Volume 3, Issue 1
An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model

Chenglong Bao, Chang Chen & Kai Jiang

CSIAM Trans. Appl. Math., 3 (2022), pp. 133-171.

Published online: 2022-03

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  • 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.

  • Keywords

Multicomponent coupled-mode Swift-Hohenberg model, stationary states, adaptive block Bregman proximal gradient algorithm, convergence analysis, adaptive step size.

  • AMS Subject Headings

54C40, 14E20

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-3-133, author = {Chenglong and Bao and and 22470 and and Chenglong Bao and Chang and Chen and and 22471 and and Chang Chen and Kai and Jiang and and 22472 and and Kai Jiang}, title = {An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2022}, volume = {3}, number = {1}, pages = {133--171}, 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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0002}, url = {http://global-sci.org/intro/article_detail/csiam-am/20292.html} }
TY - JOUR T1 - An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model AU - Bao , Chenglong AU - Chen , Chang AU - Jiang , Kai JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 133 EP - 171 PY - 2022 DA - 2022/03 SN - 3 DO - http://doi.org/10.4208/csiam-am.SO-2021-0002 UR - https://global-sci.org/intro/article_detail/csiam-am/20292.html KW - Multicomponent coupled-mode Swift-Hohenberg model, stationary states, adaptive block Bregman proximal gradient algorithm, convergence analysis, adaptive step size. AB -

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.

Chenglong Bao, Chang Chen & Kai Jiang. (2022). An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model. CSIAM Transactions on Applied Mathematics. 3 (1). 133-171. doi:10.4208/csiam-am.SO-2021-0002
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