Scalable algorithms for time-parallel optimal control based on diagonalisation (Arne Bouillon, KU Leuven)

05.03.2024 14:00 – 15:00

Optimal-control problems can give rise to boundary value problems (BVPs) whose solution contains the required control term. ParaDiag [1,2,3] and ParaOpt [4] are two algorithms to solve these optimality BVPs in a time-parallel way. The former is limited to linear problems with a symmetric system matrix, but generally scales well; the latter applies to non-linear problems, but includes a linear-system solving step that does not scale well when increasing time-parallelism.

The presentation consists of two main parts. In the first, we propose algorithmic improvements and extensions to these methods that make them more generally applicable and that improve their scaling. More specifically, we extend ParaDiag's applicability to non-symmetric linear problems and more general control objectives [5]. We then apply these new ParaDiag preconditioners to the linear system in ParaOpt, which improves its scaling [6].

The second part of the presentation discusses theoretical results about ParaDiag and ParaOpt applied to linear diffusive problems. We show how convergence for both algorithms can be studied by finding eigenvalues of related matrices and present convergence bounds that are parameterised by the time discretisation scheme(s) used. This generic approach allows us to improve the state-of-the-art coarse propagator for ParaOpt and to directly apply ParaDiag's analysis to the ParaOpt preconditioners we developed.

[1]: S.-L. Wu and T. Zhou. Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems. ESAIM: Control, Optimisa- tion and Calculus of Variations, 26, 2020.
[2]: S.-L. Wu and J. Liu. A parallel-in-time block-circulant preconditioner for optimal control of wave equations. SIAM Journal on Scientific Computing, 42(3):A1510–A1540, 2020.
[3]: M. J. Gander, J. Liu, S.-L. Wu, X. Yue, and T. Zhou. ParaDiag: parallel-in-time algorithms based on the diagonalization technique. arXiv:2005.09158 [cs, math], Apr 2021.
[4]: M. J. Gander, F. Kwok, and J. Salomon. PARAOPT: A Parareal algorithm for optimality systems. SIAM Journal on Scientific Computing, 42(5):A2773–A2802, Jan 2020.
[5]: A. Bouillon, G. Samaey, and K. Meerbergen. On generalized preconditioners for time-parallel parabolic optimal control. arxiv:2302.06406 [cs, math], Feb 2023.
[6]: A. Bouillon, G. Samaey, and K. Meerbergen. Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt. arxiv:2304.09235 [cs, math], Apr 2023.

Lieu

Conseil Général 7-9, Room 1-05, Séminaire d'analyse numérique

entrée libre