Building upon multigrid methods, the framework of multilevel optimization methods was developed to solve structured optimization problems, including problems in optimal control, image processing, etc. In this paper, we give a broader view of the multilevel framework and establish some connections between multilevel algorithms and the other approaches. An interesting case of the so called Galerkin model is further studied. By studying three different case studies of the Galerkin model, we take the first step to show how the structure of optimization problems could improve the convergence of multilevel algorithms.
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