Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation. (English) Zbl 1273.35027
Summary: 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.
MSC:
35A35 | Theoretical approximation in context of PDEs |
35K55 | Nonlinear parabolic equations |
35K65 | Degenerate parabolic equations |
49M25 | Discrete approximations in optimal control |
65K10 | Numerical optimization and variational techniques |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |