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Threshold dynamics for subdiffusive grain growth
Marvin Fritz
TL;DR
The paper asks how threshold dynamics can represent subdiffusive interface motion with temporal memory. It combines L1 history averaging with Helmholtz-resolvent diffusion and thresholding; in tested fixed-step runs, smaller fractional orders show faster coarsening.
Problem
The paper addresses how threshold dynamics for interface motion can incorporate the temporal memory of subdiffusion.
Source: §1 · p. 1Method
At each step, the scheme applies a Helmholtz resolvent to a convex L1 history average of past thresholded states, then thresholds the diffused field.
Source: §1 · p. 1, §6 · p. 15Results
In the tested fixed-step regime, simulations exhibit faster coarsening for smaller fractional orders, while fixed-smoothing comparisons examine the influence of L1 history across successive updates.
Source: §6 · pp. 15–16Takeaways & Limitations
The convex L1 history representation and resolvent properties yield a well-defined time-discrete construction.
Source: §6 · p. 15The sharp-interface interpretation is formal under smoothness and transport assumptions; rigorous analysis requires compactness of thresholded interfaces and control of a canonical memory term.
Source: §4.4 · p. 8
Abstract
from arXiv · showhide
We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For $\alpha=1$, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for $0<\alpha<1$ it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.
1. INTRODUCTION
The paper introduces a threshold-dynamics scheme that combines L1 temporal memory with Helmholtz-resolvent diffusion and thresholding. It establishes properties of the diffusion step and examines multiphase grain growth under changes in fractional order and smoothing length.
1. INTRODUCTION
- Classical MBO alternates heat diffusion and thresholding to approximate mean-curvature motion in the sharp-interface limit.It can also be interpreted formally as a singular operator splitting of Allen–Cahn dynamics.Source: §1 · p. 1
- The scheme applies the Helmholtz resolvent to a convex L1 history average of past thresholded states, then thresholds the result; at α = 1 it is a resolvent analogue of MBO.For 0 < α < 1, the scheme introduces subdiffusive temporal memory.Source: §1 · p. 1
- Unlike diffuse-interface approaches, the method inserts L1 history averaging into threshold dynamics and directly enforces pure phases after diffusion.The introduction places the method alongside L1 Caputo discretizations and structure-preserving Allen–Cahn schemes.Source: §1 · p. 1
- The continuous Helmholtz diffusion step has a variational characterization, stability, and scalar maximum and comparison principles.The convex L1 history representation connects these resolvent properties to the fractional thresholding construction.Source: §1 · p. 1
- Periodic equal-tension multiphase experiments use an argmax rule to form new partitions and examine coarsening, grain geometry, fractional order, and smoothing length.Each phase indicator is diffused from its own L1 history average.Source: §1 · p. 1
2. SETTING AND PRELIMINARIES
The preliminaries set the periodic spatial domain and recall the Caputo derivative’s temporal nonlocality. They develop the L1 time discretization, its history representation, and a positivity estimate used in the analysis.
2. SETTING AND PRELIMINARIES
- The analysis is set on a flat torus with periodic boundary conditions, using Lebesgue measure for spatial integrals.Unless stated otherwise, functions take values in R^N.Source: §2 · p. 2
2.1. Fractional derivative and L1 time discretization.
- The Caputo derivative depends on the entire past, and the L1 formula approximates it by piecewise-linear interpolation in time.The weights b_k=(k+1)^(1−α)−k^(1−α) are positive, nonincreasing, and asymptotically proportional to (1−α)k^(−α).Source: §2.1 · p. 2
- The L1 history representation has nonnegative coefficients that sum to one, and for α = 1 its history average reduces to the previous state.Under the limiting convention, b_0=1 and b_k=0 for k≥1, giving the implicit-Euler Helmholtz-threshold reference scheme.Source: §2.1 · p. 2
- The L1 convolution satisfies a positivity estimate for arbitrary discrete histories, with a Hilbert-space version obtained by applying the scalar inequality coordinatewise.The cited scalar inequality is from [32, Lemma 3.1].Source: §2.1 · pp. 2–3
3. A HISTORY-AWARE FRACTIONAL MBO SCHEME
The scheme evolves phase indicators by diffusing a convex L1 history average with a Helmholtz resolvent, then thresholding the result. The diffusion step is variational and satisfies stability, maximum, comparison, and mass-preservation properties.
