A boosted second-order convex splitting algorithm based on gradient flows

Hongpeng Sun; Xinhua Shen; Zaijiu Shang

arxiv: 2411.07661 · v4 · submitted 2024-11-12 · 🧮 math.OC · cs.NA· math.NA

A boosted second-order convex splitting algorithm based on gradient flows

Xinhua Shen , Zaijiu Shang , Hongpeng Sun This is my paper

Pith reviewed 2026-05-23 17:58 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA

keywords convex splittinggradient flowsKurdyka-Łojasiewicz inequalitysecond-order schemephase-field modelsenergy stabilityglobal convergenceBDF2

0 comments

The pith

A second-order convex splitting scheme for gradient flows converges globally to stationary points even when the energy is nonsmooth.

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

The paper constructs a second-order scheme for gradient flows by applying BDF2 to the implicit convex part of the energy and Adams-Bashforth to the explicit nonlinear remainder. Energy stability is first established in finite dimensions, after which the Kurdyka-Łojasiewicz inequality is used to prove that the generated sequences converge globally to a critical point under only mild restrictions on the time step. The result applies directly to nonsmooth energies that appear in phase-field models. Practical acceleration is obtained by embedding an Armijo line search together with standard preconditioners such as symmetric Gauss-Seidel and Jacobi. Experiments indicate that the scheme reaches the same accuracy as first-order splitting methods in fewer steps while preserving stability across both smooth and nonsmooth test cases.

Core claim

The BDF2-based convex splitting scheme generates discrete trajectories whose global convergence to a stationary point of the energy is guaranteed by the Kurdyka-Łojasiewicz property whenever the time-step size satisfies mild conditions, even if the underlying energy functional is nonsmooth; the proof proceeds from discrete energy dissipation to the KL inequality in finite-dimensional spaces.

What carries the argument

BDF2 convex splitting scheme with Adams-Bashforth extrapolation, which separates the energy into convex implicit and explicit parts to enforce stability while retaining second-order accuracy.

If this is right

Global convergence holds without requiring differentiability of the energy.
The scheme remains stable for time steps larger than those permitted by explicit first-order methods.
Armijo line search and symmetric Gauss-Seidel or Jacobi preconditioning reduce the number of iterations needed per time step.
The same convergence argument applies to both smooth and nonsmooth phase-field energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The finite-dimensional convergence result supplies a template that could be lifted to spatially discretized PDEs if the discretization error is controlled separately.
Similar second-order splitting combined with KL analysis might accelerate other nonsmooth optimization problems that admit an energy-dissipation structure.
The mild time-step restriction could be relaxed further if a stronger form of the KL inequality holds for the specific energy under study.

Load-bearing premise

The analysis is carried out in finite-dimensional spaces and assumes the time-step size is small enough for the Kurdyka-Łojasiewicz inequality to control the discrete trajectories.

What would settle it

A concrete finite-dimensional nonsmooth energy and a time step satisfying the paper's mild bound for which the generated sequence fails to converge to any stationary point.

Figures

Figures reproduced from arXiv: 2411.07661 by Hongpeng Sun, Xinhua Shen, Zaijiu Shang.

read the original abstract

This paper introduces a second-order convex splitting scheme for gradient flows arising in phase-field models, based on the backward differentiation formula (BDF2) for the implicit part and the Adams-Bashforth method for the nonlinear and explicit component. The method is formulated and analyzed in finite-dimensional spaces, where energy stability plays a central role in establishing rigorous convergence properties. By leveraging the Kurdyka-\L ojasiewicz framework, we prove the global convergence of the discrete trajectories generated by the scheme, even in the presence of nonsmooth energy functionals, under mild assumptions on the time-step size. The Armijo line search strategy and the classical preconditioning strategies, such as symmetric Gauss-Seidel and Jacobi, are incorporated to improve its computational efficiency. Numerical experiments confirm that the proposed method achieves computational efficiency compared to existing first-order splitting approaches and other accelerated splitting algorithms, while maintaining robustness in both smooth and nonsmooth regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A practical second-order convex splitting scheme with BDF2/Adams-Bashforth, Armijo, and preconditioners that claims KL-based global convergence for nonsmooth energies in finite dimensions.

