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.

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

图/表 8

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模型	$ \lambda_1 $	$ \lambda_2 $	$ \lambda_3 $	$ \lambda_4 $	$ \lambda_5 $	$ \lambda_6 $
LB	$ - $7.64$ \times 10^{-16} $	$ - $6.49$ \times 10^{-16} $	0.72	0.72	0.98	1.46
LP	$ - $6.63$ \times 10^{-12} $	$ - $6.63$ \times 10^{-12} $	6.63$ \times 10^{-13} $	6.72$ \times 10^{-13} $	0.076	0.076

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