Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces

Akwum Onwunta; Ludwig Striet; Patrick Dondl; Stephan Wojtowytsch

REVIEW 4 major objections 5 minor 1 cited by

Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Convex-concave splitting for the Allen-Cahn equation moves interfaces on an ε² time scale, not the nominal step size τ.

desk verdict Solid new MBO link for the τ=∞ quadratic case, but the universal ε²-slowness claim is only partly proved; refereeing it with a toned-down abstract is the right call. read the letter →

arxiv 2506.18869 v1 pith:FTHPOKQA submitted 2025-06-23 math.NA cs.NAmath.AP

Patrick Dondl , Akwum Onwunta , Ludwig Striet , Stephan Wojtowytsch This is my paper · ORCID

classification math.NAcs.NAmath.AP MSC 65M1235A3549Q05

keywords Allen-Cahnequationconvex-concavesplittingMBOschememeancurvaturefloweffectivetimestepinterfacedynamicsenergystabilitythresholding

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the convex-concave splitting time-stepping scheme for the Allen-Cahn equation, popular for being unconditionally energy stable and easy to implement, suffers from a severe hidden limitation: for a range of double-well potentials, interfaces move on a time scale proportional to ε², where ε is the width of the transition layer. Choosing a large nominal time step τ does not accelerate the dynamics; stability is bought by effectively freezing the interface in place. For potentials with a quadratic convex part, the authors prove this by showing that the infinite-step iteration is exactly a Merriman–Bence–Osher thresholding step of size ε²/2, linking the Allen-Cahn scheme to mean curvature flow. The same ε² scaling is shown by explicit example for barrier potentials and numerically for the standard potential, and a weaker $ε^{{1/(p-1)}}$ bound is proved in general.

What carries the argument

The central object is the equivalence between the infinite-step-size convex-concave splitting iteration and thresholding dynamics: (1 - τΔ)$u^{{n+1}}$ = sign(u^n) with τ = ε²/2 is the screened-Poisson (Yukawa) kernel iteration K_τ * sign(u^n), which is the MBO scheme for a first-order heat approximation. The second machinery is Lemma 3.2, an energy-dissipation estimate giving a Hölder bound ∥u^N - $u^{0}$∥_{L^p} ≲ (N $ε^{{1/(p-1)}}$)^{1-1/p}, independent of τ, which forces slowness for potentials with convex part |u|^p. The third is the explicit double-obstacle solution for the barrier potential, where r_new = r + O(ε²).

What would settle it

Run the convex-concave splitting scheme on a shrinking circle in a periodic square with a large fixed nominal step τ for two values of ε that differ by a factor of two, and measure the number of steps needed to shrink the radius by a fixed fraction. If the effective step scales as ε², the step counts differ by a factor of four; if the interface moves at the nominal τ pace, the counts are nearly equal.

Watch

Extended reading notes

Core claim

For the prototypical potential with quadratic convex part, W(u) = (|u|-1)², the convex-concave splitting iteration with formally infinite time step reduces to $u^{{n+1}}$ = (1 - (ε²/2)Δ)^{-1} sign(u^n), which is a first-order implicit Euler approximation to the heat equation applied to the thresholded previous state — i.e. an MBO thresholding step with time step ε²/2. On the whole space ℝ³, the Green's function of (1 - (ε²/2)Δ) is radially symmetric with finite second moment and satisfies the Ishii–Pires–Souganidis conditions, so the iteration is a time discretization of mean curvature flow at speed ε²/2. Hence the effective time step of the scheme is ε²/2 regardless of nominal τ. For barrier potentials, an explicit double-obstacle calculation in a ball shows the zero-level set shifts by O(ε²) per step; numerical experiments with the standard potential confirm the same scaling.

Load-bearing premise

The ε²-slow motion for the practically used cases — the standard potential, bounded domains with periodic or Neumann boundary conditions, and finite time steps — rests on the assertion, stated but not proved in Section 4.2, that the whole-space thresholding analysis extends to those settings; for the standard potential, the scaling is only observed in numerics, not derived.

