Insensitive Functionals, Inconsistent Gradients, Spurious Minima, and Regularized Functionals in Flow Optimization Problems

We use the simple context of Navier-Stokes flow in a channel with a bump to examine problems caused by the insensitivity of functionals with respect to design parameters, the inconsistency of functional gradient approximations, and the appearance of spurious minima in discretized functionals. We discuss how regularization can help overcome these problems. Along the way, we compare the discretize-then-differentiate and differentiate-then-discretize approaches to optimization, especially as they relate to the issue of inconsistent functional gradients. We close with a discussion of the implications that our observations have on more practical flow control and optimization problems.