[Paper Review] Computational aspects of multigrid methods for optimization in shape spaces

Martin Siebenborn, Kathrin Welker|arXiv (Cornell University)|Nov 16, 2016

3D Shape Modeling and Analysis3 citations

TL;DR

This paper proposes a mesh-independent multigrid shape optimization algorithm using quasi-Newton updates in shape spaces, with focus on accurate discrete geometric approximations like mean curvature. It achieves scalable convergence for large-scale problems, demonstrated on a biological cellular structure identification model governed by elasticity and diffusion equations.

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.

Motivation & Objective

• To develop a scalable, mesh-independent multigrid algorithm for shape optimization on supercomputers.
• To investigate the impact of discrete geometric approximations—particularly mean curvature—on convergence and accuracy.
• To enable efficient optimization in shape spaces for complex PDE-constrained problems.
• To apply the method to a realistic biological model involving minimal compliance in elasticity and diffusion equations.

Proposed method

• Employs multigrid methods to accelerate convergence in shape optimization within shape spaces.
• Uses quasi-Newton updates to improve convergence rate and reduce Hessian approximation errors.
• Implements discrete geometric quantities such as mean curvature using finite element approximations on adaptive meshes.
• Integrates shape calculus with multigrid cycles to maintain consistency across scales.
• Applies the algorithm to PDE-constrained optimization problems governed by elasticity and diffusion equations.
• Ensures mesh-independence by preserving convergence rates across refinement levels.

Experimental results

Research questions

• RQ1 How do discrete approximations of geometric quantities like mean curvature affect multigrid convergence in shape optimization?
• RQ2 Can multigrid methods achieve mesh-independent convergence in shape spaces with quasi-Newton updates?
• RQ3 What is the scalability of the proposed algorithm on supercomputers for large-scale shape optimization problems?
• RQ4 How does the method perform on complex biological models with minimal compliance constraints?
• RQ5 What is the role of consistent geometric discretization in maintaining algorithmic robustness?

Key findings

• The algorithm achieves mesh-independent convergence, enabling scalability on supercomputers.
• Accurate discrete mean curvature approximation is critical for maintaining convergence rates in multigrid shape optimization.
• Quasi-Newton updates significantly improve convergence behavior compared to standard methods.
• The method successfully identifies cellular structures in biology with minimal compliance under elasticity and diffusion constraints.
• Discrete geometric consistency ensures robustness and prevents convergence degradation across mesh refinements.
• The approach demonstrates practical applicability to complex, real-world biological modeling problems.