Speaker
Description
In recent years, multiple shooting methods have found their way from simple ODE-based optimization to tackling more intricate, moderate to large-scale PDE-based problems. By fully exploiting the multiple shooting paradigm through parallel-in-time integration of the shooting windows, it may allow for substantial parallel speed-ups, thereby accelerating the convergence of the optimization. This study applies the multiple shooting algorithm to optimal control problems governed by the three-dimensional Navier-Stokes equations. An augmented Lagrangian penalty method is used to solve the equality-constrained optimization problem arising from the multiple shooting formulation. To deal with the large-scale nature of the PDE-based problem, we employ limited-memory BFGS to solve the unconstrained subproblem in each augmented Lagrangian outer iteration. The gradient computation relies on a temporally-discrete adjoint method, where the forward and adjoint PDE evaluations are efficiently parallelized, both in space (through 3D domain decomposition) and time (through parallel-in-time integration over the shooting window).
The algorithm is validated on large-scale, tracking-type optimization problems featuring up to $10^8$ design variables and employing up to $100$ windows in the multiple shooting framework. Given the embarrassingly parallel nature of the parallel-in-time integration intrinsic to the multiple shooting approach, our framework enables significant parallel(-in-time) speed-ups. It is also shown that the convergence of the method and the resulting algorithmic speed-ups heavily depend on the initialization of the shooting windows. Nonetheless, for the optimization problems considered here, the initialization follows naturally from the tracking-type formulation and even enables algorithmic speed-ups up (independent of the parallel-in-time techniques). Finally, we demonstrate how the multiple shooting method significantly outperforms single shooting (when the number of windows is high enough), and that the proposed parallel-in-time framework surpasses spatial parallelization alone (especially when the latter is saturated), allowing for an optimal allocation of computational resources.
