Convergence of the thresholding scheme for multi-phase mean-curvature flow

Abstract

We consider the thresholding scheme, a time discretization for mean curvature flow introduced by Merriman et al. (Diffusion generated motion by mean curvature. Department of Mathematics, University of California, Los Angeles 1992). We prove a convergence result in the multi-phase case. The result establishes convergence towards a weak formulation of mean curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu et al. (Commun Pure Appl Math 68(5):808-864, 2015). This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movements scheme for an energy functional that -converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren et al. (SIAM J Control Optim 31(2):387-438, 1993) and Luckhaus and Sturzenhecker (Calculus Var Partial Differ Equ 3(2):253-271, 1995), which establish convergence of a more academic minimizing movements scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which however is not ensured by the compactness coming from the basic estimates.

Existence and Uniqueness for a Crystalline Mean Curvature Flow

“…A simple proof of convergence of the Almgrem-Taylor-Wang (ATW) to the generalized motion is shown in [20], while a consistency result was already shown in [2]. See also [22] for a similar convergence proof in a more general setting (allowing for unbounded surfaces, as in the present paper), and [35] for new proofs and a generalization to partitions. Roughly speaking, it turns out that whenever fattening does not occur, the generalized level set motion coincides with the ATW flow and is also a solution in the sense of Brakke.…”

Minimizing Movements for Mean Curvature Flow of Partitions

“…Some of those solutions (e.g., the Brakke solution [15,54], the GMM solution [11], the elliptic regularization [50]) can be adapted to treat the multiphase case at least in the Euclidean case, especially those that do not rely heavily on the comparison principle. Also, the existence of a distributional solution of mean curvature evolution of partitions on the torus using the time thresholding method introduced in [46] has been proved in [41]; see also [39].…”

Diffusion generated methods for denoising target-valued images

“…This procedure provides a proof of unconditional stability and consistency of the algorithm. In [LO16], the algorithm was rigorously proved to converge to multiphase mean curvature flow with an angle constraint at the multiple junction when τ 0. A convergence proof for T = S 1 is given in [LS17].…”