Level set methods for optimization problems involving geometry and constraints II. Optimization over a fixed surface

Authors:
Emmanuel Maitre;Fadil Santosa
Affiliations:
Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, F-38041 Grenoble Cedex 9, France;School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455, USA
Venue:
Journal of Computational Physics
Year:
2008

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Abstract

In this work, we consider an optimization problem described on a surface. The approach is illustrated on the problem of finding a closed curve whose arclength is as small as possible while the area enclosed by the curve is fixed. This problem exemplifies a class of optimization and inverse problems that arise in diverse applications. In our approach, we assume that the surface is given parametrically. A level set formulation for the curve is developed in the surface parameter space. We show how to obtain a formal gradient for the optimization objective, and derive a gradient-type algorithm which minimizes the objective while respecting the constraint. The algorithm is a projection method which has a PDE interpretation. We demonstrate and verify the method in numerical examples.