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Non-overlapping Schwarz methods in time for parabolic optimal control problems

Martin Jakob Gander, Liu-Di Lu

TL;DR

This work studies time-domain Schwarz methods for unconstrained parabolic optimal control problems by non-overlappingly decomposing the time interval into $I_1=(0,\alpha)$ and $I_2=(\alpha,T)$. It develops and analyzes four Dirichlet variants SD$_1$–SD$_4$ and four Neumann variants SN$_1$–SN$_4$, deriving explicit convergence factors (e.g., $\rho_{\mathrm{SD}_1}$ and $\rho_{\mathrm{SN}_1}$) and optimal relaxation parameters, demonstrating that preserving the forward-backward structure is crucial for robust convergence. The key finding is that only SD$_1$ and SN$_1$ maintain good convergence properties, while the other Dirichlet/Neumann variants either diverge or stagnate; relaxation can substantially improve smoothers for high-frequency modes. Numerical experiments on a 1D heat-control problem validate the theory, showing reduced iteration counts when using the optimal relaxations and highlighting the practical potential of time-domain Schwarz methods for parallelizing parabolic optimal control solvers.

Abstract

We present here the classical Schwarz method with a time domain decomposition applied to unconstrained parabolic optimal control problems. Unlike Dirichlet-Neumann and Neumann-Neumann algorithms, we find different properties based on the forward-backward structure of the optimality system. Variants can be found using only Dirichlet and Neumann transmission conditions. Some of these variants are only good smoothers, while others could lead to efficient solvers.

Non-overlapping Schwarz methods in time for parabolic optimal control problems

TL;DR

This work studies time-domain Schwarz methods for unconstrained parabolic optimal control problems by non-overlappingly decomposing the time interval into and . It develops and analyzes four Dirichlet variants SD–SD and four Neumann variants SN–SN, deriving explicit convergence factors (e.g., and ) and optimal relaxation parameters, demonstrating that preserving the forward-backward structure is crucial for robust convergence. The key finding is that only SD and SN maintain good convergence properties, while the other Dirichlet/Neumann variants either diverge or stagnate; relaxation can substantially improve smoothers for high-frequency modes. Numerical experiments on a 1D heat-control problem validate the theory, showing reduced iteration counts when using the optimal relaxations and highlighting the practical potential of time-domain Schwarz methods for parallelizing parabolic optimal control solvers.

Abstract

We present here the classical Schwarz method with a time domain decomposition applied to unconstrained parabolic optimal control problems. Unlike Dirichlet-Neumann and Neumann-Neumann algorithms, we find different properties based on the forward-backward structure of the optimality system. Variants can be found using only Dirichlet and Neumann transmission conditions. Some of these variants are only good smoothers, while others could lead to efficient solvers.
Paper Structure (4 sections, 3 theorems, 23 equations, 2 figures, 1 table)

This paper contains 4 sections, 3 theorems, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dirichlet transmission conditions
  3. Neumann transmission condition
  4. Numerical experiments and comments

Key Result

Theorem 1

Assume that $A$ is symmetric positive semi-definite (i.e., $d_i\geq0$). Then the Schwarz algorithm eq:SD1 converges for all initial guesses if (i) $\alpha \leq \frac{T}{2}$, or (ii) $\gamma=0$.

Figures (2)

  • Figure 1: One dimensional illustration of decomposing in space (left) and decomposing in time (right).
  • Figure 2: Convergence factor as a function of eigenvalues (left) and error decay as a function of the number of iterations (right).

Theorems & Definitions (4)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3