Discovering new phases via computing second-order stationary states of Landau-Brazovskii model

Chenglong Bao; Juan Zhang; Kai Deng; Kai Jiang

arxiv: 2603.03933 · v3 · submitted 2026-03-04 · 🧮 math.NA · cs.NA· math.OC

Discovering new phases via computing second-order stationary states of Landau-Brazovskii model

Chenglong Bao , Kai Deng , Kai Jiang , Juan Zhang This is my paper

Pith reviewed 2026-05-15 16:50 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC

keywords Landau-Brazovskii modelsecond-order stationary pointstrust region methodphase diagramcubic FDDD phasenonconvex optimizationcrystal phasesnumerical methods

0 comments

The pith

An implicit-explicit trust region method finds second-order stationary points in the Landau-Brazovskii model and locates a new stable cubic FDDD phase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors develop a numerical algorithm that targets second-order stationary points in the nonconvex energy functional of the Landau-Brazovskii model. Standard gradient methods only reach first-order points and can stop at saddles, but this trust-region approach is proven to reach local minima. Applying it from varied starting points reveals a cubic FDDD structure that remains stable in a specific region of parameter space. This allows construction of a revised phase diagram showing where the new phase is favored. A reader would care because it demonstrates a practical way to map out hidden stable states in models of crystal formation.

Core claim

The paper establishes that the implicit-explicit trust region method converges to second-order stationary points of the LB energy, which correspond to local minima, and uses this to identify the cubic FDDD phase as a previously unknown stable ordered structure in the model, leading to an updated phase diagram that identifies the stable region for this phase.

What carries the argument

The implicit-explicit trust region method, which solves subproblems to ensure descent directions that avoid first-order saddle points and guarantee convergence to second-order stationary states in the high-dimensional LB energy landscape.

If this is right

The Landau-Brazovskii model admits a stable cubic FDDD phase in certain parameter regimes.
The new method reliably locates local minima from different initial conditions unlike first-order algorithms.
An updated phase diagram incorporates the FDDD phase and its stability region.
Targeting second-order stationary points provides a systematic way to explore complex free-energy landscapes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar optimization techniques might uncover additional unknown phases in related models of soft matter or materials.
Previous phase diagrams for the LB model computed with gradient methods may have missed or misclassified some structures.
Extending the method to time-dependent or stochastic versions could study phase transition dynamics more accurately.

Load-bearing premise

That convergence from multiple random initial conditions in the discretized model is enough to prove the FDDD structure is a genuine local minimum and not caused by numerical artifacts.

What would settle it

A direct comparison of the energy of the computed FDDD configuration against all other candidate phases across the same parameter values, or observation of the phase in a laboratory experiment corresponding to the LB model parameters, would confirm or refute its stability.

Figures

Figures reproduced from arXiv: 2603.03933 by Chenglong Bao, Juan Zhang, Kai Deng, Kai Jiang.

**Figure 4.1.** Figure 4.1: Final physical phases and energy. IMEX-TR identifies the (meta-)stable [PITH_FULL_IMAGE:figures/full_fig_p017_4_1.png] view at source ↗

**Figure 4.2.** Figure 4.2: Evolution of the four smallest eigenvalues of the Hessian matrix. The dashed [PITH_FULL_IMAGE:figures/full_fig_p018_4_2.png] view at source ↗

**Figure 4.3.** Figure 4.3: Trajectories of IMEX-TR activated at different stages of first-order meth [PITH_FULL_IMAGE:figures/full_fig_p021_4_3.png] view at source ↗

**Figure 4.4.** Figure 4.4: (a) Phase diagram of the LB model. The purple shaded area corresponds [PITH_FULL_IMAGE:figures/full_fig_p021_4_4.png] view at source ↗

read the original abstract

In this work, we report a stable ordered structure -- the cubic FDDD phase -- that has not previously been identified in the Landau-Brazovskii (LB) model, a fundamental and important model for studying crystals and their phase transitions. The key to this discovery is the proposed implicit-explicit trust region method for computing second-order stationary points in the high-dimensional nonconvex energy landscape of the LB model. Different from existing first-order gradient-based algorithms, which only guarantee convergence to first-order stationary points and may therefore stagnate at saddle points, the proposed method is theoretically guaranteed to converge to second-order stationary points corresponding to local minima. Numerical experiments verify the theoretical properties of the algorithm and demonstrate its robustness in locating stable phases from different initial conditions. Based on the discovered FDDD phase, we further construct an updated phase diagram of the LB model and identify its stable region. These results show that targeting second-order stationary points provides an effective computational paradigm for exploring complex free-energy landscapes and uncovering new stable phases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds a new cubic FDDD phase in the LB model with a trust-region method that targets second-order points, but lacks proof that the discrete solution converges to a continuous critical point.

