Efficient Numerical Methods for Computing Stationary States of Spherical Landau-Brazovskii Model

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.