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

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