read the letter

The core contribution is the identification of a cubic FDDD phase in the Landau-Brazovskii model that prior work missed. The authors introduce an implicit-explicit trust-region scheme that is shown to converge to second-order stationary points on the discretized problem, and their runs from varied initial conditions locate this structure consistently enough to redraw the phase diagram and mark its stability region. That algorithmic shift from first-order methods is the practical advance here, and the numerics back up its ability to escape saddles in this setting. The theoretical guarantee for the discrete case is stated clearly and the experiments appear reproducible from the description. The main limitation is the missing link to the continuous functional. Convergence is established only after discretization, with no a-priori error estimates, consistency result, or mesh-refinement study showing that the computed critical point approaches a true stationary point of the original energy as the grid is refined. Without that, the reported stability window could still be tied to the chosen mesh size or periodic cell. The work is aimed at people doing numerical exploration of nonconvex phase-field models in materials science. It is coherent on its own terms and the discovery claim is specific, so it merits a serious referee who can press on the discretization analysis and ask for quantitative checks on mesh independence. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper proposes an implicit-explicit trust-region algorithm that is proved to converge to second-order stationary points of the discretized Landau-Brazovskii energy. Using this method, the authors numerically locate a previously unreported cubic FDDD phase from multiple initial conditions, map its stability region in parameter space, and present an updated phase diagram for the LB model.

Significance. If the reported FDDD structure is shown to be a genuine local minimum of the continuous LB functional, the work would provide both a new computational paradigm for non-convex phase exploration and a concrete addition to the known phase diagram of a standard model in soft-matter physics. The algorithmic guarantee for second-order points is a clear technical strength.

major comments (2)

[§3] §3 (Algorithm and convergence analysis): The global convergence theorem is stated only for the finite-dimensional discretization; no consistency result, a-priori error bound, or mesh-refinement argument is given showing that discrete second-order critical points converge in H¹ (or any Sobolev norm) to a critical point of the continuous functional as h→0. This gap directly affects the claim that the discovered FDDD phase is a stable structure of the continuous LB model rather than a grid artifact.
[§4.2] §4.2 (Numerical experiments and phase diagram): The stability region for the FDDD phase is delineated solely from discrete runs on a fixed periodic cell and quadrature rule. Without quantitative convergence diagnostics (e.g., energy or gradient residuals under successive h-refinement), it is impossible to confirm that the reported boundaries remain unchanged in the continuum limit.

minor comments (2)

Notation for the trust-region radius and the implicit-explicit splitting should be introduced once and used consistently; several paragraphs reuse symbols without redefinition.
Figure captions for the phase diagrams should explicitly state the discretization parameters (mesh size, cell size, quadrature order) used to generate each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the concerns about the lack of consistency analysis and mesh refinement studies by providing additional explanations and committing to revisions that include numerical diagnostics to better link the discrete results to the continuous Landau-Brazovskii model.

read point-by-point responses

Referee: [§3] §3 (Algorithm and convergence analysis): The global convergence theorem is stated only for the finite-dimensional discretization; no consistency result, a-priori error bound, or mesh-refinement argument is given showing that discrete second-order critical points converge in H¹ (or any Sobolev norm) to a critical point of the continuous functional as h→0. This gap directly affects the claim that the discovered FDDD phase is a stable structure of the continuous LB model rather than a grid artifact.

Authors: We appreciate the referee highlighting this theoretical gap. Our convergence theorem establishes global convergence to second-order stationary points for the discretized problem, which is the setting in which the algorithm is implemented and the numerical experiments are performed. The discovery of the FDDD phase is thus rigorously a local minimum in the discrete energy landscape. To connect to the continuous model, we note that the Fourier spectral discretization converges spectrally for sufficiently smooth functions, and the LB energy is smooth. In the revised version, we will add a remark in Section 3 discussing the discretization and its expected convergence properties, along with numerical evidence from h-refinement showing that the FDDD structure and its energy remain consistent as the mesh is refined. While a complete a priori error analysis for second-order points is a substantial undertaking that we consider outside the primary scope of this algorithmic paper, the added numerical support will mitigate concerns about grid artifacts. revision: partial
Referee: [§4.2] §4.2 (Numerical experiments and phase diagram): The stability region for the FDDD phase is delineated solely from discrete runs on a fixed periodic cell and quadrature rule. Without quantitative convergence diagnostics (e.g., energy or gradient residuals under successive h-refinement), it is impossible to confirm that the reported boundaries remain unchanged in the continuum limit.

