SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 476-499, February 2024.
Abstract. Finding stationary states of Landau free energy functionals has to solve a nonconvex infinite-dimensional optimization problem. In this paper, we develop a Bregman distance based optimization method for minimizing a class of Landau energy functionals and focus on its convergence analysis in the function space. We first analyze the regularity of the stationary states and show the weakly sequential convergence results of the proposed method. Furthermore, under the Łojasiewicz–Simon property, we prove a strongly sequential convergent property and establish the local convergence rate in an appropriate Hilbert space. In particular, we analyze the Łojasiewicz exponent of three well-known Landau models, the Landau–Brazovskii, Lifshitz–Petrich, and Ohta–Kawasaki free energy functionals. Finally, numerical results support our theoretical analysis.

中文翻译：

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最小化一类朗道自由能泛函的 Bregman 迭代的收敛性分析

SIAM 数值分析杂志，第 62 卷，第 1 期，第 476-499 页，2024 年 2 月。
摘要。寻找朗道自由能泛函的稳态必须解决非凸无限维优化问题。在本文中，我们开发了一种基于 Bregman 距离的优化方法，用于最小化一类 Landau 能量泛函，并重点关注其在函数空间中的收敛性分析。我们首先分析了稳态的规律性，并展示了该方法的弱顺序收敛结果。此外，在Łojasiewicz-Simon性质下，我们证明了强顺序收敛性质，并在适当的希尔伯特空间中建立了局部收敛率。我们特别分析了三个著名的朗道模型（Landau-Brazovskii、Lifshitz-Petrich 和 Ohta-Kawasaki 自由能泛函）的 Łojasiewicz 指数。最后，数值结果支持我们的理论分析。