Communications in Mathematical Sciences

Volume 9 (2011)

Number 3

Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation

Pages: 637 – 662

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n3.a1

Authors

Adam Oberman (Department of Mathematics, Simon Fraser University, Canada)

Stanley Osher (Department of Mathematics, University of California at Los Angeles)

Ryo Takei (Department of Mathematics, University of California at Los Angeles)

Richard Tsai (Department of Mathematics, University of Texas, Austin)

Abstract

Numerical methods for planar anisotropic mean curvature flow are presented for smooth and crystalline anisotropies. The methods exploit the variational level-set formulation of A. Chambolle, in conjunction with the split Bregman algorithm (equivalent to the augmented Lagrangian method and the alternating directions method of multipliers). This induces a decoupling of the anisotropy, resulting in a linear elliptic PDE and a generalized shrinkage (soft thresholding) problem. In the crystalline anisotropy case, an explicit formula for the shrinkage problem is derived. In the smooth anisotropy case, a system of nonlinear evolution equations, called inverse scale space flow, is solved. Numerical results are presented.

Keywords

Anisotropic mean curvature flow, Wulff shapes, total variation minimization, split Bregman method, shrinkage, soft thresholding, inverse scale space

2010 Mathematics Subject Classification

35K55, 35K65, 49M25, 65K10, 65M06, 65M12

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