Operations Research Transactions >
2025 , Vol. 29 >Issue 3: 243 - 266
DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2025.03.012
A survey on the Bregman iteration in computing Landau's free functional minimization problems
Received date: 2025-03-31
Online published: 2025-09-09
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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.
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 . DOI: 10.15960/j.cnki.issn.1007-6093.2025.03.012
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