Convergence of diffusion generated motion to motion by mean curvature

Abstract

We provide a new proof of convergence to motion by mean curvature (MMC) for the Merriman-Bence-Osher (MBO) thresholding algorithm. The proof is elementary and does not rely on maximum principle for the scheme. The strategy is to construct a natural ansatz of the solution and then estimate the error. The proof thus also provides a convergence rate. Only some weak integrability assumptions of the heat kernel, but not its positivity, is used. Currently the result is proved in the case when smooth and classical solution of MMC exists.

Brakke’s inequality for the thresholding scheme

“…, which by (29) vanishes as h → 0. That means the approximate energies converge to the same limit in L 1 (0, T ) and therefore we obtain the L 1 -convergence (32) and -after the possible passage to a further subsequence -the pointwise convergences (30) and (31). The convergence of u h to χ = lim χ h can now be proven by the very same argument as the one following (43).…”

“…Since the scheme preserves the geometric comparison principle of mean curvature flow, they were able to prove convergence towards the viscosity solution of mean curvature flow. Recently, Swartz and Yip [31] proved convergence for a smooth evolution by establishing consistency and stability of the scheme, very much in the flavor of classical numerical analysis. They prove explicit bounds on the curvature and injectivity radius of the approximations and get a good understanding of the transition layer.…”

De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

“…The convergence assumption (38) is motivated by a similar assumption on the implicit time discretization in the seminal paper [25] by Luckhaus and Sturzenhecker, and has also appeared in previous work in the context of the thresholding scheme [19][20][21]. As of now, this assumption can be verified only in particular cases, such as before the first singularity [36] or for certain types of singularities, namely mean convex ones, meaning H > 0. This was shown for the implicit time discretization in [6] and a proof in the case of the thresholding scheme will appear in a forthcoming work by Fuchs and the first author.…”

“…Theorem 1 Given χ 0 ∈ A and such that ∇χ 0 is a bounded measure and a sequence h ↓ 0; let χ h be defined by (36). Assume that there exists χ :…”

An Adaptive Threshold Dynamics Method for Three-Dimensional Wetting on Rough Surfaces

“…The method is motivated by the work of S. Esedoglu and F. Otto [17] and can originate from the well-known MBO method for motion of interfaces driven by the mean curvature(socalled mean curvature flow) [30]. Theoretical studies show that the MBO method has first order convergence when the surface is smooth [20,22,27,39]. The threshold dynamics method has also been applied to many other problems [15,19,31,37,40].…”