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New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms

Martin Jakob Gander, Liu-Di Lu

TL;DR

This work develops and analyzes nine Neumann–Neumann time-domain decomposition variants for unconstrained parabolic optimal-control problems, following a spatial semi-discretization that yields a forward–backward ODE system. By solving the error equations and deriving explicit iteration matrices, the authors identify optimal relaxation parameters and demonstrate that many natural NN formulations act mainly as smoothers rather than robust solvers. The results reveal that variants NN2a/NN3a (and NN2c/NN3c) with carefully chosen $\theta$ offer fast convergence and strong smoothing, while several other configurations diverge or perform poorly. Numerical experiments corroborate the theoretical findings and highlight the critical role of transmission conditions and update rules in ensuring convergence. The approach provides efficient parallel-in-time solvers for parabolic optimal control and paves the way for extending to multi-subdomain settings and broader parabolic constraints.

Abstract

We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.

New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms

TL;DR

This work develops and analyzes nine Neumann–Neumann time-domain decomposition variants for unconstrained parabolic optimal-control problems, following a spatial semi-discretization that yields a forward–backward ODE system. By solving the error equations and deriving explicit iteration matrices, the authors identify optimal relaxation parameters and demonstrate that many natural NN formulations act mainly as smoothers rather than robust solvers. The results reveal that variants NN2a/NN3a (and NN2c/NN3c) with carefully chosen offer fast convergence and strong smoothing, while several other configurations diverge or perform poorly. Numerical experiments corroborate the theoretical findings and highlight the critical role of transmission conditions and update rules in ensuring convergence. The approach provides efficient parallel-in-time solvers for parabolic optimal control and paves the way for extending to multi-subdomain settings and broader parabolic constraints.

Abstract

We present new Neumann-Neumann algorithms based on a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semi-discretization, the Lagrange multiplier approach provides a coupled forward-backward optimality system, which can be solved using a time domain decomposition. Due to the forward-backward structure of the optimality system, nine variants can be found for the Neumann-Neumann algorithms. We analyze their convergence behavior and determine the optimal relaxation parameter for each algorithm. Our analysis reveals that the most natural algorithms are actually only good smoothers, and there are better choices which lead to efficient solvers. We illustrate our analysis with numerical experiments.
Paper Structure (24 sections, 9 theorems, 81 equations, 5 figures, 1 table)

This paper contains 24 sections, 9 theorems, 81 equations, 5 figures, 1 table.

Table of Contents

  1. Keywords:
  2. MSCcodes:
  3. Introduction
  4. Neumann-Neumann algorithms
  5. Convergence analysis
  6. Category I
  7. Algorithm NN$_{1\text{a}}$
  8. Algorithm NN$_{1\text{b}}$
  9. Algorithm NN$_{1\text{c}}$
  10. Category II
  11. Algorithm NN$_{2\text{a}}$
  12. Algorithm NN$_{2\text{b}}$
  13. Algorithm NN$_{2\text{c}}$
  14. Category III
  15. Algorithm NN$_{3\text{a}}$
  16. ...and 9 more sections

Key Result

Theorem 1

Algorithm NN$_{1\text{a}}$eq:NN1a-eq:NN1atran converges if and only if where $\lambda(A)$ is the spectrum of the matrix $A$.

Figures (5)

  • Figure 1: Illustration of the forward-backward system.
  • Figure 2: Convergence factor with $\theta=0.25$ of NN$_{2\text{b}}$ and NN$_{3\text{b}}$ as a function of the eigenvalues $d_i \in [10^{-2}, 10^2 ]$. Left: case A for NN$_{2\text{b}}$. Right: case B for NN$_{3\text{b}}$.
  • Figure 3: Convergence factor with different relaxation parameters $\theta$ of NN$_{1\text{a}}$ as a function of the eigenvalues $d_i \in [10^{-2}, 10^2 ]$. Left: case A. Right: case B.
  • Figure 4: Convergence factor with $\theta=1/2$ of the six algorithms as a function of the eigenvalues $d_i \in [10^{-2}, 10^2 ]$. Left: case A. Right: case B.
  • Figure 5: Convergence factor with optimal relaxation parameter $\theta^{\star}$ of the six algorithms as a function of the eigenvalues $d_i \in [10^{-2}, 10^2 ]$. Left: case A. Right: case B.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • Remark 5
  • Theorem 3
  • Remark 6
  • Remark 7
  • ...and 11 more