We come now to the last issue which is the construction of a relaxation method for the optimization problem, with a property that it smoothes the error. This together with a coarse grid correction which effectively reduce the low frequency errors will result in a one-shot multigrid method for the optimization problem.
For the solution of elliptic differential equations it is well known that the discrete system should obey the so called property. Roughly speaking it means that high frequency are magnified significantly by the action of the discrete operator. This may not be true for any discretization of elliptic equations as we saw in the examples of a previous section.
If we want to construct a multigrid method for the full optimization problem we have to understand the role of relaxation in that process and how to construct the proper smoothers. The relaxation will have in general three steps,
1. Relax the state variable
2. Relax the adjoint variable
3. Relax the design variables
The first two steps here are some standard relaxation for the state and the adjoint variables which we do not discuss here. The last step which involves the update of the design variables is of a special interest. This is due to the requirement that we want to achieve a one-shot solver, meaning that the total computer time for the full optimization problem will not be larger than solving the state equations just a few times. Notice that this step requires updating the state variable as well as the adjoint variable.
In order to achieve an efficient solver we must have those updates being done very efficiently. For example, by using one or two relaxations only, and actually better if done with less than that. For a variety of problems this is indeed possible. One class of such problems is the one in which design variables introduce smooth changes in the state and adjoint variables, enabling their update on those coarse levels. Such updates on coarse levels require only a small fraction of the computer time to relax the state or costate variables on the finest level of discretization, thus resulting in fast updates for design variables. Another class of problems is that of boundary control and boundary shape design for elliptic problems. In such problems the changes in the design variables produce changes in the state and adjoint variable which are localized to a vicinity of the boundary. In that case the updates of the state and adjoint variables are done by local relaxation near the boundary only, and its cost is only a fraction of a full relaxation for the state or adjoint equation.