Elementary discrete diffusion/redistancing schemes for the mean curvature flow

1. Problem Statement

The paper addresses the numerical approximation of the Mean Curvature Flow (MCF) for boundaries of sets in $\mathbb{R}^N$. MCF is a geometric evolution equation where a hypersurface moves in the direction of its normal vector with a speed proportional to its mean curvature.

The authors focus on fully discrete schemes (discretized in both space and time). While the celebrated Merriman-Bence-Osher (MBO) scheme is a standard approach, it relies on characteristic functions and discrete heat kernels. The paper identifies limitations in existing fully discrete approaches:

• Pinning Phenomena: If the spatial discretization is not refined sufficiently relative to the time step, the interface can get "stuck" (pinned) on the grid.
• Accuracy: Characteristic functions are difficult to represent accurately at sub-grid resolutions.
• Lack of Rigorous Convergence: There is a scarcity of rigorous convergence proofs for fully discrete schemes that provide precise bounds on the ratio between the time step ($h$) and the space step ($\varepsilon$).

The goal is to design an elementary, explicit, fully discrete scheme that converges to the viscosity solution of the MCF under a standard CFL condition ($h \sim \varepsilon^2$) and to extend this to general convolution kernels and nonlinear settings (relevant to deep learning).

2. Methodology

The proposed algorithm alternates between two distinct steps on a regular grid $\varepsilon \mathbb{Z}^N$:

A. Diffusion Step (Explicit Heat Equation)

Instead of using the discrete heat kernel directly (as in MBO), the authors use the explicit Euler scheme for the discrete heat equation.
Given a signed distance function $u^k$ at time step $k$:
$$\frac{u^{k+1} - u^k}{h} = \Delta_\varepsilon u^k$$
where $\Delta_\varepsilon$ is the standard discrete Laplacian. This step is explicit and computationally simple.

B. Redistancing Step (Inf/Sup-Convolution)

After diffusion, the function $u^{k+1}$ is no longer a signed distance function. The authors apply a redistancing operator to restore the signed distance property. Unlike standard PDE-based methods (e.g., Fast Marching) which solve a Hamilton-Jacobi equation, this paper uses a "brute-force" inf/sup-convolution approach defined on the lattice:

• $d^+[u]_i = \inf_{j: u_j < 0} (u_j + |j - i|)$
• $sd^+[u]_i = \sup_{j: u_j \ge 0} (d^+_j - |j - i|)$

This operation computes the "true" discrete distance on the lattice graph (interpreted as a complete graph) rather than a graph distance on a fixed grid. This ensures the function remains 1-Lipschitz and accurately represents the interface position even at sub-grid scales.

C. Generalization to Kernels and Nonlinearity

The framework is generalized in two directions:

1. General Kernels: The diffusion step is replaced by a convolution $u^{k+1} = K_\varepsilon * u^k$, where $K_\varepsilon$ is a symmetric, non-negative kernel with mass 1. The time step $h$ is intrinsically linked to the second moment of the kernel.
2. Nonlinear Redistancing: The scheme is adapted to evolve profiles of the form $\gamma(sd)$, where $\gamma$ is a nonlinear profile (e.g., the Modica-Mortola/Cahn-Hilliard profile $\tanh$). This connects the discrete scheme to Allen-Cahn approximations and explains the success of "learned" convolution-redistancing networks in deep learning.

3. Key Contributions

• Rigorous Convergence Proof: The paper provides a complete convergence proof for a fully discrete scheme towards the viscosity solution (and distributional solution) of the Mean Curvature Flow.
• CFL Condition: The proof establishes that convergence holds under the condition $h \gtrsim \varepsilon^2$. This is shown to be the necessary threshold to prevent pinning and ensure the redistancing error does not accumulate faster than the diffusion step.
• Elementary Redistancing Analysis: The authors prove that their specific inf/sup-convolution redistancing operator is "essentially exact" on the lattice, satisfying the consistency requirements needed for convergence proofs, which are often lacking in standard numerical redistancing methods.
• Connection to Deep Learning: The paper offers a theoretical justification for fully learned approaches (neural networks mimicking convolution-redistancing) for MCF. It proves that if the learned filter approximates a valid kernel (symmetric, mass 1, correct second moment), the scheme converges to MCF even when operating on nonlinear profiles (Allen-Cahn type).
• Extension to Implicit Schemes: The framework accommodates implicit schemes (Implicit Euler) and discrete heat flow semigroups, allowing for larger time steps while maintaining convergence.

4. Main Results

• Theorem 2.3 (Basic Scheme): The sequence of signed distance functions generated by the explicit Euler + inf/sup-convolution scheme converges (up to subsequences) to the signed distance function of the unique MCF solution, provided $h \sim \varepsilon^2$.
• Theorem 4.6 (General Kernels): The convergence result extends to any convolution kernel $K_\varepsilon$ satisfying specific moment conditions (mass 1, symmetry, vanishing second moment scaling as $h$). The proof relies on the stability-monotonicity-consistency framework.
• Theorem 5.4 (Nonlinear/Deep Learning): The scheme converges even when the state variable is a nonlinear transformation $\gamma(u)$ (e.g., $\tanh(u)$), provided the convolution kernel approximates the Laplacian. This validates the use of such architectures in data-driven geometric flow simulations.
• Theorem 8.2 (Strip Restriction): The authors show that the redistancing computation can be restricted to a local "strip" around the interface (to reduce computational cost) without affecting the convergence rate, provided the strip width $M$ is sufficiently large relative to $\varepsilon$.

5. Significance and Impact

• Bridging Theory and Practice: The paper bridges the gap between the heuristic success of MBO-type algorithms and rigorous mathematical analysis. It clarifies why these schemes work and when they fail (pinning).
• Resolution of Pinning: By using signed distance functions and precise redistancing, the method avoids the pinning artifacts common in characteristic-function-based MBO schemes, provided the time step is not too small relative to the grid size.
• Theoretical Foundation for AI in Geometry: The work provides a rare rigorous convergence result for "learned" geometric flows. It demonstrates that neural networks designed to mimic diffusion-redistancing are not just black boxes but can be mathematically guaranteed to approximate MCF if their weights satisfy specific spectral properties.
• Computational Efficiency: While the "brute-force" redistancing is computationally expensive ($O(N^2)$ or similar depending on implementation), the paper suggests practical optimizations (strip restriction) and notes that the method is robust. It also highlights that standard Fast Marching methods lack the necessary error estimates for this specific theoretical framework.

In summary, this paper establishes a robust, elementary, and mathematically rigorous framework for simulating Mean Curvature Flow on discrete grids, resolving long-standing issues regarding convergence rates and pinning, and providing a theoretical basis for modern deep learning approaches to geometric evolution.