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On generalized preconditioners for time-parallel parabolic optimal control

Arne Bouillon, Giovanni Samaey, Karl Meerbergen

TL;DR

This work proposes three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives.

Abstract

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.

On generalized preconditioners for time-parallel parabolic optimal control

TL;DR

This work proposes three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives.

Abstract

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.
Paper Structure (24 sections, 9 theorems, 71 equations, 9 figures, 2 tables, 3 algorithms)

This paper contains 24 sections, 9 theorems, 71 equations, 9 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. ParaDiag for tracking objectives
  3. Existing method
  4. The small-T limit and alpha-circulants
  5. Analytic eigenvalue expressions
  6. Interpreting the eigenvalue results
  7. Generalizing past self-adjoint problems
  8. ParaDiag for terminal-cost objectives
  9. A new preconditioner
  10. Analytic eigenvalue expressions
  11. Interpreting the eigenvalue results
  12. Parallel-scaling analysis for self-adjoint problems
  13. Increasing the time horizon
  14. Decreasing the time step
  15. Numerical results
  16. ...and 9 more sections

Key Result

Theorem 1

\newlabelthm:pd-track:eigs0 Let $\widehat{L}>3$, $\alpha=\pm1$ and $\varphi,\psi\in\mathbb R\backslash\{0\}$. The $2\widehat{L}\times2\widehat{L}$ matrix has only two potentially non-zero eigenvalues $\omega_{\{1,2\}}$. Specifically, where

Figures (9)

  • Figure 1: Eigenvalues $\theta$ of $P^{-1}{\widehat{A}}$ and iteration count $k_g$ of ParaDiag for the example in \ref{['sec:pd-track:alpha']}, using gmres with relative tolerance $10^{-6}$. \ref{['fig:pd-track:eigs-1:1']} mimics wuDiagonalizationbasedParallelintimeAlgorithms2020b, but \ref{['fig:pd-track:eigs-0.0001:1']} discovers issues when $T$ is small.
  • Figure 1: The non-unity preconditioned eigenvalues $\theta$ of $P(0)^{-1}A$ with $L=1000$
  • Figure 1: Ratio $\abs{\theta(L=10^4)}/\abs{\theta(L=10^3)}$ of the preconditioned-eigenvalue magnitudes when scaling $L$ from $10^3$ to $10^4$ through $T$, for different preconditioners
  • Figure 1: Initial condition $y_\mathrm{init}$ from \ref{['eq:num:intro:yinit-rough']}
  • Figure 1: $\beta_j(\cdot, \cdot)$ for different parameters when $\widehat{L}=3$
  • ...and 4 more figures

Theorems & Definitions (16)

  • Theorem 1
  • Proof 1
  • Corollary 2
  • Corollary 3
  • Proof 2
  • Theorem 4
  • Proof 3
  • Theorem 1
  • Proof 4
  • Corollary 2
  • ...and 6 more