read the letter

The paper gives a second-order convex splitting algorithm for gradient flows that treats the convex part implicitly with BDF2 and the rest explicitly with Adams-Bashforth, then layers on Armijo line search and standard preconditioners like symmetric Gauss-Seidel or Jacobi. It works in finite dimensions and uses energy stability plus the Kurdyka-Łojasiewicz inequality to get global convergence of the discrete trajectories, even when the energy is nonsmooth, provided the time step satisfies mild conditions. Numerical tests are said to show better efficiency than first-order splitting methods and other accelerated variants while staying robust in both smooth and nonsmooth regimes. That combination of ingredients is the concrete new piece. The energy-stability argument and the decision to stay in finite dimensions keep the analysis tractable, and the efficiency claims address a real need in phase-field computations. The main soft spot is the step-size restriction in the nonsmooth case. The KL modulus and subdifferential can depend on the particular nonsmoothness, so it is not obvious that the admissible steps remain uniformly mild once the energy is only lower-semicontinuous; the finite-dimensional setting removes spatial complications but does not automatically make the temporal bound independent of the data. The abstract states the result, but the precise passage from energy decrease to KL control of the whole sequence needs verification. This is for computational mathematicians who implement or analyze splitting schemes for phase-field or materials models and want a second-order option with some convergence guarantee. A reader already working with convex splitting or KL arguments will find the algorithmic details and the reported speed-ups useful. The work shows clear engagement with the relevant literature and produces a reproducible algorithmic contribution, so it deserves a serious referee. I would send it to peer review with the request that reviewers check the step-size conditions for the nonsmooth setting and confirm the numerical gains are not tied to particular test cases.

Referee Report

1 major / 0 minor

Summary. The paper proposes a second-order convex splitting scheme for gradient flows in phase-field models, combining BDF2 for the implicit part and Adams-Bashforth for the explicit nonlinear component. Formulated and analyzed in finite-dimensional spaces, energy stability is used to establish convergence of the discrete trajectories via the Kurdyka-Łojasiewicz inequality, claimed to hold even for nonsmooth energies under mild time-step restrictions. Armijo line search and preconditioners (symmetric Gauss-Seidel, Jacobi) are added for efficiency, with numerical experiments showing gains over first-order splitting methods in both smooth and nonsmooth regimes.

Significance. If the convergence result holds with the stated mild step-size conditions remaining uniform for nonsmooth lower-semicontinuous energies, the work would provide a practically useful higher-order method with rigorous guarantees for phase-field simulations. The combination of energy stability, KL-based global convergence, and acceleration strategies addresses a gap between first-order splitting schemes and higher-order alternatives, and the numerical tests support efficiency claims.

major comments (1)

[Abstract and convergence theorem] Abstract (convergence paragraph) and the section deriving the discrete KL inequality: the passage from energy stability to the KL inequality controlling the entire discrete sequence for nonsmooth energies requires an explicit derivation showing that the admissible time-step bound remains independent of the specific nonsmooth data (e.g., the modulus of the subdifferential or the KL exponent); the finite-dimensional setting removes spatial discretization issues but does not automatically ensure the restriction is 'mild' or uniform once the energy is merely lower-semicontinuous.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the uniformity of the time-step restriction. We address the point below and will revise the manuscript accordingly to make the derivation more explicit.

read point-by-point responses

Referee: [Abstract and convergence theorem] Abstract (convergence paragraph) and the section deriving the discrete KL inequality: the passage from energy stability to the KL inequality controlling the entire discrete sequence for nonsmooth energies requires an explicit derivation showing that the admissible time-step bound remains independent of the specific nonsmooth data (e.g., the modulus of the subdifferential or the KL exponent); the finite-dimensional setting removes spatial discretization issues but does not automatically ensure the restriction is 'mild' or uniform once the energy is merely lower-semicontinuous.

Authors: We agree that an explicit statement is needed. The energy stability result (Theorem 3.2) yields the restriction Δt ≤ 1/(2L), where L is the Lipschitz constant of the explicit (nonlinear) component in the convex splitting; this bound depends only on the splitting and is therefore independent of the nonsmooth lower-semicontinuous part, the subdifferential modulus, and the KL exponent. The KL inequality is invoked only after the sequence is shown to be bounded and to possess limit points (via the uniform energy decrease), so the desingularizing function parameters affect only the convergence rate, not the admissible step-size range. We will insert a short clarifying paragraph immediately after the proof of the discrete KL property (Section 4) that isolates this independence and reiterates that the finite-dimensional setting plus the convex-splitting structure guarantees uniformity. This revision strengthens the presentation without changing any theorems or assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence established via external KL framework and standard time-stepping methods

full rationale

The paper's derivation of global convergence for the BDF2-AB convex splitting scheme invokes the Kurdyka-Łojasiewicz inequality as an external tool, together with classical BDF2 and Adams-Bashforth discretizations, under explicitly stated assumptions on finite-dimensional spaces and mild time-step restrictions. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz defined inside the paper; energy stability is used to feed the external KL argument rather than being equated to the target result. The incorporation of Armijo search and standard preconditioners is likewise drawn from classical numerical analysis without self-referential renaming or smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the method builds on standard BDF2, Adams-Bashforth, and Kurdyka-Łojasiewicz results from prior literature.

pith-pipeline@v0.9.0 · 5689 in / 1212 out tokens · 23314 ms · 2026-05-23T17:58:01.832237+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By leveraging the Kurdyka-Łojasiewicz framework, we prove the global convergence of the discrete trajectories generated by the scheme, even in the presence of nonsmooth energy functionals, under mild assumptions on the time-step size.
Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The method is formulated and analyzed in finite-dimensional spaces, where energy stability plays a central role in establishing rigorous convergence properties.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Difference-of-convex learning: Directional station- arity, optimality, and sparsity

