ALGORITHMIC ASPECTS OF MULTIGRID METHODS FOR OPTIMIZATION IN SHAPE SPACES

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newton updates is investigated. Both multigrid and quasi-Newton methods are necessary to achieve mesh-independent convergence and, thus, scalable algorithms for supercomputers. For the purpose of illustration, we consider a complex model for the identification of cellular structures in biology with minimal compliance in terms of elasticity and diffusion equations.