A Structure-Exploiting Implicit-Explicit Trust Region Method for Computing Second-Order Stationary Points of the Landau-Brazovskii Model

Abstract

This work focuses on the reliable computation of second-order stationary points in the high-dimensional nonconvex energy landscape of the Landau-Brazovskii (LB) model, a fundamental model for studying phases and phase transitions. For this purpose, we develop an efficient implicit-explicit trust region (IMEX-TR) method. Trust region (TR) methods can avoid saddle-point stagnation and guarantee convergence to second-order stationary points under appropriate conditions. However, their direct application to the LB model has been impractical because the Hessian is dense if treated directly. The proposed IMEX-TR method overcomes this difficulty by exploiting the Hessian's special structure: the linear interaction part is diagonal in reciprocal space, whereas the nonlinear bulk-energy part is diagonal in physical space. Based on this structure, we design an efficient solver for the TR subproblem that with globally convergent guarantee and enjoys FFT-based acceleration, with $\mathcal O(N \log N)$ complexity per iteration. Existing first-order gradient-based methods for the LB model only guarantee convergence to first-order stationary points and may stagnate at saddle points. In contrast, the proposed IMEX-TR method inherits the theoretical guarantee of converging to second-order stationary points while remaining computationally practical. Numerical experiments verify the theoretical properties of the algorithm and demonstrate its robustness in locating stable phases from different initial conditions. Numerical results also show that IMEX-TR can escape unstable stationary states reached by first-order schemes and converge to physically meaningful second-order stationary points. These results suggest that targeting second-order stationary points provides an effective computational paradigm for exploring complex free-energy landscapes and identifying stable or metastable states.