PDE-constrained optimization with error estimation and control

This paper describes an algorithm for PDE-constrained optimization that controls numerical errors using error estimates and grid adaptation. A key aspect of the algorithm is the use of adjoint variables to estimate errors in the first-order optimality conditions. Multilevel optimization is used to drive the optimality conditions and their estimated errors below a specified tolerance. The error estimate requires two additional adjoint variables, but only at the beginning and end of each optimization cycle. Moreover, the adjoint systems can be formed and solved with limited additional infrastructure. The approach is general and can accommodate both reduced-space and full-space (i.e. one-shot) formulations of the optimization problem. The algorithm is illustrated using the inverse design of a nozzle using the quasi-one-dimensional Euler equations.