Editorial extensions

If this is right

For potentials with quadratic convex part, the scheme is literally a first-order approximation to the MBO thresholding scheme, so its interface motion is mean curvature flow on the ε² scale, no matter how large the nominal step τ.
The ε² scaling is independent of τ, so increasing the time step cannot buy faster interface dynamics; the energy-decreasing property is obtained by freezing the interface.
For barrier potentials, the explicit double-obstacle example shows the same ε² slowdown, and the numerical experiment with the standard potential indicates the behavior is universal.
A general energy estimate implies that even when the convex part grows like |u|^p, the effective step is at most ε^{1/(p-1)}, so a wide family of convex-concave splittings are slow compared to the desired mean-curvature time scale.
The scheme does not improve on a simpler semi-implicit treatment (implicit Laplacian, explicit double-well) with a sufficiently small step, and it can be slower in practice because each step may require a new convex minimization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The MBO equivalence suggests that quantitative convergence-rate results for thresholding dynamics (for example, in L¹) could transfer to this Allen-Cahn scheme, with an additional error of order ε² per step, once the missing analysis on bounded domains is supplied.
The ε² slowness likely holds for any convex-concave splitting whose convex part has positive second derivative at the wells: evaluating the concave force at the previous iterate produces a fixed force that resists translation of the profile quadratically in the shift, an effect that does not vanish with larger τ.
A testable design consequence is that a semi-implicit scheme with the double-well term treated explicitly and the Laplacian implicitly would move interfaces at the same ε²-limited speed but at a fraction of the per-step cost, making the convex-concave splitting strictly dominated for computing real interface dynamics.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the paper gives a genuinely new structural identification: for the quadratic convex part and τ=∞, the convex-concave splitting iteration is exactly an MBO thresholding step of size ε²/2. On R³ they verify the IPS99 conditions in Appendix B, so that link is rigorous and it is the best part of the paper. Second, the headline claim—that the effective time step always scales like ε²—is not proved at that level of generality. Bounded domains, finite τ, and the standard potential are asserted or numerical. The authors flag most of this in the body, but the abstract sells the universal conclusion.

Credit where due: Lemma 3.2 and Corollary 3.3 give a clean, correct energy-dissipation bound: each step dissipates ε/2∥δu∥²_H¹ + (ε/τ)∥δu∥²_L² + (c̄/ε)∥δu∥^p_{L^p}, and the Hölder argument yields the ε^{1/(p-1)} slowness bound independent of τ. That's a solid, citable estimate. The barrier-potential example in Section 5 is also a real computation: for the radial double obstacle, r_new − r = O(ε²), shown by an explicit inner-variation argument. It gives a nontrivial data point in a regime where the curvature conditions fail.

Soft spots, in proportion. The bounded-domain claim in Section 4.2 is an assertion, not a proof: the resolvent kernel is not translation-invariant on a periodic or Neumann domain, and IPS99 is whole-space theory. The finite-τ update has effective step τ/(1+2τ/ε²), so saturation at ε²/2 is plausible, but no theorem shows the interface speed is ε²/2 there; the authors admit they haven't ruled out a sweet spot. For the standard potential, the ε²-scaling is a numerical fit with prefactor 0.25·ε², and no code or data is shipped. I'd call these addressable, not fatal. The practical message—large τ does not buy speed—is supported by Figure 3 and the numerics, but the universal statement needs a rewrite or a proof.

Who is it for: anyone doing phase-field simulations with unconditional stability schemes, and especially people working on MBO-type thresholding. The MBO link is the sort of bridge that will generate follow-up work.

Recommendation: send it to peer review. The rigorous core is new and sound; the framing needs work. I'd push for either proving the bounded-domain/finite-τ statements or labeling them explicitly as conjectural, and for shipping the numerics. My own verdict is 'conditional'—engage with it, but push for honesty in the abstract.

Referee Report

4 major / 5 minor

Summary. The paper studies the convex-concave splitting time discretization of the Allen-Cahn equation, in which the double-well potential is decomposed into a convex and a concave part and the two parts are treated implicitly and explicitly, respectively. The main claimed result is that the scheme's 'effective time step' scales as ε^2, where ε is the interface width, so that unconditional energy stability is achieved only at the price of effectively freezing the interface. The paper contains three strands: (i) a general energy-dissipation estimate (Lemma 3.2 and Corollary 3.3) showing that for potentials with a uniform p-curvature condition, N steps cannot move the solution by more than O((N ε^{1/(p-1)})^{1-1/p}); (ii) for potentials with quadratic convex part, a link between the infinite-step-size iteration (4.4) and the Merriman-Bence-Osher thresholding scheme, with a rigorous identification on the whole space R^3 via the IPS99 convergence theorem, giving effective step ε^2/2; and (iii) evidence for the same ε^2 scaling for barrier potentials (via an explicit radial example) and for the standard quartic potential (via numerical experiments). The authors are explicit that the analysis is only fully rigorous on R^3 and that the standard-potential claim is numerical, but the abstract and title present the ε^2-slowness as a universal conclusion.

