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Efficient numerical methods for computing stationary states of spherical Landau-Brazovskii model

Qun Qiu, Wei Si, Guanghua Ji, Kai Jiang

TL;DR

This work targets the efficient computation of stationary states for the spherical Landau-Brazovskii model. It adopts a discretize-then-optimize approach by expanding the order parameter in spherical harmonics and minimizing the resulting finite-dimensional energy $E_h(\{\hat{\varphi}_{\ell,m}\})=G_h+F_h$ under the mass constraint $\hat{\varphi}_{0,0}=0$. A suite of optimization algorithms—AA-BPG, Nesterov, ANesterov, AGD, and ACG—are developed with line-search and convergence guarantees, complemented by a PMA method to generate high-quality initial guesses and radii via dominant spherical modes and the relation $R=\sqrt{\ell(\ell+1)}$. Numerical experiments on spotted and striped phases show that AA-BPG and Nesterov drastically reduce iterations and CPU time compared to gradient-flow baselines, while PMA markedly improves initialization success rates. The results validate the approach and suggest directions for exploring richer pattern formation on spherical surfaces.

Abstract

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time

Efficient numerical methods for computing stationary states of spherical Landau-Brazovskii model

TL;DR

This work targets the efficient computation of stationary states for the spherical Landau-Brazovskii model. It adopts a discretize-then-optimize approach by expanding the order parameter in spherical harmonics and minimizing the resulting finite-dimensional energy under the mass constraint . A suite of optimization algorithms—AA-BPG, Nesterov, ANesterov, AGD, and ACG—are developed with line-search and convergence guarantees, complemented by a PMA method to generate high-quality initial guesses and radii via dominant spherical modes and the relation . Numerical experiments on spotted and striped phases show that AA-BPG and Nesterov drastically reduce iterations and CPU time compared to gradient-flow baselines, while PMA markedly improves initialization success rates. The results validate the approach and suggest directions for exploring richer pattern formation on spherical surfaces.

Abstract

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time
Paper Structure (18 sections, 3 theorems, 70 equations, 10 figures, 7 tables, 5 algorithms)

This paper contains 18 sections, 3 theorems, 70 equations, 10 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Spherical LB model
  4. Spherical harmonics discretization
  5. Numerical methods
  6. Optimization methods
  7. Line search
  8. AGD and ACG algorithms
  9. AA-BPG and (A)Nesterov algorithms
  10. Principal mode analysis (PMA) method
  11. Numerical results
  12. The efficiency of the PMA method
  13. Spotted phase
  14. Striped phase
  15. The efficiency of optimization methods
  16. ...and 3 more sections

Key Result

Theorem 3.1

Consider the AGD scheme $\hat{\varphi}^{n+1}=\hat{\varphi}^{n}+\alpha_{n} p_{n}$ with $p_{n}:=-\nabla E_{h}(\hat{\varphi}^{n})$. The following assumptions hold where $0<c_{1}<c_{2}<1$. Then we have

Figures (10)

  • Figure 1: A stationary spotted phase with 32 spots when $\xi=1.0$, $\epsilon=-0.4$, $\lambda=0.4$. The initial configuration and sphere radius are given by the PMA method
  • Figure 2: A stationary striped phase with $16$ strips when $\xi=1.0$, $\epsilon=-0.2$, $\lambda=0.0$. The initial configuration and sphere radius are given by the PMA method
  • Figure 3: Initial and stationary structures for the spotted phase with $60$ spots. $\xi=1.0$, $\epsilon=-1.0$, $\lambda=0.8$
  • Figure 4: Comparison of numerical behavior of the AA-BPG-2, AA-BPG-4, Nesterov, ANesterov, ACG and AGD methods as well as SIS, ASIS methods for computing the spotted phase on a sphere of radius $R=\sqrt{240}$. The $\times$ markers denote the restart steps in AA-BPG-2 and AA-BPG-4 algorithms
  • Figure 5: Step behavior of AA-BPG-2, AA-BPG-4 and other adaptive methods for computing the stationary spotted phase. The step sizes of the AGD and ACG method are $\alpha_{0}=0.002$, $\alpha_{\min}=1.0\times 10^{-5}$ and $\alpha_{\max}=5.0$. Meanwhile, the AA-BPG-2/4 methods have $\alpha_{0}=0.02$, $\alpha_{\max}=5.0$ and $\alpha_{\min}=0.01$, while $\alpha_{\max}=20.0$ for the ASIS method and $\alpha_{\min}=1.0\times 10^{-5}$ for the ANesterov method
  • ...and 5 more figures

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.2
  • Theorem 3.3
  • Remark 4.1