3.1. Fractional Helmholtz-threshold scheme (two-phase).
- Each time step forms a convex combination of past thresholded states, solves a Helmholtz diffusion problem from that history average, and thresholds the resulting field.The L1 history average encodes the discrete memory associated with the Caputo derivative.Source: §3.1 · pp. 3–4, §3 · p. 3
- The Helmholtz diffusion solution is the unique minimizer of a variational functional.Strict convexity and coercivity give uniqueness, and the Euler–Lagrange equation is the diffusion equation.Source: §3.1 · p. 3
- The scalar diffusion step preserves bounds between zero and one and preserves ordering between input histories through the diffused fields and the fixed 1/2 threshold.The comparison result follows from the nonnegative L1 coefficients and monotonicity of the threshold map.Source: §3.1 · p. 4
3.2. The L1 subdiffusion structure behind thresholding.
- The subdiffusion resolvent preserves spatial averages, is L2-contractive, and satisfies maximum and comparison principles.Its solution is also characterized as a unique minimizer over H1(Ω).Source: §3.2 · pp. 4–5
- Thresholding can change phase volumes even though the preceding Helmholtz resolvent preserves the input’s spatial average.Volume-preserving variants require an additional volume constraint or a modified threshold rule.Source: §3.2 · p. 5
4. STRUCTURAL PROPERTIES AND THE ROLE OF MEMORY
The section connects the Helmholtz-resolvent threshold map to L1 temporal memory, characterizing its projection, smoothing, and history dependence. It gives memory-aware pinning criteria and a first-step curvature interpretation, with the latter remaining formal under stated interface assumptions.
4.1. Projection and resolvent structure.
- The Helmholtz step is a positivity-preserving, constant-preserving contraction, while thresholding projects the diffused field onto scalar or multiphase pure states.In the multiphase case, projection chooses an index attaining the pointwise argmax.Source: §4.1 · pp. 5–6
4.2. Where the fractional parameter enters.
- At fixed time step, changing α alters both the L1 history weights and the Helmholtz smoothing length; for h = 5 · 10−4, smaller α gives a larger length in the displayed regime.As α approaches 1, the history average approaches the most recent state, uniformly over prescribed bounded histories; extending this estimate to evolving thresholded trajectories requires control near the threshold level.Source: §4.1 · p. 6, §4.2 · p. 6
- After q unchanged recent states, the memory-tail estimate bounds the remaining influence of older states by the tail weight b_q, whose asymptotic decay is b_q ∼ (1 − α)q^-α.For phase indicators, the history values have diameter at most one.Source: §4.2 · pp. 6–7
- As α decreases to zero at fixed time step and finite time level, the initial state dominates the history average, but convergence of thresholded states requires control of the limiting tie set.The smoothing length is then comparable to the side length of the unit computational torus.Source: §4.2 · p. 7
4.3. Flat interfaces and finite-dimensional pinning.
- Exactly flat interfaces are fixed by the Helmholtz-threshold map, while a finite-dimensional pinning criterion is established under a resolvent-contraction assumption.The criterion uses a graph or finite-difference Laplacian and a sufficient bound that keeps the diffused field within one-half of the indicator in the infinity norm.Source: §4.3 · p. 7
- With q recent states unchanged, combining the memory-tail estimate with the finite-dimensional criterion gives a sufficient condition for the next thresholding step to remain pinned.For a nearest-neighbour grid Laplacian, the criterion expresses that smoothing length must be comparable to grid scale before thresholding can move an interface.Source: §4.3 · p. 7
4.4. First-step normalization and formal sharp-interface heuristic.
- The first Helmholtz-threshold step has the mean-curvature displacement corresponding to effective time ℓ²/2.This normalization follows from the local boundary-layer expansion of the Helmholtz resolvent.Source: §4.4 · p. 8
- The fractional sharp-interface interpretation is formal: it assumes smooth past interfaces transported as normal graphs, and rigorous analysis requires compactness and controlled identification of the memory term.The Caputo derivative of an interface parametrization depends on how past interfaces are transported.Source: §4.4 · p. 8
5. NUMERICAL EXPERIMENTS
Periodic finite-element simulations apply an equal-tension multiphase Helmholtz-threshold update across fractional orders, then examine coarsening, grain geometry, and history effects.