Significance. If the ε^2-slowness claim is correct in the generality stated, it is an important and somewhat sobering message for the phase-field community: the popular convex-concave splitting scheme, often advertised as unconditionally stable, achieves that stability by drastically limiting the effective time step, so large nominal time steps do not accelerate interface motion. The rigorous components of the paper are clean and valuable: the energy-dissipation bound in Lemma 3.2/Corollary 3.3 is elementary but sharp in its ε-scaling, Lemma 4.3 gives a variational characterization of the thresholding step, and Appendix B carefully verifies the IPS99 hypotheses for the screened Poisson kernel on R^3. The paper is also commendably honest in flagging its own limitations, explicitly stating that the MBO equivalence is only rigorous on R^3 and that no finite-τ 'sweet spot' has been excluded. However, the headline universal claim is not supported by the proofs as written: the rigorous kernel analysis covers one special potential, τ=∞, and the whole space, while the standard-potential claim is purely numerical and the barrier-potential claim rests on a single explicit example.

major comments (4)

[§4.2 and Appendix B] The ε^2-thresholding equivalence is rigorously established only for the potential W(u)=(|u|-1)^2, for formally infinite step size τ=∞, and on the whole space R^3. In that setting, (4.4) is rewritten as convolution with the screened Poisson kernel and Appendix B verifies the hypotheses of [IPS99] for that kernel. The sentence immediately after (4.4) asserting that 'standard proofs apply' to bounded domains with periodic or Neumann boundary conditions, and to finite τ, is not a proof: on a bounded domain the Green's function of (1-ε^2/2 Δ)^{-1} is not translation-invariant, and the IPS99 theorem is formulated on R^d. For finite τ the update is u^{n+1}=(1+2τ/ε^2-τΔ)^{-1}(u^n+(2τ/ε^2)sign u^n), whose natural diffusion scale is τ/(1+2τ/ε^2), and the paper itself states that no finite-τ sweet spot has been rigorously excluded. Since the abstract and title assert ε^2-slowness for the full class of potentials and domains, this gap is load-bearing. I request either a proof or a precise theorem for bounded domains and finite τ, or a reformulation of the title, abstract, and conclusions so that the rigorous claim is restricted to the setting actually proved, with the bounded-domain and finite-τ statements explicitly labeled as conjectural.
[§6, Figure 7] The claim that the standard potential W(u)=(u^2-1)^2 is ε^2-slow rests entirely on numerical fits, with an effective step of 0.25 ε^2, at a single spatial resolution n=512 and with Newton-Raphson solves for the convex subproblem. No data, code, error bars, or systematic convergence study in ε are provided. Since the standard potential is the practically most important case and is presented in the abstract as one of the three scenarios, this evidence is not sufficient to support the universal statement. Please either supply reproducible data/code and a convergence study, or explicitly downgrade the standard-potential claim to a numerical observation in the abstract and conclusions.
[§5 and Appendix D] For barrier potentials, the ε^2 conclusion is supported by one explicit τ=∞ radial example (Example 5.2), not by a convergence theorem for the iteration. The text says the authors 'demonstrate by example' and Figure 6 shows the displacement r_new-r as O(ε^2), but the abstract counts barrier potentials among the scenarios where the effective time step scales as ε^2. Moreover, the derivation in Appendix D of r_new-r=O(ε^2) uses unquantified 'err' terms and informal 'easy to see' boundedness statements, so even this single example is not fully rigorous as written. Please state precisely what is proved for barrier potentials, what is observed numerically, and what remains conjectural.
[§1 and §4.2] The central notion of 'effective time step size' is never defined formally. In the rigorous MBO setting it can be identified with the thresholding time h=ε^2/2, but in the general slow-motion bound (Corollary 3.3) and in the numerical sections it is inferred informally from the number of iterations required to move the interface a distance of order one. Without a precise definition, statements such as 'the effective time step size scales as ε^2' are not quantitatively falsifiable and cannot be compared across the three potential classes. I recommend giving a formal definition, for example in terms of the number of iterations needed to achieve a prescribed interface displacement, and then stating each theorem in those terms.

