One way to look at the solution process for the optimization problem is as a solver for the necessary conditions. These are usually a set of nonlinear partial differential equation. It is expected that one can construct efficient solver for such a system. The general framework is therefore the following. We have a relaxation method for the state, costate and design variables which is a smoother (for all). The objective is to construct what we call a one-shot multigrid method. That is, a method that solves the full optimization problem in a computation cost which is 2-3 times that of solving the constraints. This efficiency should be independent of the number of design variables.
A two level algorithm has the form outlined below. A multilevel version of this method is essentially a recursive application of it.
Algorithm: Two Level
(1) Relax (smooth) the state, adjoint and design variables
(2) Accelerate convergence using a coarse grid optimization problem
The rest of this lecture will be devoted to the one-shot multigrid methods.