An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model

Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, China Chang Chen Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China Kai Jiang School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, 411105, China

Numerical Analysis and Scientific Computing mathscidoc:2206.25001

CSIAM Trans. Appl. Math., 3, (1), 133-171, 2022.3
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.
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@inproceedings{chenglong2022an,
  title={An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model},
  author={Chenglong Bao, Chang Chen, and Kai Jiang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613221106243986354},
  booktitle={CSIAM Trans. Appl. Math.},
  volume={3},
  number={1},
  pages={133-171},
  year={2022},
}
Chenglong Bao, Chang Chen, and Kai Jiang. An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model. 2022. Vol. 3. In CSIAM Trans. Appl. Math.. pp.133-171. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613221106243986354.
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