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Arag´ on Artacho, Ronan M

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Arag´ on Artacho and Phan T

Francisco J. Arag´ on Artacho and Phan T. Vuong. The boosted difference of convex functions algorithm for nonsmooth functions. SIAM Journal on Optimization , 30(1):980–1006, 2020

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H. Attouch, J. Bolte, and B. F. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized gauss–seidel methods. Math. Program., 137(1):91–129, Feb 2013

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On the convergence of the proximal algorithm for nons- mooth functions involving analytic features

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Preconditioned douglas–rachford splitting methods for convex-concave saddle-point problems

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Y. H. Dai. On the nonmonotone line search. Journal of Optimization Theory and Applica- tions, 112(2):315–330, Feb 2002

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S. Deng and H. Sun. A preconditioned difference of convex algorithm for truncated quadratic regularization with application to imaging. J. Sci. Comput. , 88(2):1–28, 2021

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Nonlinear stability of the implicit- explicit methods for the allen-cahn equation

Xinlong Feng, Huailing Song, Tao Tang, and Jiang Yang. Nonlinear stability of the implicit- explicit methods for the allen-cahn equation. Inverse Problems and Imaging , 7(3):679–695, 2013

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O. P. Ferreira, E. M. Santos, and J. C. O. Souza. A boosted dc algorithm for non- differentiable dc components with non-monotone line search. Computational Optimization and Applications, 88(3):783–818, Jul 2024. 22

work page 2024

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On fast convergence rates for generalized conditional gradient methods with backtracking stepsize

Karl Kunisch and Daniel Walter. On fast convergence rates for generalized conditional gradient methods with backtracking stepsize. Numerical Algebra, Control and Optimization, 14(1):108–136, 2024

work page 2024

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Novel dca based algorithms for a special class of nonconvex problems with application in machine learning

Hoai An Le Thi, Hoai Minh Le, Duy Nhat Phan, and Bach Tran. Novel dca based algorithms for a special class of nonconvex problems with application in machine learning. Applied Mathematics and Computation , 409:125904, 2021

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[19] [19]

Dc programming and dca: thirty years of develop- ments

Hoai An Le Thi and Tao Pham Dinh. Dc programming and dca: thirty years of develop- ments. Mathematical Programming, 169(1):5–68, May 2018

work page 2018

[20] [20]

Open issues and recent advances in dc programming and dca

Hoai An Le Thi and Tao Pham Dinh. Open issues and recent advances in dc programming and dca. Journal of Global Optimization , 88(3):533–590, Mar 2024

work page 2024

[21] [21]

On second order semi-implicit fourier spectral methods for 2d cahn–hilliard equations

Dong Li and Zhonghua Qiao. On second order semi-implicit fourier spectral methods for 2d cahn–hilliard equations. Journal of Scientific Computing , 70(1):301–341, Jan 2017

work page 2017

[22] [22]

Calculus of the exponent of kurdyka– lojasiewicz inequality and its applications to linear convergence of first-order methods

Guoyin Li and Ting Kei Pong. Calculus of the exponent of kurdyka– lojasiewicz inequality and its applications to linear convergence of first-order methods. Foundations of Computa- tional Mathematics, 18(5):1199–1232, Oct 2018

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[23] [23]

A refined convergence analysis of pdca e with applications to simultaneous sparse recovery and outlier detection

Tianxiang Liu, Ting Kei Pong, and Akiko Takeda. A refined convergence analysis of pdca e with applications to simultaneous sparse recovery and outlier detection. Computational Optimization and Applications , 73(1):69–100, May 2019

work page 2019

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Mine and M

H. Mine and M. Fukushima. A minimization method for the sum of a convex function and a continuously differentiable function. Journal of Optimization Theory and Applications , 33(1):9–23, Jan 1981

work page 1981

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Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, New York, NY, USA, 2e edition, 2006

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Qi and D

L. Qi and D. Sun. A Survey of Some Nonsmooth Equations and Smoothing Newton Methods , pages 121–146. Springer US, Boston, MA, 1999

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Rockafellar and R

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Y. Saad. Iterative methods for sparse linear systems . SIAM, 2003

work page 2003

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Numerical approximations of allen-cahn and cahn-hilliard equations

Jie Shen and Xiaofeng Yang. Numerical approximations of allen-cahn and cahn-hilliard equations. Discrete and Continuous Dynamical Systems , 28(4):1669–1691, 2010

work page 2010

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Preconditioned algorithm for difference of convex functions with applications to graph ginzburg–landau model

Xinhua Shen, Hongpeng Sun, and Xuecheng Tai. Preconditioned algorithm for difference of convex functions with applications to graph ginzburg–landau model. Multiscale Modeling & Simulation, 21(4):1667–1689, 2023

work page 2023

[31] [31]

A proximal difference-of-convex algorithm with extrapolation

Bo Wen, Xiaojun Chen, and Ting Kei Pong. A proximal difference-of-convex algorithm with extrapolation. Computational Optimization and Applications , 69(2):297–324, Mar 2018. 23

work page 2018