minor comments (5)

[Page 2, Section 1] The phrase 'certain applications certain applications' contains a duplicated word; it should read 'certain applications'.
[Title and running header] The title contains spacing and hyphenation artifacts: 'CONVEX-CONCA VE' and 'EQUA TION' should be cleaned up.
[§7, heuristic discussion] In the curvature heuristic, the text says 'If κ>0, there is an incentive to move the transition where u'≫1 to the left; while κ>0 incentivizes transitions further to the right'; the second occurrence should presumably be κ<0, since the two statements are contradictory as written.
[Figure 1 and Figure 3] The effective-step prefactors 0.29 and 0.5 are stated without explanation of how they were obtained; a sentence describing the fitting procedure would improve reproducibility.
[Appendix D] The notation in the displayed inner-variation calculation would benefit from a definition of the error term 'err' and from a short justification of why the neglected terms are higher order in ε; as written the O(ε^2) bound is plausible but not fully quantified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ε²/2 time scale is derived from scheme algebra plus the external IPS99 thresholding theorem, not from the conclusion being fitted or defined in.

full rationale

The paper's central derivation is self-contained against an external benchmark. For the quadratic-convex potential W=(|u|−1)², Section 4.2 algebraically computes the τ=∞ convex-concave splitting iterate as u^{n+1} = (1 − (ε²/2)Δ)^{-1} sign(u^n), and then observes that on R³ this operator is convolution with the screened Poisson kernel K_τ/τ for τ=ε²/2. Appendix B explicitly verifies the hypotheses of the external thresholding convergence theorem of Ishii–Pires–Souganidis (IPS99), showing the iteration is an MBO-type discretization of mean curvature flow with time step ε²/2. This is not circular: the ε²/2 step size is a computed consequence of the scheme and a cited external theorem, not an input or a fitted parameter. The self-citations ([ADG+24], [Woj23]) are background references for convex-concave splitting and basic gradient-descent bounds, and are not load-bearing for the ε² conclusion. The paper honestly discloses the main limitation: the rigorous MBO identification is carried out only on the whole space R³ and for formally infinite step size, while bounded domains, finite τ, and the standard potential rely on an asserted extension and numerical observation. Those are completeness and rigor caveats, not circular reductions. The numerical prefactors 0.29, 0.25, and 0.5 used in the figures are empirical plotting conventions chosen to compare with the analytic shrinking-circle solution; they do not force the ε² scaling exponent, and they are not presented as derived predictions. In summary, no step in the derivation chain is equivalent to its own input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard phase-field assumptions (double-well potential, self-adjoint elliptic resolvents), the external IPS99 thresholding convergence theorem, an unproved bounded-domain/finite-tau extension, and three empirical prefactors used only in the numerical displays. No new entities are postulated.

free parameters (3)

effective step prefactor for quadratic convex part = 0.29
Used in Figure 1 to rescale time as 0.29·epsilon^2 so that the numerical interface shrinkage matches the analytic MCF circle solution. The constant is chosen to match the known solution, not derived.
effective step prefactor for standard potential = 0.25
Used in Figure 7 (left) with effective step 0.25·epsilon^2 to display agreement with MCF for W=(u^2-1)^2.
effective step prefactor for barrier potential = 0.5
Used in Figure 7 (right) with effective step 0.5·epsilon^2 to display agreement with MCF for the barrier potential.

