A parallel-in-time multiple shooting algorithm for large-scale PDE-constrained optimal control problems

•Novel multiple shooting algorithm for PDE optimal control problems.•Use of augmented Lagrangian method for equality constrained optimization.•Efficiency strongly depends on initial guess for matching conditions.•Efficiency strongly depends on proper scaling of gradients in AL subproblems.•Significant algorithmic speed-ups for up to 50 shooting windows. Multiple shooting methods for solving optimal control problems governed by ODEs have been extensively studied in past decades. However, their application for solving large-scale PDE-based optimal control problems still faces many challenges, including the difficulty of solving large scale equality constrained optimization problems in an efficient parallelizable way. The current work proposes and analyzes a new parallel-in-time multiple shooting algorithm for large scale optimal control problems governed by parabolic PDEs. We solve the equality constrained optimization problems introduced by the multiple shooting strategy by using the augmented Lagrangian method, in which the unconstrained subproblems are solved using a classical limited-memory BFGS quasi-Newton method. An optimal control problem governed by the Nagumo equation is employed to validate the proposed algorithm and analyze its efficiency. The results demonstrate that substantial accelerations can be achieved for multiple shooting approaches when proper starting guesses of controls are provided, and when the control variables are scaled appropriately. A second test case consists of a two-dimensional velocity tracking problem that is governed by the Navier–Stokes equations. The influence of the flow complexity on the optimization method is studied, and the results illustrate that for a fluid field with more complex structures, the efficiency of the algorithm further increases. Overall, for the different cases considered, we find algorithmic speed-ups of up to 6 versus single shooting, depending on the starting guess, and the specific tracking problem.