Simultaneous Analysis and Design in Pde-constrained Optimization a Dissertation Submitted to the Institute for Computational and Mathematical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

New methods for solving certain types of PDE-constrained optimization problems are presented in this thesis. The approach taken is to augment state-of-the-art PDE methods. The PDE variables and the optimization variables are solved for simultaneously. This impacts the PDE method by changing the core from solving a system of nonlinear equations to that of finding a nonlinear saddle point. This in turn alters the character of the linear equations that are solved at each iteration. In the problem we address, the objective function has to match a given target state. Both volume and boundary controls are considered in order to match the target. Regularization is added to the objective function to aid stability and to facilitate computing the solution of the linear systems that arise within the algorithm. Solving such linear systems has been the focus of much research, with many methods being proposed. How to do this simultaneously and efficiently for the specific systems that arise in the PDE-constrained optimization problems of interest is the main focus of our work. The new methods have been implemented by modifying the cutting-edge software AERO-S. Numerical results are presented for a variety of problems including a flapping wing, a robotic control problem, and a thermal control problem.