@article{52cc9e71fefe45fb8f61e1861aeee3ca,
title = "Paraopt: A parareal algorithm for optimality systems",
abstract = "The time parallel solution of optimality systems arising in PDE constrained optimization could be achieved by simply applying any time parallel algorithm, such as Parareal, to solve the forward and backward evolution problems arising in the optimization loop. We propose here a different strategy by devising directly a new time parallel algorithm, which we call ParaOpt, for the coupled forward and backward nonlinear partial differential equations. ParaOpt is inspired by the Parareal algorithm for evolution equations and thus is automatically a two-level method. We provide a detailed convergence analysis for the case of linear parabolic PDE constraints. We illustrate the performance of ParaOpt with numerical experiments for both linear and nonlinear optimality systems. ",
keywords = "Optimal control, Parareal algorithm, Preconditioning",
author = "GANDER, {MARTIN J.} and KWOK, {Wing Hong Felix} and JULIEN SALOMON",
note = "Funding Information: \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section October 9, 2019; accepted for publication (in revised form) July 6, 2020; published electronically September 17, 2020. https://doi.org/10.1137/19M1292291 Funding: This work was supported by ANR Cin\e'-Para (ANR-15-CE23-0019) and ANR/RGC ALLOWAP (ANR-19-CE46-0013/A-HKBU203/19), by Swiss National Science Foundation grant 200020 178752, and by Hong Kong Research Grants Council (ECS 22300115 and GRF 12301817). \dagger Section of Mathematics, University of Geneva, 1211 Geneva 4, Switzerland (Martin.Gander@ unige.ch). \ddagger D\e'partement de math\e'matiques et de statistique, Universit\e' Laval, Qu\e'bec G1V 0A6, Canada, and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong (felix.kwok@mat.ulaval.ca). \S INRIA Paris, ANGE Project-Team, 75589 Paris Cedex 12, France, and Sorbonne Universit\e', CNRS, Laboratoire Jacques-Louis Lions, 75005 Paris, France (julien.salomon@inria.fr).",
year = "2020",
doi = "10.1137/19M1292291",
language = "English",
volume = "42",
pages = "A2773--A28020",
journal = "SIAM Journal of Scientific Computing",
issn = "1064-8275",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "5",
}