Algorithmic aspects of multigrid methods for optimization in shape spaces

TL;DR

This paper explores how multigrid methods can be effectively integrated into shape optimization algorithms to achieve scalable, mesh-independent convergence, with applications demonstrated in biological cellular structure identification.

Contribution

It introduces a multigrid shape optimization framework that maintains mesh independence and investigates the effects of geometric approximation on algorithm performance.

Findings

Achieved mesh-independent convergence in shape optimization.

Analyzed impact of geometric quantity approximations on algorithm stability.

Applied methods to biological cellular structure identification.

Abstract

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. Our aim is a scalable algorithm for application on supercomputers, which can only be achieved by mesh-independent convergence. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newton updates is investigated. 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.