Main innovation: the newly developed ParaOpt methods directly work on the coupled forward-backward problem, leading to enhancement of convergence and higher speedup ratios.
We developed two new PinT methods for solving optimal control problems, or more generally, time dependent PDE constraint optimization problems. In contrast to parallelize such problems in time by simply applying any PinT method both to the forward and backward problem in the optimization loop, we developed and analysed two new PinT methods called ParaOpt1 and ParaOpt2 which use the structure of the coupled forward and backward problems directly.
ParaOpt1 uses the fundamental understanding that the parareal algorithm is an approximation of a multiple shooting method applied to the evolution problem with an approximate Jacobian on a coarse discretization or a coarse model, and the new ParaOpt1 algorithm solves the coupled forward and backward evolution problems in the optimization loop using a parareal type correction scheme with a coarse discretization or coarse model to approximate the Jacobian. There is a technical issue we had to overcome for the coarse problem, using the derivative parareal technique, since otherwise the coarse problem would have become too costly to solve. We obtained a complete converge analysis of the new ParaOpt1 algorithm, which appeared in a joint paper in SISC with Julien Salomon and Felix Kwok. At KU Leuven, TIME-X enabled an additional contribution, unforeseen at the time of writing the proposal, that complements the above work, involving a collaboration between Stefan Vandewalle and application experts in turbulent flow simulations (Johan Meyers and Liang Fang).
For the new ParaOpt2 algorithm, we directly decomposed the forward and backward evolution problems in time first using a Schwarz domain decomposition technique. This makes sense because the problems are coupled forward and backward in time. We obtained new time parallel algorithms for time dependent optimal control and PDE constraint optimization problems based on this idea. We now also have a complete convergence analysis of this approach, and a journal manuscript is in preparation for publication with Felix Kwok. We also started working on using Dirichlet-Neumann and Neumann-Neumann techniques in the time direction for such problems.
Publications
- L. Fang, S. Vandewalle, J. Meyers (2022). “A parallel-in-time multiple shooting algorithm for large-scale PDE-constrained optimal control problems”. Journal of Computational Physics, Vol. 452 (1), Art.No. 110926.
- M. J. Gander, F. Kwok, J. Salomon (2020). “PARAOPT: A parareal algorithm for optimality systems”, SIAM Journal on Scientific Computing 42.5 (2020): A2773-A2802.