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.
Bibliographical noteFunding Information:
The authors are grateful to Professor Robert Gulliver for helpful discussions on this work. This work was conducted while E.M. was visiting the University of Minnesota. He thanks the department for the warm hospitality and support. The research of E.M. is partially supported by the French Ministry of Education through ACI program NIM (ACI MOCEMY contract # 04 5 290). The research of F.S. is partially supported by NSF Grant DMS-0504185.
- Constrained optimization
- Differential geometry
- Inverse problem
- Level set method
- Optimal design