An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model (CSIAM Transactions on Applied Mathematics, 2022)
In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. Theoriginal infinite-dimensional non-convex minimization problem is approximated by afinite-dimensional constrained non-convex minimization problem after an appropriatespatial discretization. To efficiently solve the above optimization problem, we proposea so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fullyexploits the problem’s block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministicor random manner. Besides, we choose the step size by developing a practical linearsearch approach such that the generated sequence either keeps energy dissipation orhas a controllable subsequence with energy dissipation. The convergence property ofthe proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. Thenumerical results on computing stationary ordered structures in binary, ternary, andquinary component coupled-mode Swift-Hohenberg models have shown a significantacceleration over many existing methods.
Keywords: Multicomponent coupled-mode Swift-Hohenberg model; stationary states; adaptive block Bregman proximal gradient algorithm; convergence analysis; adaptive step size.