Operations Research Transactions ›› 2025, Vol. 29 ›› Issue (3): 243-266.doi: 10.15960/j.cnki.issn.1007-6093.2025.03.012

Special Issue: 第九届中国运筹学会科学技术奖获奖者专辑

• Research Article • Previous Articles    

A survey on the Bregman iteration in computing Landau's free functional minimization problems

Chenglong BAO1,2,*(), Chang CHEN1,3   

  1. 1. Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
    2. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
    3. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
  • Received:2025-03-31 Online:2025-09-15 Published:2025-09-09
  • Contact: Chenglong BAO E-mail:clbao@tsinghua.edu.cn

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

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