variables and permuting to a time-major block-tridiagonal form, it enables a trivially parallelizable block Jacobi smoother, with a GMRES coarse-grid solve preconditioned by symmetric Gauss-Seidel inside a flexible GMRES outer loop. The method is integrated with a matrix-free SQP framework and validated on the van der Pol oscillator and viscous Burgers’ equation, showing strong performance when problem scaling is appropriate. Findings indicate significant reductions in iteration counts and robust scalability, including extensions to invariant hyperbolic problems. The approach preserves the full augmented-system structure and aligns closely with multigrid principles, offering a path toward scalable, time-parallel optimization workflows.

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.
Paper Structure (20 sections, 3 theorems, 58 equations, 7 figures, 6 tables, 1 algorithm)

This paper contains 20 sections, 3 theorems, 58 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Augmented Systems
  4. A Case for Multigrid in Time
  5. Discretization
  6. Scaling
  7. Preconditioner
  8. Virtual Variables
  9. Permutation of the Augmented System
  10. Multigrid
  11. Interpolation and Restriction
  12. Smoother
  13. Control Scaling
  14. Coarse-Grid Solver
  15. Numerical Results
  16. ...and 5 more sections

Key Result

Proposition 1

Let $U^*$ be $\mathbb{R}^{pn}$ equipped with the inner product for some matrix $\Sigma_{U^*}$. If the primal-dual pairing $\langle\cdot,\cdot\rangle_{U,U^*}$ is the Euclidean inner product between vectors in $\mathbb{R}^{pn}$, then $\Sigma_{U^*} := \Sigma_U^{-1}$

Figures (7)

  • Figure 1: Schematic of parallel-in-time integration applied to optimal control. Black squares indicate serialization points, belonging to the coarse correction/propagation schemes. Top: Conventional approaches remain intrinsically serial, consisting of forward time stepping, 1, to evaluate the objective function, followed by backward time stepping, 2, to evaluate derivatives. Bottom: Our preconditioner is, in contrast, a parallel-in-time method tailored to the coupled bidirectional flow of time in optimality systems arising from optimal control. Parallel-in-time execution involves concurrent solutions of optimal control problems on the time subdomains.
  • Figure 2: Augmented system when $n = 4$.
  • Figure 3: Time domain decomposition.
  • Figure 4: Augmented system with virtual variables when $n = 4$.
  • Figure 5: Block-tridiagonal permutation of the augmented system when $n = 4$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Theorem 2
  • proof
  • Proposition 3
  • proof