On generalized preconditioners for time-parallel parabolic optimal control

TL;DR

This paper advances the ParaDiag algorithms for time-parallel solutions of parabolic optimal control problems by introducing new preconditioners, extending to non-self-adjoint cases, and analyzing eigenvalues for improved parallel scalability.

Contribution

The paper introduces alpha-circulant preconditioners, generalizes ParaDiag for non-self-adjoint equations, and develops algorithms for terminal-cost objectives, supported by eigenvalue analysis and numerical validation.

Findings

Eigenvalue analysis shows favorable properties of alpha-circulant preconditioners.

Parallel-scaling analysis indicates improved efficiency for self-adjoint problems.

Numerical tests confirm the theoretical predictions and extend to non-self-adjoint cases.

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.