  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.
