Simultaneous Optimization with Unsteady Partial Differential Equations

Optimization problems subject to unsteady partial differential equations (PDEs) comprise one of the most challenging areas of applied mathematics. Gradient-based optimization schemes are typically employed, where an time-averaged quantity is reduced by iterative updates of design parameters. The adjoint approach provides a powerful tool for gradient evaluation. Yet significant computational complexities arise from repeatedly solving the unsteady dynamics and the adjoint equation in each step of conventional optimization methods. The problem becomes even more challenging when chaotic systems are considered, as adjoint-based gradients often involve the solution of a space-time boundary value problem.In order to reduce the overall runtime of conventional optimization methods, this work focuses on integrating existing unsteady PDE solvers into a simultaneous optimization framework. In particular, the simultaneous One-shot approach is pursued. The One-shot optimization algorithm incorporates adjoint-based design updates towards optimality into the process of simulating the underlying PDEs. Since common unsteady simulation codes resolve the unsteady dynamics in a forward time-marching manner, solving nonlinear equations iteratively at each time step, the transition from simulation to One-shot optimization is non-trivial. Three novel approaches to achieve this are presented in this thesis.The first approach embeds design updates into a sequence of time-marching schemes that adopt approximate solutions at each time step. In each iteration of an outer optimization cycle, the reduced time-marching scheme is enhanced by approximate adjoint and design update steps, such that feasibility and optimality are reached simultaneously. An application of the method is demonstrated for an optimal active flow control problem using an unsteady Reynolds-averaged Navier-Stokes solver. A speedup factor of three is obtained in comparison to a conventional optimization method. The second approach is concerned with the parallel-in-time One-shot optimization method. This scheme utilizes an iterative, non-intrusive multigrid algorithm applied to the time domain of unsteady time-marching schemes. Adjoint and design updates are then incorporated after each multigrid iteration. The parallel-in-time One-shot method draws its efficiency from distributing computational workload to multiple processors along the time domain. The potential of the method is demonstrated for an advection-dominated model problem. Here, a significantly higher speedup factor of $24$ is achieved in comparison to a conventional time-serial method. The third approach addresses optimization with chaotic PDEs. A reformulation of the unsteady time-marching scheme is devised that enables forward- and backward-in-time information propagation. The new formulation is able to compensate changes in the design by adjusting the initial conditions. The resulting boundary value problem is solved iteratively in space-time, such that adjoint-based design updates can be integrated naturally