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Block $ω$-circulant preconditioners for parabolic optimal control problems

Po Yin Fung, Sean Hon

TL;DR

The authors develop block ω-circulant preconditioners for all-at-once linear systems arising from parabolic optimal control problems discretized in time with the θ-method. They establish spectral properties showing GMRES with $\mathcal{P}_S$ yields eigenvalues near $\pm1$ and MINRES with the ideal or modified preconditioners yields rapid convergence, aided by parallel-in-time FFT diagonaliation and multigrid for shifted-Laplacian solves. A key contribution is the design of two practical preconditioners, $|\mathcal{P}_S|$ and $\mathcal{P}_{MS}$, with proven spectral clustering and efficient PinT implementation. Numerical experiments corroborate fast, robust performance across a range of regularization parameters, discretizations, and boundary conditions, highlighting the method’s potential for scalable PDE-constrained optimization. The work also outlines future directions, including ε-circulant hybrids and Runge-Kutta-based all-at-once preconditioning strategies.

Abstract

In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block $ω$-circulant based preconditioners for the all-at-once linear system arising from the concerned optimal control problem, where both first order and second order time discretization methods are considered. The proposed preconditioners can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix are clustered around $\pm 1$, which leads to rapid convergence when the minimal residual method is used. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners are justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.

Block $ω$-circulant preconditioners for parabolic optimal control problems

TL;DR

The authors develop block ω-circulant preconditioners for all-at-once linear systems arising from parabolic optimal control problems discretized in time with the θ-method. They establish spectral properties showing GMRES with yields eigenvalues near and MINRES with the ideal or modified preconditioners yields rapid convergence, aided by parallel-in-time FFT diagonaliation and multigrid for shifted-Laplacian solves. A key contribution is the design of two practical preconditioners, and , with proven spectral clustering and efficient PinT implementation. Numerical experiments corroborate fast, robust performance across a range of regularization parameters, discretizations, and boundary conditions, highlighting the method’s potential for scalable PDE-constrained optimization. The work also outlines future directions, including ε-circulant hybrids and Runge-Kutta-based all-at-once preconditioning strategies.

Abstract

In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block -circulant based preconditioners for the all-at-once linear system arising from the concerned optimal control problem, where both first order and second order time discretization methods are considered. The proposed preconditioners can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix are clustered around , which leads to rapid convergence when the minimal residual method is used. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners are justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.
Paper Structure (10 sections, 14 theorems, 92 equations, 1 figure, 4 tables)

This paper contains 10 sections, 14 theorems, 92 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Toeplitz matrices
  3. Main results
  4. GMRES - block $\omega$-circulant based preconditioner
  5. MINRES - ideal preconditioner
  6. MINRES - block $\omega$-circulant based preconditioner
  7. MINRES - modified block $\omega$-circulant based preconditioner
  8. Implementation
  9. Numerical examples
  10. Conclusions

Key Result

Proposition 3.1

Let $\mathcal{\widehat{A}} \in \mathbb{R}^{2mn \times 2mn} ,\mathcal{P}_{S} \in \mathbb{C}^{2mn \times 2mn}$ be defined by (eqn:matrix_A_hat) and (eqn:matrix_P_GMRES), respectively. Then, where $I_{mn}$ is the $mn$ by $mn$ identity matrix and $\mathrm{rank}(\widetilde{\mathcal{R}}_{1}) \leq 4m$.

Figures (1)

  • Figure 1: Eigenvalues of $\mathcal{P}_{MS}^{-1}|\mathcal{P}_{S}|$ with $n=16$, $m=15$, and various $\gamma$.

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 22 more