assumptions (6)

domain assumption The double-well potential W satisfies W>=0, W=0 only at u=±1, and admits a convex-concave splitting W=W_vex+W_conc with W_conc concave and the joint uniform curvature condition (4) of Section 3.1.
This is the class of potentials for which the theorems in Section 3 are proven; the results are conditional on this assumption.
standard math The operator (a-b Delta)^{-1} is densely defined, positive semi-definite, and self-adjoint on L^2(Ω) for a,b>=0, a+b>0 under Dirichlet, Neumann, or periodic boundary conditions.
Stated in Section 1.1 and used throughout to define the time-step map and the thresholding iteration.
standard math The thresholding scheme with radially symmetric kernels converges to mean curvature flow when the kernel satisfies the IPS99 conditions.
External theorem of Ishii-Pires-Souganidis [IPS99], invoked in Section 4.2 and verified for the screened-Poisson kernel in Appendix B.
ad hoc to paper On bounded domains and for finite tau, the iteration (4.4) still behaves as an MBO scheme with effective step epsilon^2/2.
Asserted in Section 4.2 with 'standard proofs' and explicitly left as beyond the scope of the study. This unproved extension carries the abstract's universal epsilon-squared conclusion.
standard math The explicit double-obstacle solution in Example 5.2 and Appendix D relies on the well-posedness and regularity theory of double obstacle problems.
The paper cites [KS00], [LPS19], and [Ger85] for existence and C^{1,1} regularity of the obstacle problem.
domain assumption The numerical solvers (Newton-Raphson for the standard potential, OSQP for the barrier potential) converge to the true minimizer on the 512x512 grid.
Solver convergence is not verified beyond visual inspection; this is standard practice in numerical experiments.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces." pith.science (2026). https://pith.science/paper/FTHPOKQA

@misc{pith2026250618869,
  author       = {Pith},
  title        = {Pith review of: Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FTHPOKQA}},
  note         = {Machine review of arXiv:2506.18869}
}

abstract

The convex-concave splitting discretization of the Allen-Cahn is easy to implement and guaranteed to be energy decreasing even for large time-steps. We analyze the time-stepping scheme for a large class of potentials which includes the standard potential as well as two extreme settings: Potentials with quadratic convex part (uniform positive curvature), and potentials which are concave between the potential wells and either linear or infinite outside (highly concentrated curvature). In all three scenarios, the 'effective time step size' of the scheme scales with the square of the small parameter $\varepsilon$ governing the width of transition layers. A weaker 'slow motion' result is proved under much more general assumptions. Thus, stability is achieved by effectively 'freezing' the interfaces in place. The time step limitation is not geometric in origin, but depends on the phase-field parameter $\varepsilon$. Along the way, we establish a new link between an Allen-Cahn type equation and a thresholding approximation of mean curvature flow.

Figures

Figures reproduced from arXiv: 2506.18869 by the authors.

**Figure 1.** Comparing the (normalized) Modica-Mortola energies c −1 W Eε along the convex-concave splitting time-stepping scheme for τ = 100 (left) and τ = 100, 000 (right). The initial condition is 2 · χE − 1 where E is a circle of radius r0 = 0.4 in the square (0, 1)2 with periodic boundary conditions. The potential is WR for R = 100 as in Appendix A. In each plot, the times are computed not according to the ‘nominal step-siz… view at source ↗

**Figure 2.** Left: f(x) + 1 2τ ∥x − z∥ 2 for f(x) = − cos(πx), τ = 1.5 (red line) and z = 0.5 (vertical green line). The minimizer x ≈ 0.03 (marked in purple) is the solution of the ‘minimizing movements’ scheme. Middle: x + τf ′ (x) as well as z = 0.5 (blue line). The intersections of the lines are the spurious solutions to the implicit Euler gradient descent equation x+τf′ (x) = z starting at z (local extrema in the left plot)… view at source ↗

**Figure 3.** The number of convex-concave splitting gradient descent iterations required until Eε falls below a given level with ε = 0.1 (left), ε = 0.05 (middle) and ε = 0.01 (right) as a function of time step size. Initially, increasing the step size reduces the number of iterations required, but the effect tapers of at a (surprisingly small) finite threshold. The threshold is lower for smaller ε. The setting is otherwise as … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Solutions to the time-stepping problem of Example 5.2 for the evolution of a ball with radius 2 in radial direction in dimension d = 3 (left two) and dimension d = 10 (right two) with varying values of ε. In the first and third plot, we select ε = 0.2 large, in the sec…

**Figure 5.** Figure 5: Solutions to the time-stepping problem of Example 5.2 with barrier potentials, zoomed in to length scale ε, as well as suitably scaled versions of the optimal transition, centered at the x-axis intercept of u. The initial condition is a ball of radius r = 2 in three di…

