Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations

Abstract

In the past decades, multiple shooting methods have proven to be a promising direction to speed up the optimization process, especially in the context of ODE-based optimization. Very recently, Fang et al. (Journal of Computational Physics, vol. 452, 110926, 2022) proposed a multiple shooting algorithm for large-scale PDE-constrained optimization. The current paper continues this line of work and explores the potential of multiple shooting methods for optimal control problems governed by the three-dimensional Navier–Stokes equations. Similar to Fang et al., the augmented Lagrangian (AL) method is used to solve the resulting equality-constrained optimization problem, and we employ the classical limited-memory BFGS method for the unconstrained subproblems inside the AL loop. In the current work, we exploit the multiple shooting paradigm in full by processing the shooting windows parallel-in-time, allowing for significant parallel speed-ups compared to single shooting. The proposed method is validated on a velocity tracking case, using up to 100 windows. Our analysis shows that the multiple shooting algorithm allows for considerable algorithmic and parallel speed-ups. While algorithmic speed-up depends on the exact tracking case and initialization of the shooting windows, the multiple shooting algorithm always outperforms single shooting in terms of computational time (if the number of windows is sufficiently high) due to the parallel-in-time implementation. For a given amount of resources, we also show that the proposed parallel-in-time strategy can outperform spatial parallelization alone, especially when the spatial parallelization is saturated.