An adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model

TL;DR

This paper introduces an adaptive block Bregman proximal gradient method to efficiently compute stationary states in multicomponent phase-field crystal models, demonstrating significant acceleration over existing methods.

Contribution

It develops a novel adaptive block Bregman proximal gradient algorithm tailored for block-structured non-convex minimization problems in phase-field crystal models.

Findings

Significant acceleration in computing stationary states.

Effective energy dissipation control during iterations.

Successful application to binary, ternary, and quinary models.

Abstract

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.