A matrix-free algorithm for reduced-space PDE-constrained optimization

Abstract

This thesis present a matrix-free method for partial differential equation (PDE) constrained optimization problems formulated in the reduced space. When many state-based constraints are present in the reduced-space formulation, the constraint Jacobian can become prohibitively expensive to compute explicitly, because each constraint gradient requires the solution of a distinct adjoint PDE. This leads many practitioners to use constraint aggregation, which can produce overly conservative solutions. To avoid conservative solutions as well as the expense of forming the constraint Jacobian, we adopt a matrix-free inexact-Krylov optimization framework. This choice introduces additional challenges related to globalization and preconditioning. To address globalization, the proposed method uses a homotopy continuation approach and a predictor-corrector algorithm to trace the solution curve. The predictor and corrector linear systems are solved using a Krylov iterative method with the necessary matrix-vector products evaluated via second-order adjoints. To cope with the poorly conditioned primal-dual system, a matrix-free preconditioner is proposed that uses a low-rank approximation of the Schur complement of the primal- dual matrix; the low-rank approximation is constructed using a fixed number of iterations of the Lanczos method. The algorithm is verified using analytical problems, a subset of CUTEr problems, a stress-constrained mass minimization problem, and an aerodynamic shape optimization problem. The method shows promising performance relative to a state-of-the-art matrix-based active-set algorithm, particularly for large numbers of design variables.