**Figure 6.** Figure 6: r −ri,ε (left), ro,ε −r (middle) and r −rnew,ε (right) as a function of ε for various dimensions. The dashed black line ψ(ε) = ε is given for reference. Clearly, the distance of both ri and ro from r grows linearly with ε, independently of dimension. This is to be expe…

**Figure 7.** Figure 7: Left: The convex-concave splitting time-stepping scheme the standard potential W(u) = (u 2 − 1)2 . Times are computed according to an ‘effective step size’ 0.25 · ε 2 . Right: The convex-concave splitting time-stepping scheme the barrier potential W(u) = 1 − u 2 + ∞ · …

**Figure 8.** Figure 8: Left: The doublewell potential WR for varying values of R. Right: WR and its first two derivatives for R = 1. = u 2 + 1 − 2 lim R→∞ r R(R + 1) R2 u 2 + R + 1 R2 ! = u 2 + 1 − 2|u|, i.e. we can see WR as a smooth approximation to W – see [PITH_FULL_IMAGE:figures/full_f…

**Figure 9.** Figure 9: The auxiliary function ξε for fixed ε and varying dimension (left two) and fixed dimension, and varying ε (right two). The vertical lines represent ri , the intersection with the x-axis. which yields that (ri/r) d + (ro/r) d = 2. The continuity condition becomes 1 + r …

**Figure 5.** Figure 5: Since r − 2ε < ri,ε < rnew,ε < r, this yields the additional guarantee that |rnew,ε − r| < 2ε, i.e. the motion is ε-slow. No better result can be expected from estimating ri,ε since the transition between the potential wells 1, −1 must happen on a length-scale ro,ε − r…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs
math.AP 2024-12 conditional novelty 7.0 of 10

The accelerated Allen-Cahn equation formally converges to the hyperbolic interface law ∂_t v = (1-v^2)(h-αv), and a large-step FISTA discretization empirically accelerates Ginzburg-Landau minimization.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages · cited by 1 Pith paper

[1]

Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs

Oluwatosin Akande, Patrick Dondl, Kanan Gupta, Akwum Onwunta, and Stephan Wojtowytsch. Momentum-based minimization of the ginzburg-landau functional on euclidean spaces and graphs. arXiv preprint arXiv:2501.00389 , 2024

work page Pith review arXiv 2024
[2]

Acceleration by stepsize hedging: Silver stepsize schedule for smooth convex optimization

Jason M Altschuler and Pablo A Parrilo. Acceleration by stepsize hedging: Silver stepsize schedule for smooth convex optimization. Mathematical Programming , pages 1--14, 2024

work page 2024
[3]

Acceleration by stepsize hedging: Multi-step descent and the silver stepsize schedule

Jason M Altschuler and Pablo A Parrilo. Acceleration by stepsize hedging: Multi-step descent and the silver stepsize schedule. Journal of the ACM , 72(2):1--38, 2025

work page 2025
[4]

Numerical methods for nonlinear partial differential equations , volume 47

S \"o ren Bartels. Numerical methods for nonlinear partial differential equations , volume 47. Springer, 2015

work page 2015
[5]

Motion by mean curvature as the singular limit of ginzburg-landau dynamics

Lia Bronsard and Robert V Kohn. Motion by mean curvature as the singular limit of ginzburg-landau dynamics. Journal of differential equations , 90(2):211--237, 1991

work page 1991
[6]

Generalizing diffuse interface methods on graphs: nonsmooth potentials and hypergraphs

Jessica Bosch, Steffen Klamt, and Martin Stoll. Generalizing diffuse interface methods on graphs: nonsmooth potentials and hypergraphs. SIAM Journal on Applied Mathematics , 78(3):1350--1377, 2018

work page 2018
[7]

Approssimazione variazionale di funzionali con curvatura

Giovanni Bellettini and Maurizio Paolini. Approssimazione variazionale di funzionali con curvatura. In Seminario di Analisi Matematica , pages 87--97. Tecnoprint, 1993

work page 1993
[8]

On the slowness of phase boundary motion in one space dimension

Lia Bronsard. On the slowness of phase boundary motion in one space dimension. NASA STI/Recon Technical Report N , 91:21480, 1990

work page 1990

Show all 33 references

[9]

Graph mbo as a semi-discrete implicit euler scheme for graph allen--cahn

Jeremy Budd and Yves van Gennip. Graph mbo as a semi-discrete implicit euler scheme for graph allen--cahn. arXiv preprint arXiv:1907.10774 , 2019

