Discrete Gradient Flows for Shape Optimization and Applications

Ricardo Nochetto, University of Maryland

Abstract. We present a variational framework for shape optimization problems that hinges on devising energy decreasing flows based on shape differential calculus followed by suitable space and time discretizations (discrete gradient flows). A key ingredient is the flexibility in choosing appropriate descent directions by varying the scalar products used for computation of normal velocity on the deformable domain boundary. We discuss applications to image segmentation, optimal shape design for PDE, and surface diffusion, along with several simulations exhibiting large deformations as well as pinching and topological changes in finite time. This work is joint with E. Baensch, G. Dogan, P. Morin, and M. Verani.