Implicit-Explicit Trust Region Method for Computing Second-Order Stationary Points of A Class of Landau Models
This work addresses the challenge of finding physically stable phases in materials science models, offering a more reliable optimization method for researchers in computational physics and materials engineering, though it is incremental as it builds on trust region methods with specific adaptations.
The authors tackled the problem of computing second-order stationary points for Landau-type free energy functionals, such as the Landau-Brazovskii model, by proposing an implicit-explicit trust region method that efficiently escapes saddle points and outperforms existing first-order schemes, successfully identifying a previously unreported stable phase structure.
We propose an implicit-explicit trust region method for computing second-order stationary points of a class of Landau-type free energy functionals, which correspond to physically (meta-)stable phases. The proposed method is demonstrated through the Landau-Brazovskii (LB) model in this work, while broader applicability to more Landau models of the similar type is straightforwardly extended. The LB energy functional is discretized via the Fourier pseudospectral method, which yields a finite-dimensional nonconvex optimization problem. By exploiting the Hessian structure, specifically, that the interaction potential is diagonal in reciprocal space whereas the bulk energy is diagonal in physical space, we design an adaptive implicit-explicit solver for the trust region subproblem. This solver utilizes the fast Fourier transform to perform efficient matrix-vector products, significantly reducing computational complexity while ensuring provable convergence to the global minimizer of the subproblem. In contrast to existing algorithms that target first-order stationary points, our proposed method can converge to a second-order stationary state, corresponding to a local minimum with theoretical convergence guarantees. Numerical experiments on the LB model demonstrate that the proposed approach efficiently escapes saddle points and significantly outperforms existing first-order schemes. Furthermore, we successfully identify the stable region of the FDDD phase, a structure previously unreported in the LB phase diagram.