Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient parallel inversion of ParaOpt preconditioners

Corentin Bonte, Arne Bouillon, Giovanni Samaey, Karl Meerbergen

TL;DR

This work advances parallel-in-time optimal control by extending ParaOpt with a nonlinear preconditioner and a direct inner-system inversion. It generalizes a diagonalization-based preconditioner to nonlinear settings using averaging and a novel alpha-circulant factorization, enabling efficient even black-box inversion of many small inner systems. The core contribution is a direct inversion method that preserves the black-box nature of propagators while reducing the number of coarse solves and GMRES iterations, as demonstrated on a nonlinear Burgers' equation. The results offer a scalable approach for solving nonlinear ParaOpt problems in parallel, with practical impact for time-parallel optimal-control applications.

Abstract

Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.

Efficient parallel inversion of ParaOpt preconditioners

TL;DR

This work advances parallel-in-time optimal control by extending ParaOpt with a nonlinear preconditioner and a direct inner-system inversion. It generalizes a diagonalization-based preconditioner to nonlinear settings using averaging and a novel alpha-circulant factorization, enabling efficient even black-box inversion of many small inner systems. The core contribution is a direct inversion method that preserves the black-box nature of propagators while reducing the number of coarse solves and GMRES iterations, as demonstrated on a nonlinear Burgers' equation. The results offer a scalable approach for solving nonlinear ParaOpt problems in parallel, with practical impact for time-parallel optimal-control applications.

Abstract

Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.

Paper Structure

This paper contains 7 sections, 1 theorem, 21 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The ParaOpt algorithm
  3. ParaOpt preconditioners
  4. Preconditioning linear ParaOpt
  5. Generalization to the nonlinear case
  6. Efficiently solving the smaller systems of the preconditioner
  7. Numerical example

Key Result

Lemma 1

Consider the $\alpha$-circulant matrix $C(\alpha) \in \mathbb{C}^{L\times L}$ and its diagonalization $C(\alpha) = \Gamma^{-1}_{\alpha}\mathbb{F}^*D(\alpha)\mathbb{F}\Gamma_{\alpha}$. If the first column of $C(\alpha)$ only has one nonzero element $a$ and if $|\alpha| = 1$, then it follows that the

Figures (2)

  • Figure 1: Total GMRES iterations and coarse BVP solves in one subinterval in tracking ParaOpt (without and with the proposed improvements)
  • Figure 2: Total GMRES iterations and coarse BVP solves in one subinterval in final value ParaOpt (without and with the proposed improvements)

Theorems & Definitions (2)

  • Lemma 1
  • proof