Multigrid for the Full Optimization Problem

Another approach to use multigrid method for the efficient solution of constrained optimization problems is to use it for the full problem. This means not to accelerate an already existing algorithm by accelerating one of its step but to use multigrid in a genuine way. The idea is to construct a relaxation process for the optimization problem that will have the smoothing property. That is, high frequency in the design variables will be fast to converge with this relaxation. Smooth components of the design variables will be accelerated by coarse grids. This requires the construction of a sequence of optimization problems given on a sequence of coarse grids and a proper interaction between the grids that will achieve the fast solution process required.

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.