[Paper Review] Numerical Methods for Biomembranes: conforming subdivision methods versus non-conforming PL methods

Jingmin Chen, Thomas Yu+4 more|arXiv (Cornell University)|Jan 28, 2019

Advanced Numerical Analysis Techniques64 references4 citations

TL;DR

This paper compares conforming subdivision surface (SS) methods and non-conforming piecewise linear (PL) methods for solving geometric variational problems in biomembrane modeling, particularly the Willmore energy minimization. It proves that PL methods based on discrete Willmore energies fail to converge to smooth Willmore minimizers, while SS-based minimization succeeds due to higher-order accuracy and conformality, leading to a proposed regularization for PL methods using conformal geometry.

ABSTRACT

The Canham-Helfrich-Evans models of biomembranes consist of a family of geometric constrained variational problems. In this article, we compare two classes of numerical methods for these variational problems based on piecewise linear (PL) and subdivision surfaces (SS). Since SS methods are based on spline approximation and can be viewed as higher order versions of PL methods, one may expect that the only difference between the two methods is in the accuracy order. In this paper, we prove that a numerical method based on minimizing any one of the `PL Willmore energies' proposed in the literature would fail to converge to a solution of the continuous problem, whereas a method based on minimization of the bona fide Willmore energy, well-defined for SS but not PL surfaces, succeeds. Motivated by this analysis, we propose also a regularization method for the PL method based on techniques from conformal geometry. We address a number of implementation issues crucial for the efficiency of our solver. A software package called Wmincon accompanies this article, provides parallel implementations of all the relevant geometric functionals. When combined with a standard constrained optimization solver, the geometric variational problems can then be solved numerically. To this end, we realize that some of the available optimization algorithms/solvers are capable of preserving symmetry, while others manage to break symmetry; we explore the consequences of this observation.

Motivation & Objective

• To analyze the convergence behavior of piecewise linear (PL) and subdivision surface (SS) methods in solving geometric variational problems for biomembranes.
• To identify why PL-based Willmore energy minimization fails to converge to smooth solutions, despite theoretical consistency.
• To propose a regularization technique for PL methods using conformal geometry to improve convergence properties.
• To develop and release a parallel software package, Wmincon, for computing geometric functionals on PL and SS surfaces.
• To explore symmetry preservation and breaking in optimization solvers applied to Willmore-type problems.

Proposed method

• Formalize the Willmore energy as a continuous geometric functional and define discrete analogues for PL and SS surfaces.
• Prove that any consistent PL Willmore energy minimization fails to converge to a smooth Willmore minimizer, due to lack of conformality and insufficient regularity.
• Introduce a regularization for PL methods based on conformal parametrization and discrete conformal geometry to stabilize convergence.
• Implement geometric functionals (Willmore, area, volume, mean curvature) in a parallel C++ software package called Wmincon.
• Use standard constrained optimization solvers with Wmincon to numerically solve Canham-Helfrich-Evans problems.
• Analyze symmetry behavior of optimization solvers, distinguishing between symmetry-preserving and symmetry-breaking algorithms.

Experimental results

Research questions

• RQ1 Why do PL-based discrete Willmore energy minimizations fail to converge to smooth Willmore minimizers, despite consistency?
• RQ2 Can a regularization based on conformal geometry restore convergence for PL methods in Willmore energy minimization?
• RQ3 How do different optimization solvers affect the symmetry of solutions in geometric variational problems?
• RQ4 What is the role of conformality and higher-order approximation in ensuring convergence to smooth solutions?
• RQ5 Can the proposed regularization technique be generalized to other non-conforming finite element methods in geometric PDEs?

Key findings

• Any consistent discrete Willmore energy defined on PL surfaces fails to converge to a smooth Willmore minimizer, as proven in Proposition 3.9 and 3.12.
• Subdivision surface methods, due to their higher-order approximation and conformality, successfully converge to smooth Willmore minimizers, unlike PL methods.
• The proposed regularization for PL methods, based on conformal geometry, restores convergence behavior and improves numerical stability.
• The software package Wmincon enables efficient parallel computation of geometric functionals on both PL and SS surfaces, facilitating numerical solution of variational problems.
• Some optimization solvers preserve symmetry and converge to symmetric minimizers, while others break symmetry and converge to asymmetric solutions, affecting the reliability of numerical results.
• Theoretical analysis suggests that the space of PL surfaces is too large and irregular to support consistent convergence to smooth Willmore minimizers, even for genus 0.