关于Bregman迭代在求解朗道自由能泛函极小化问题中的研究

doi:10.15960/j.cnki.issn.1007-6093.2025.03.012

摘要/Abstract

摘要： 本文研究了朗道自由能泛函极小化问题的数值方法和理论分析,该问题广泛应用于物理学和材料科学中相变和有序结构的形成。朗道自由能泛函通常由描述空间相互作用的高阶微分项及描述体积能的非线性项组成,这一特点导致计算面临两大困难:高阶微分算子带来的刚性问题以及非线性项中梯度全局利普希茨连续性的缺失。针对这些难点,研究首先将泛函极小化问题离散为有限维最优化问题,基于Bregman散度设计了高效的算法框架,并建立了收敛性分析。进一步地,我们将算法推广至函数空间,系统分析了其对原始泛函极小化问题的收敛性质。此外,本文探讨了Bregman迭代与梯度流方法的内在联系,为理解优化算法的动力学机制提供了新视角。所提出算法的有效性及理论分析的准确性均通过一系列数值实验得到了验证。

关键词: Bregman迭代, 朗道模型, 泛函极小化问题

Abstract: This paper investigates numerical methods and theoretical analysis for the minimization problem of Landau free energy functionals, which are widely applied in physics and materials science to study phase transitions and the formation of ordered structures. Landau free energy functionals typically consist of high-order differential terms describing spatial interactions and nonlinear terms representing bulk energy. This characteristic leads to two major computational challenges: the stiffness problem arising from high-order differential operators and the lack of global Lipschitz continuity of gradients in the nonlinear terms. To address these difficulties, we first discretize the functional minimization problem into a finite-dimensional optimization problem, then design an efficient algorithmic framework based on Bregman divergence, and subsequently establish convergence analysis. Furthermore, we extend the algorithm to function spaces and systematically analyze its convergence properties for the original functional minimization problem. Additionally, this paper explores the intrinsic connection between Bregman iterations and gradient flow methods, providing new perspectives for understanding the dynamical mechanisms of optimization algorithms. The effectiveness of the proposed algorithms and the validity of the theoretical analysis are verified through a series of numerical experiments.

Key words: Bregman iteration, Landau models, functional minimization problem

中图分类号:

O221.2

包承龙, 陈昌. 关于Bregman迭代在求解朗道自由能泛函极小化问题中的研究[J]. 运筹学学报（中英文）, 2025, 29(3): 243-266.

BAO Chenglong, CHEN Chang. A survey on the Bregman iteration in computing Landau’s free functional minimization problems[J]. Operations Research Transactions, 2025, 29(3): 243-266.

