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A Parallel in Time Algorithm Based on ParaExp for Optimal Control Problems

Felix Kwok, Djahou N Tognon

TL;DR

A new parallel-in-time algorithm for solving optimal control problems constrained by discretized partial differential equations is proposed and a preconditioner is proposed to speed up the convergence of GMRES in the special cases of the heat and wave equations.

Abstract

We propose a new parallel-in-time algorithm for solving optimal control problems constrained by discretized partial differential equations. Our approach, which is based on a deeper understanding of ParaExp, considers an overlapping time-domain decomposition in which we combine the solution of homogeneous problems using exponential propagation with the local solutions of inhomogeneous problems. The algorithm yields a linear system whose matrix-vector product can be fully performed in parallel. We then propose a preconditioner to speed up the convergence of GMRES in the special cases of the heat and wave equations. Numerical experiments are provided to illustrate the efficiency of our preconditioners.

A Parallel in Time Algorithm Based on ParaExp for Optimal Control Problems

TL;DR

A new parallel-in-time algorithm for solving optimal control problems constrained by discretized partial differential equations is proposed and a preconditioner is proposed to speed up the convergence of GMRES in the special cases of the heat and wave equations.

Abstract

We propose a new parallel-in-time algorithm for solving optimal control problems constrained by discretized partial differential equations. Our approach, which is based on a deeper understanding of ParaExp, considers an overlapping time-domain decomposition in which we combine the solution of homogeneous problems using exponential propagation with the local solutions of inhomogeneous problems. The algorithm yields a linear system whose matrix-vector product can be fully performed in parallel. We then propose a preconditioner to speed up the convergence of GMRES in the special cases of the heat and wave equations. Numerical experiments are provided to illustrate the efficiency of our preconditioners.
Paper Structure (8 sections, 1 theorem, 48 equations, 2 figures, 5 tables)

This paper contains 8 sections, 1 theorem, 48 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Linear optimal control problem
  3. Parallel-in-time algorithm
  4. Preconditioner design
  5. Heat equation
  6. Wave equation
  7. Conclusion
  8. Acknowledgments

Key Result

Theorem IV.1

Let $N$ be given and ${\@fontswitch\mathcal{R}}_{\ell}$ be approximated using implicit Euler with $N$ fine sub-intervals over each $[T_{\ell-1}, T_{\ell}]$. Then any eigenvalue $\mu$ of ${\@fontswitch\mathcal{M}}\widehat{\@fontswitch\mathcal{M}}^{-1}$ satisfies where $\delta t=T/LN$.

Figures (2)

  • Figure 1: Parallel distribution on the processors. Djahou :We should propose another more consistent figure.
  • Figure 2: The maximal eigenvalue of ${\@fontswitch\mathcal{M}}\widehat{\@fontswitch\mathcal{M}}^{-1}$ and the upper bound in \ref{['eq:estimate']} for various $\delta t$ when $\alpha=10^{-4}$.

Theorems & Definitions (3)

  • Theorem IV.1
  • proof
  • Remark 1