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Implicit-Explicit Trust Region Method for Computing Second-Order Stationary Points of A Class of Landau Models

Chenglong Bao, Kai Deng, Kai Jiang, Juan Zhang

Abstract

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.

Implicit-Explicit Trust Region Method for Computing Second-Order Stationary Points of A Class of Landau Models

Abstract

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.
Paper Structure (12 sections, 10 theorems, 85 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 12 sections, 10 theorems, 85 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Discretizing the LB model with Fourier pseudospectral method
  3. Implicit-Explicit Trust Region Method
  4. IMEX-TR Method and Convergence Analysis
  5. Convergence to Second-Order Stationary Points
  6. Efficient Global Solver for the Nonconvex TR Subproblem
  7. Global Convergence of the IMEX solver for the TR subproblem
  8. Numerical Experiments
  9. Convergence
  10. Robustness to initial conditions
  11. Construction of new phase diagram
  12. Conclusion

Key Result

Theorem 3.1

A vector $\bm{d}_*$ is a global minimizer of the subproblem eq:tr_subproblem_with_subscript if and only if there exists a scalar $\lambda_*\geq 0$ such that the following conditions are satisfied. If $\bm{H}_j + \lambda_* \bm{I}$ is positive definite, then $\bm{d}_*$ is the unique global minimizer of eq:tr_subproblem_with_subscript.

Figures (4)

  • Figure 1: Final physical phases and energy. IMEX-TR identifies the (meta-)stable HEX, BCC, and cubic FDDD phases across the three examples, consistently achieving lower energy states compared to standard first-order schemes.
  • Figure 2: Evolution of the four smallest eigenvalues of the Hessian matrix. The dashed line indicates the zero threshold. IMEX-TR consistently reaches the SP-II region (non-negative eigenvalues), whereas first-order methods often stagnate at SDPs with negative curvature.
  • Figure 3: Trajectories of IMEX-TR activated at different stages of first-order methods. Whether started from the initial point, an intermediate state, or the converged saddle point, IMEX-TR consistently converges to the same (meta-)stable SP-II (HEX phase), demonstrating its efficacy in exploiting negative curvature directions to escape unstable stationary points.
  • Figure 4: (a) Phase diagram of the LB model. The purple shaded area corresponds to the stable region of the FDDD phase. (b)-(f) illustrate the SP-II phases: LAM, HEX, BCC, DG, and FDDD.

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 3.1: conn2000trust, Corollary 7.2.2
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • Lemma 3.6
  • proof
  • ...and 10 more