Convergence Analysis for Bregman Iterations in Minimizing a Class of Landau Free Energy Functionals

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.