Parallel-in-time adjoint-based optimization – application to unsteady incompressible flows

Gradient-based optimization algorithms, where gradient information is extracted using adjoint equations, are efficient but can quickly slow down when applied to unsteady and nonlinear flow problems. This is mainly due to the sequential nature of the algorithm, where the primal problem is first integrated forward in time, providing the initial condition for the adjoint problem, which is then integrated backward. In order to address the sequential nature of this optimization procedure parallel-in-time algorithms can be employed. However, the characteristics of the governing equations of interest in this work, and in particular, the divergence-free constraint (incompressibility effect) as well as the nonlinearity and the unsteadiness of the flow, make direct application of existing parallel-in-time algorithms less than straightforward. In this work, we introduce a parallel-in-time procedure, applied to the integration of the adjoint problem, which addresses all the existing constraints and allows quick access to local gradients. The performance of the proposed algorithm is assessed for both steady and unsteady actuation; in both cases it readily outperforms the sequential algorithm.