Blended Cured Quasi-Newton for Geometric Optimization

TL;DR

This paper introduces BCQN, a new geometry optimization algorithm that combines innovative techniques to improve speed, robustness, and applicability across various deformation energy problems in mesh-based simulations.

Contribution

The paper presents three novel improvements—barrier-aware line-search, energy proxy blending, and a characteristic gradient norm—that collectively form the BCQN algorithm, advancing the state of the art in geometric optimization.

Findings

BCQN is generally the fastest and most robust method across diverse problems.

It enables previously intractable problems to become practical.

Offers up to tenfold improvement in some cases.

Abstract

Optimizing deformation energies over a mesh, in two or three dimensions, is a common and critical problem in physical simulation and geometry processing. We present three new improvements to the state of the art: a barrier-aware line-search filter that cures blocked descent steps due to element barrier terms and so enables rapid progress; an energy proxy model that adaptively blends the Sobolev (inverse-Laplacian-processed) gradient and L-BFGS descent to gain the advantages of both, while avoiding L-BFGS's current limitations in geometry optimization tasks; and a characteristic gradient norm providing a robust and largely mesh- and energy-independent convergence criterion that avoids wrongful termination when algorithms temporarily slow their progress. Together these improvements form the basis for Blended Cured Quasi-Newton (BCQN), a new geometry optimization algorithm. Over a wide range of problems over all scales we show that BCQN is generally the fastest and most robust method available, making some previously intractable problems practical while offering up to an order of magnitude improvement in others.