# The one-shot method: function space analysis and algorithmic extension by adaptivity

• Die One-Shot-Methode : Analysis im Funktionenraum und algorithmische Erweiterung durch Adaptivität

Kaland, Lena; Gauger, Nicolas Ralph (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2013)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2013

Abstract

This thesis is concerned with the function space analysis of the one-shot method as well as its algorithmic extension by adaptivity. We consider optimization problems, where the constraint is given in terms of a partial differential equation. Based on a fixed-point formulation of the state equation, it is possible to compute the first-order optimality system of the optimization problem. An adequate fixed-point formulation of the adjoint equation is given. Introducing an iterative procedure, the one-shot method can be set up. The updates of state, adjoint state and design variable operate fully simultaneously. We review the convergence analysis in finite dimensions and extend the method algorithmically by an additional adaptive step. Simultaneous to the optimization, the mesh is refined or coarsened. The variation in the dimension of the variables during the optimization asks for an analysis of the method in function spaces. We present a formulation of the one-shot method as well as a convergence analysis in a Hilbert space setting. The convergence proof follows the same idea as in finite dimensions. An augmented Lagrangian is defined, which acts as an exact penalty function. Furthermore, it can be shown, that one step of the one-shot method yields descent on the augmented Lagrangian. We consider the behavior for several model problems in more detail. The distributed control of the solid fuel ignition model enables the analysis of the method in an easy setting in terms of the fixed-point operators as well as the chosen preconditioner, which is initially chosen as a constant. The choice of the fixed-point formulation is improved for the distributed control of the viscous Burgers equation. As a last model problem the incompressible Navier-Stokes equations are analyzed. Here, the fixed-point formulation as well the preconditioner are improved. Based on a reduced SQP method coupled with a BFGS update for the reduced Hessian, we obtain the structure for the one-shot method by dropping further linearization terms. Next to the distributed control case, we also analyze the boundary control problem. All three model problems are discretized by finite elements and investigated numerically. For the solid fuel ignition model and the viscous Burgers equation various preconditioning constants and regularization parameters are tested. Clearly, the one-shot method shows a mesh-independent behavior. Therefore, for several degrees of freedom the optimization results in a similar number of iterations in total. An additional adaptive step is included in the algorithm and solutions on the adapted grids are presented. Also for the incompressible Navier-Stokes equations the one-shot method develops a mesh-independent behavior. The numerical behavior is studied for the lid driven cavity problem. Finally, the adaptive one-shot method is applied to an aerodynamic optimization problem. In particular, we consider the shape optimization of an airfoil with the state equations given by the compressible Navier-Stokes equations. First, the relevant aspects of the given simulation code PADGE are introduced. In detail, we introduce the discretization by the discontinuous Galerkin method as well as the adjoint Navier-Stokes equations. Afterwards, an iterative adjoint solver necessary for the one-shot optimization is set up and the computation of the shape derivative explained. The one-shot method is tested for the shape optimization of a NACA0012 airfoil by minimizing the drag. The results underline the positive effect of adaptivity during the optimization due to the increasing accuracy.