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

TL;DR

This paper introduces a novel parallel-in-time algorithm based on ParaExp for solving optimal control problems constrained by PDEs, enabling efficient parallel computation through overlapping time-domain decomposition and specialized preconditioning.

Contribution

The paper develops a new parallel-in-time algorithm utilizing overlapping decomposition and exponential propagation, with tailored preconditioners for heat and wave equations, advancing computational efficiency in optimal control.

Findings

Algorithm achieves fully parallel matrix-vector products.

Preconditioners significantly improve GMRES convergence.

Numerical results demonstrate efficiency gains.

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.