Numerical Sensitivity Analysis for the Quantity of Interest in PDE‐Constrained Optimization

In this paper, we consider the efficient computation of derivatives of a functional (the quantity of interest) which depends on the solution of a PDE-constrained optimization problem with inequality constraints and which may be different from the cost functional. The optimization problem is subject to perturbations in the data. We derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters. An algorithm for the efficient evaluation of these derivatives is developed, with considerable savings over a direct approach, especially in the case of high-dimensional parameter spaces. The computational cost is shown to be small compared to that of the overall optimization algorithm. Numerical experiments involving a parameter identification problem for Navier-Stokes flow and an optimal control problem for a reaction-diffusion system are presented which demonstrate the efficiency of the method.