5.1. Discrete multiphase scheme used in the computations.
- Each phase is diffused from its convex L1 history average and assigned by an argmax rule, yielding an equal-tension multiphase Helmholtz-threshold scheme.The periodic computations solve one Helmholtz problem per label; the pinning criterion applies only when the discrete resolvent contraction condition is verified.Source: §5.1 · pp. 8–9, §5.2 · p. 9
5.2. Implementation details.
- The periodic finite-element runs use a structured 200 × 200 mesh, h = 5 · 10^-4, 60 initial grains, and five Voronoi seeds.Runs use 300 steps to T = 0.15; equal-physical-time snapshots share seed 42, while averaged coarsening curves use all five seeds.Source: §5.2 · p. 9
5.3. Single-grain benchmark.
- At fixed h = 5 · 10^-4, the first recorded zero cell area occurs at t = 0.028, 0.0155, 0.004, and 0.001 for α = 1, 0.9, 0.7, and 0.5, respectively.The faster loss at smaller α accompanies both increased smoothing length and changed L1 history weights; for α = 0.5, ℓ/R0 ≃ 0.782.Source: §5.3 · pp. 9–10, Figure 1 · p. 10
5.4. Structural diagnostics for the thresholding step.
- The constructed older-disk history attains the memory-tail bound exactly, and smaller orders retain a more persistent older-state contribution.At common smoothing length, the polycrystal comparison examines L1-history effects over successive updates, but order-dependent time steps give different physical time grids.Source: §5.4 · pp. 10–11, Figure 3 · p. 11
5.5. Polycrystalline coarsening.
- At fixed h, decreasing α produces faster coarsening, and by T = 0.15 every run for α ≤ 0.7 has one active grain.At T = 0.15, α = 1 averages 13.8 active grains, while α = 0.9 averages 6.4. Changing α at fixed h changes both the L1 history weights and the smoothing length.Source: §5.5 · pp. 11–12, Table 2 · p. 12, Figure 4 · p. 12
- At step 300, the initial state still has weight about 12.7% for α = 0.3 and 50.9% for α = 0.1, while one-grain partitions form in these runs.Their smoothing lengths are comparable to the unit-torus scale, placing these additional runs in a large-smoothing regime.Source: §5.5 · p. 13, Table 3 · p. 13
5.6. Grain-area and side-number statistics.
- Conditional mean area generally rises with side number, but for α = 0.7 it falls from approximately 2.094 at nine sides to 1.635 at ten sides.The ten-side class contains one grain, while the nine-side class contains eight.Source: §5.6 · p. 13
5.7. Fractional von Neumann–Mullins diagnostic.
- The computed fractional area derivative tends to be larger for grains with more sides; for 210 active grains at α = 0.3, the fitted slope is approximately 0.06374.The proposed formal fractional von Neumann–Mullins relation is conditional on a suitable discrete chain rule and a Caputo-type representation of interfacial memory.Source: §5.7 · pp. 14–15
6. CONCLUSION AND OUTLOOK
The conclusion presents a history-aware fractional MBO scheme with a well-defined time-discrete construction and reports faster coarsening for smaller fractional orders in the tested fixed-step regime. It identifies the limiting geometric law and its convergence theory as open challenges.
6. CONCLUSION AND OUTLOOK
- The scheme applies the Helmholtz resolvent to a convex L1 history average of thresholded states, then thresholds the field back to pure phases.The Helmholtz step is exactly one implicit L1 subdiffusion step.Source: §6 · p. 15
- Coarsening is faster for smaller fractional orders in the tested fixed-step regime, where both history and smoothing vary.Comparisons at fixed smoothing length examine the influence of L1 history across successive updates on order-dependent time grids.Source: §6 · pp. 15–16
- The continuous diffusion step has a variational characterization, scalar maximum/comparison principles, and resolvent stability; with deterministic thresholding, these yield a well-defined time-discrete construction.The memory-tail estimate quantifies older-state influence and gives a finite-dimensional pinning condition under a contractive discrete resolvent.Source: §6 · p. 15
- Identifying the limiting geometric law as h ↓ 0 requires a transport representation of interfacial history, compactness, a no-loss condition for interfacial area, and identification of the limiting memory term.For interfaces with memory, the geometric term I1−αV must be defined through that transport representation.Source: §6 · p. 16
- A complementary direction is sharp-interface analysis linking time-fractional Allen–Cahn dynamics to memory-dependent interface motion.Other proposed extensions include volume-preserving fractional thresholding, anisotropic kernels and surface tensions, adaptive time steps, and variational convergence.Source: §6 · p. 16