1907 arXiv
[10]

Classification and image processing with a semi-discrete scheme for fidelity forced allen--cahn on graphs

Jeremy Budd, Yves van Gennip, and Jonas Latz. Classification and image processing with a semi-discrete scheme for fidelity forced allen--cahn on graphs. GAMM-Mitteilungen , 44(1):e202100004, 2021

2021
[11]

Metastable patterns in solutions of u_t= ^2u_ xx - f (u)

Jack Carr and Robert L Pego. Metastable patterns in solutions of u_t= ^2u_ xx - f (u) . Communications on pure and applied mathematics , 42(5):523--576, 1989

1989
[12]

Phase field models for thin elastic structures with topological constraint

Patrick W Dondl, Antoine Lemenant, and Stephan Wojtowytsch. Phase field models for thin elastic structures with topological constraint. Archive for Rational Mechanics and Analysis , 223(2):693--736, 2017

2017
[13]

Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen

Manfred Dobrowolski. Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen . Springer-Verlag, 2010

2010
[14]

Uniform regularity and convergence of phase-fields for willmore’s energy

Patrick W Dondl and Stephan Wojtowytsch. Uniform regularity and convergence of phase-fields for willmore’s energy. Calculus of Variations and Partial Differential Equations , 56(4):90, 2017

2017
[15]

Threshold dynamics for networks with arbitrary surface tensions

Selim Esedoglu and Felix Otto. Threshold dynamics for networks with arbitrary surface tensions. Communications on pure and applied mathematics , 68(5):808--864, 2015

2015
[16]

Slow-motion manifolds, dormant instability, and singular perturbations

Giorgio Fusco and Jack K Hale. Slow-motion manifolds, dormant instability, and singular perturbations. Journal of Dynamics and Differential Equations , 1:75--94, 1989

1989
[17]

Traveling waves as limits of solutions on bounded domains

Giorgio Fusco, Jack K Hale, and Jianping Xun. Traveling waves as limits of solutions on bounded domains. SIAM Journal on Mathematical Analysis , 27(6):1544--1558, 1996

1996
[18]

Convergence rates of the A llen-- C ahn equation to mean curvature flow: A short proof based on relative entropies

Julian Fischer, Tim Laux, and Theresa M Simon. Convergence rates of the A llen-- C ahn equation to mean curvature flow: A short proof based on relative entropies. SIAM Journal on Mathematical Analysis , 52(6):6222--6233, 2020

2020
[19]

Global c^ 1, 1 -regularity for solutions of quasilinear variational inequalities

Claus Gerhardt. Global c^ 1, 1 -regularity for solutions of quasilinear variational inequalities. Archive for rational mechanics and analysis , 89(1):83--92, 1985

1985
[20]

Accelerated objective gap and gradient norm convergence for gradient descent via long steps

Benjamin Grimmer, Kevin Shu, and Alex L Wang. Accelerated objective gap and gradient norm convergence for gradient descent via long steps. INFORMS Journal on Optimization , 2025

2025
[21]

Convergence of the A llen- C ahn equation to B rakke's motion by mean curvature

Tom Ilmanen. Convergence of the A llen- C ahn equation to B rakke's motion by mean curvature. Journal of Differential Geometry , 38(2):417--461, 1993

1993
[22]

Threshold dynamics type approximation schemes for propagating fronts

Hitoshi Ishii, Gabriel E Pires, and Panagiotis E Souganidis. Threshold dynamics type approximation schemes for propagating fronts. Journal of the Mathematical Society of Japan , 51(2):267--308, 1999

1999
[23]

A generalization of the bence, merriman and osher algorithm for motion by mean curvature

Hitoshi Ishii. A generalization of the bence, merriman and osher algorithm for motion by mean curvature. Curvature flows and related topics , pages 111--127, 1995

1995
[24]

An introduction to variational inequalities and their applications

David Kinderlehrer and Guido Stampacchia. An introduction to variational inequalities and their applications . SIAM, 2000

2000
[25]

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

Tim Laux and Felix Otto. Convergence of the thresholding scheme for multi-phase mean-curvature flow. Calculus of Variations and Partial Differential Equations , 55(5):129, 2016

2016
[26]

The thresholding scheme for mean curvature flow and de G iorgi's ideas for minimizing movements

