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Register a new userAn adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model
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Added by
Kai Jiang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 problems 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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As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted $H^{-1}$ norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.
In this paper, we develop an inexact Bregman proximal gradient (iBPG) method based on a novel two-point inexact stopping condition, and establish the iteration complexity of $mathcal{O}(1/k)$ as well as the convergence of the sequence under some proper conditions. To improve the convergence speed, we further develop an inertial variant of our iBPG (denoted by v-iBPG) and show that it has the iteration complexity of $mathcal{O}(1/k^{gamma})$, where $gammageq1$ is a restricted relative smoothness exponent. Thus, when $gamma>1$, the v-iBPG readily improves the $mathcal{O}(1/k)$ convergence rate of the iBPG. In addition, for the case of using the squared Euclidean distance as the kernel function, we further develop a new inexact accelerated proximal gradient (iAPG) method, which can circumvent the underlying feasibility difficulty often appearing in existing inexact conditions and inherit all desirable convergence properties of the exact APG under proper summable-error conditions. Finally, we conduct some preliminary numerical experiments for solving a relaxation of the quadratic assignment problem to demonstrate the convergence behaviors of the iBPG, v-iBPG and iAPG under different inexactness settings.
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