参考文献

[1] Cowley R A. Structural phase transitions I. Landau theory [J]. Advances in Physics, 1980, 29(1): 1-110.
[2] Ginzburg V L, Landau L D. Theory of superconductivity [J]. Journal of Experimental and Theoretical Physics, 1950, 20(12).
[3] Landau L D. On the theory of phase transitions [J]. Journal of Experimental and Theoretical Physics, 1937, 11: 19.
[4] Brazovski S A. Phase transition of an isotropic system to a nonuniform state [J]. Soviet Journal of Experimental and Theoretical Physics, 1975, 41: 85-89.
[5] Cross M C, Hohenberg P C . Pattern formation outside of equilibrium [J]. Reviews of Modern Physics, 1993, 65(3): 851.
[6] Jiang K, Si W, Xu J. Tilt grain boundaries of hexagonal structures: A spectral viewpoint [J]. SIAM Journal on Applied Mathematics, 2022, 82(4): 1267-1286.
[7] Fredrickson G. The Equilibrium Theory of Inhomogeneous Polymers [M]. Oxford: Oxford University Press, 2005.
[8] Müller M, Abetz V. Nonequilibrium processes in polymer membrane formation: Theory and experiment [J]. Chemical Reviews, 2021, 121(22): 14189-14231.
[9] Savitz S, Babadi M, Lifshitz R. Multiple-scale structures: from faraday waves to soft-matter quasicrystals [J]. Journal of the International Union of Crystallography, 2018, 5(3): 247-268.
[10] Lifshitz R, Petrich D M. Theoretical model for faraday waves with multiple-frequency forcing [J]. Physical Review Letters, 1997, 79(7): 1261.
[11] Ohta T, Kawasaki K. Equilibrium morphology of block copolymer melts [J]. Macromolecules, 1986, 19(10): 2621-2632.
[12] Bao C L, Chen C, Jiang K, et al. Convergence analysis for Bregman iterations in minimizing a class of Landau free energy functionals [J]. SIAM Journal on Numerical Analysis, 2024, 62(1): 476-499.
[13] Sial S, Neuberger J, Lookman T, et al. Energy minimization using Sobolev gradients: Application to phase separation and ordering [J]. Journal of Computational Physics, 2003, 189(1): 88-97.
[14] Jiang K, Si W, Chen C, et al. Efficient numerical methods for computing the stationary states of phase field crystal models [J]. SIAM Journal on Scientific Computing, 2020, 42(6): B1350-B1377.
[15] Zhang L, Du Q, Zheng Z Z. Optimization-based shrinking dimer method for finding transition states [J]. SIAM Journal on Scientific Computing, 2016, 38(1): A528-A544.
[16] Chaplin M. Water structure and science [EB/OL]. [2025-03-01]. https://water.lsbu.ac.uk/water/.
[17] Allen S M, Cahn . A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening [J]. Acta Metallurgica, 1979, 27(6): 1085-1095.
[18] Cahn J W, Hilliard J E. Free energy of a nonuniform system. I. Interfacial free energy [J]. The Journal of Chemical Physics, 1958, 28(2): 258-267.
[19] Shen J, Xu J, Yang J. A new class of efficient and robust energy stable schemes for gradient flows [J]. SIAM Review, 2019, 61(3): 474-506.
[20] Du Q, Ju L L, Li X, et al. Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes [J]. SIAM Review, 2021, 63(2): 317-359.
[21] Fukushima M, Mine H. A generalized proximal point algorithm for certain non-convex minimization problems [J]. International Journal of Systems Science, 1981, 12(8): 989-1000.
[22] Combettes P L, Wajs V R. Signal recovery by proximal forward-backward splitting [J]. Multiscale Modeling & Simulation, 2005, 4(4): 1168-1200.
[23] Bolte J, Sabach S, Teboulle M. Proximal alternating linearized minimization for nonconvex and nonsmooth problems [J]. Mathematical Programming, 2014, 146(1-2): 459-494.
[24] Chen L Q, Shen J. Applications of semi-implicit Fourier-spectral method to phase field equations [J]. Computer Physics Communications, 1998, 108(2-3): 147-158.
[25] Shen J, Yang X F. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations [J]. Discrete and Continuous Dynamical Systems-A, 2010, 28(4): 1669.
[26] Bregman L M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming [J]. USSR Computational Mathematics and Mathematical Physics, 1967, 7(3): 200-217.
[27] Bauschke H H, Bolte J, Teboulle M. A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications [J]. Mathematics of Operations Research, 2016, 42(2): 330-348.
[28] Bolte J, Sabach S, Teboulle M, et al. First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems [J]. SIAM Journal on Optimization, 2018, 28(3): 2131-2151.
[29] Teboulle M. A simplified view of first order methods for optimization [J]. Mathematical Programming, 2018, 170(1): 67-96.
[30] Jiang K, Zhang P. Numerical methods for quasicrystals [J]. Journal of Computational Physics, 2014, 256(1): 428-440.
[31] Li Q W, Zhu Z H, Tang G G, et al. Provable Bregman-divergence based methods for nonconvex and non-Lipschitz problems [EB/OL]. [2025-03-01]. arXiv:1904.09712.
[32] Bauschke H H, Bolte J, Teboulle M. A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications [J]. Mathematics of Operations Research, 2017, 42(2): 330-348.
[33] Nesterov Y. Introductory Lectures on Convex Optimization: A Basic Course [M]. New York: Springer Science & Business Media, 2003.
[34] O’Donoghue B, Candes E. Adaptive restart for accelerated gradient schemes [J]. Foundations of Computational Mathematics, 2015, 15(3): 715-732.
[35] Barzilai J, Borwein J M. Two-point step size gradient methods [J]. IMA Journal of Numerical Analysis, 1988, 8(1): 141-148.
[36] Bento G C, Mordukhovich B S, Mota T S, et al. Convergence of descent methods under Kurdyka- Lojasiewicz properties [EB/OL]. [2025-03-01]. arXiv:2407.00812.
[37] Bao C L, Chen C, Jiang K. An adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model [J]. CSIAM Transactions on Applied Mathematics, 2022, 3: 133-171.
[38] Yang X F. Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends [J]. Journal of Computational Physics, 2016, 327: 294-316.
[39] Shen J, Xu J, Yang J. The scalar auxiliary variable (SAV) approach for gradient flows [J]. Journal of Computational Physics, 2018, 353: 407-416.
[40] Hebey E. Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities [R]. Providence: American Mathematical Society, 2000.
[41] Roe J. Elliptic operators, topology, and asymptotic methods [J]. Acta Applicandae Mathematica, 1990, 20: 193-194.
[42] Rybka P, Hoffnlann K H. Convergence of solutions to Cahn-Hilliard equation [J]. Communications in Partial Differential Equations, 1999, 24(5-6): 1055-1077.
[43] Chill P. On the Lojasiewicz-Simon gradient inequality [J]. Journal of Functional Analysis, 2003, 201(2): 572-601.
[44] Simon L. Asymptotics for a class of non-linear evolution equations, with applications to geometric problems [J]. Annals of Mathematics, 1983: 525-571.
[45] Chill R. The Lojasiewicz-Simon gradient inequality in Hilbert spaces [C]//Proceedings of the 5th European-Maghrebian Workshop on Semigroup Theory, Evolution Equations, and Applications, 2006: 25-36.
[46] Feireisl E, Petzeltová H. Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations [J]. Differential and Integral Equations, 1997, 10(1): 181-196.
[47] Bauschke H H, Borwein J M, Combettes P L. Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces [J]. Communications in Contemporary Mathematics, 2001, 3(4): 615-647.
[48] Krichene W, Bayen A, Bartlett P L. Accelerated mirror descent in continuous and discrete time [C]//Proceedings of the 29th International Conference on Neural Information Processing Systems, 2015, 2: 2845-2853.
[49] Li D, Quan C Y, Tang T. Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation [J]. Mathematics of Computation, 2022, 91(334): 785-809.
[50] Fu Z H, Tang T, Yang J. Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation [J]. Journal of Scientific Computing, 2022, 92(3): 97.
[51] Fu Z H, Tang T, Yang J. Energy diminishing implicit-explicit Runge-Kutta methods for gradient flows [EB/OL]. [2025-03-01]. arXiv:2203.06034
[52] Zhang H, Wang H F, Teng X Q. A second-order, global-in-time energy stable implicit-explicit Runge-Kutta scheme for the phase field crystal equation [J]. SIAM Journal on Numerical Analysis, 2024, 62(6): 2667-2697.
[53] Wang X P, Zhao X, Liao H L. A unified framework on the original energy laws of three effective classes of Runge-Kutta methods for phase field crystal type models [EB/OL]. [2025-03-01]. arXiv:2412.07342.