Tim Laux and Felix Otto. The thresholding scheme for mean curvature flow and de G iorgi's ideas for minimizing movements. In The role of metrics in the theory of partial differential equations , volume 85, pages 63--94. Mathematical Society of Japan, 2020

2020
[27]

The regularity theory for the double obstacle problem

Ki-Ahm Lee, Jinwan Park, and Henrik Shahgholian. The regularity theory for the double obstacle problem. Calculus of Variations and Partial Differential Equations , 58:1--19, 2019

2019
[28]

Convergence of the A llen- C ahn equation to multiphase mean curvature flow

Tim Laux and Theresa M Simon. Convergence of the A llen- C ahn equation to multiphase mean curvature flow. Communications on Pure and Applied Mathematics , 71(8):1597--1647, 2018

2018
[29]

Un esempio di -convergenza

Luciano Modica and Stefano Mortola. Un esempio di -convergenza. Boll. Un. Mat. Ital. B , 14:285--299, 1977

1977
[30]

The gradient theory of phase transitions and the minimal interface criterion

Luciano Modica. The gradient theory of phase transitions and the minimal interface criterion. Archive for Rational Mechanics and Analysis , 98:123--142, 1987

1987
[31]

o ger and Reiner Sch \

Matthias R \"o ger and Reiner Sch \"a tzle. On a modified conjecture of de giorgi. Mathematische Zeitschrift , 254:675--714, 2006

2006
[32]

Osqp: an operator splitting solver for quadratic programs

Bartolomeo Stellato, Goran Banjac, Paul Goulart, Alberto Bemporad, and Stephen Boyd. Osqp: an operator splitting solver for quadratic programs. Mathematical Programming Computation , 12(4):637–672, February 2020

2020
[33]

Stochastic gradient descent with noise of machine learning type part i: Discrete time analysis

Stephan Wojtowytsch. Stochastic gradient descent with noise of machine learning type part i: Discrete time analysis. Journal of Nonlinear Science , 33(3):45, 2023

2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs

Oluwatosin Akande, Patrick Dondl, Kanan Gupta, Akwum Onwunta, and Stephan Wojtowytsch. Momentum-based minimization of the ginzburg-landau functional on euclidean spaces and graphs. arXiv preprint arXiv:2501.00389 , 2024

work page Pith review arXiv 2024

[2] [2]

Acceleration by stepsize hedging: Silver stepsize schedule for smooth convex optimization

Jason M Altschuler and Pablo A Parrilo. Acceleration by stepsize hedging: Silver stepsize schedule for smooth convex optimization. Mathematical Programming , pages 1--14, 2024

work page 2024

[3] [3]

Acceleration by stepsize hedging: Multi-step descent and the silver stepsize schedule

Jason M Altschuler and Pablo A Parrilo. Acceleration by stepsize hedging: Multi-step descent and the silver stepsize schedule. Journal of the ACM , 72(2):1--38, 2025

work page 2025

[4] [4]

Numerical methods for nonlinear partial differential equations , volume 47

S \"o ren Bartels. Numerical methods for nonlinear partial differential equations , volume 47. Springer, 2015

work page 2015

[5] [5]

Motion by mean curvature as the singular limit of ginzburg-landau dynamics

Lia Bronsard and Robert V Kohn. Motion by mean curvature as the singular limit of ginzburg-landau dynamics. Journal of differential equations , 90(2):211--237, 1991

work page 1991

[6] [6]

Generalizing diffuse interface methods on graphs: nonsmooth potentials and hypergraphs

Jessica Bosch, Steffen Klamt, and Martin Stoll. Generalizing diffuse interface methods on graphs: nonsmooth potentials and hypergraphs. SIAM Journal on Applied Mathematics , 78(3):1350--1377, 2018

work page 2018

[7] [7]

Approssimazione variazionale di funzionali con curvatura

Giovanni Bellettini and Maurizio Paolini. Approssimazione variazionale di funzionali con curvatura. In Seminario di Analisi Matematica , pages 87--97. Tecnoprint, 1993

work page 1993

[8] [8]

On the slowness of phase boundary motion in one space dimension

Lia Bronsard. On the slowness of phase boundary motion in one space dimension. NASA STI/Recon Technical Report N , 91:21480, 1990

work page 1990