Authors: We agree that explicit convergence diagnostics under mesh refinement would strengthen the reliability of the reported stability region. In the revised manuscript, we will include quantitative results from successive h-refinements for the FDDD phase at key parameter values within and near the boundaries. These will show the stabilization of the energy value and the decay of gradient residuals, confirming that the phase remains a local minimum and that the relative energies compared to other phases do not alter the stability conclusions. This addition will be placed in Section 4.2 or a new appendix. The original delineation uses the same discretization for all competing phases, ensuring fair comparison, but we recognize the importance of verifying continuum behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: discovery via independent numerical search on discretized model

full rationale

The manuscript introduces an implicit-explicit trust-region algorithm whose convergence guarantees apply to the finite-dimensional discretization of the LB energy. It then runs the method from varied initial conditions to locate a new critical point (FDDD) and maps its stability region. No equation defines the reported phase in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation whose content is merely the present claim. The central result is therefore an output of the search procedure rather than an input restated by construction. Absence of a mesh-convergence theorem is a separate rigor issue, not a circularity defect.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim depends on standard nonconvex optimization theory for trust-region convergence to second-order points and on the LB energy functional itself; no new free parameters or invented entities are introduced beyond the existing model.

axioms (1)

standard math Standard assumptions of trust-region methods guarantee convergence to second-order stationary points under sufficient decrease and curvature conditions.
Invoked to justify that the algorithm reaches local minima rather than saddles.

pith-pipeline@v0.9.0 · 5481 in / 1105 out tokens · 51288 ms · 2026-05-15T16:50:18.165595+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Agarwal, Z

N. Agarwal, Z. Allen-Zhu, B. Bullins, E. Hazan, and T. Ma. Finding approximate local minima faster than gradient descent. In STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing , pages 1195–1199. ACM, New York, 2017

work page 2017
[2]

Attouch, J

H. Attouch, J. Bolte, and B. F. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss- Seidel methods. Math. Program., 137(1-2):91–129, 2013

work page 2013
[3]

C. Bao, C. Chen, and K. Jiang. An adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model. CSIAM Trans. Appl. Math., 3(1):133–171, 2022

work page 2022
[4]

C. Bao, C. Chen, K. Jiang, and L. Qiu. Convergence analysis for Bregman iterations in minimizing a class of Landau free energy functionals. SIAM J. Numer. Anal. , 62(1):476– 499, 2024

work page 2024
[5]

Bolte, S

J. Bolte, S. Sabach, and M. Teboulle. Proximal alternating linearized minimization for non- convex and nonsmooth problems. Math. Program., 146(1-2):459–494, 2014

work page 2014
[6]

S. A. Brazovskii. Phase transition of an isotropic system to a nonuniform state. Sov. Phys. JETP, 41:85, 1975

work page 1975
[7]

S. A. Brazovski˘ ı. Phase transition of an isotropic system to a nonuniform state. In 30 Years Of The Landau Institute—Selected Papers , pages 109–113. World Scientific, 1996

work page 1996
[8]

M. E. Caplan and C. J. Horowitz. Colloquium: Astromaterial science and nuclear pasta. Rev. Modern Phys., 89:041002, 2017

work page 2017
[9]

Gradient descent finds the cubic-regularized nonconvex Newton step

Y. Carmon and J. C. Duchi. First-order methods for nonconvex quadratic minimization. SIAM Rev., 62(2):395–436, 2020. Revised reprint of “Gradient descent finds the cubic-regularized nonconvex Newton step” [4000224]

work page 2020
[10]

Cartis, N

C. Cartis, N. I. M. Gould, and P. L. Toint. Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints. SIAM J. Optim., 30(1):513–541, 2020

work page 2020
[11]

L. Q. Chen and J. Shen. Applications of semi-implicit fourier-spectral method to phase field equations. Computer Physics Communications , 108(2-3):147–158, 1998

work page 1998
[12]

A. R. Conn, N. I. M. Gould, and P. L. Toint. Trust-region methods. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2000

work page 2000
[13]

Cui and K

G. Cui and K. Jiang. Spring pair method of finding saddle points using the minimum energy 19 path as a compass. Phys. Rev. E , 110(6):Paper No. 064123, 11, 2024

work page 2024
[14]

G. Cui, K. Jiang, and T. Zhou. An efficient saddle search method for ordered phase transitions involving translational invariance. Computer Physics Communications , 306:109381, 2025

work page 2025
[15]

F. E. Curtis, D. P. Robinson, and M. Samadi. A trust region algorithm with a worst-case iteration complexity of O(ϵ−3/2) for nonconvex optimization. Math. Program., 162(1-2):1– 32, 2017

work page 2017
[16]

Fu and J

Z. Fu and J. Yang. Energy-decreasing exponential time differencing Runge-Kutta methods for phase-field models. J. Comput. Phys. , 454:Paper No. 110943, 11, 2022

work page 2022
[17]

V. L. Ginzburg and L. D. Landau. On the theory of superconductivity. In On superconductivity and superfluidity: a scientific autobiography , pages 113–137. Springer, 2009

work page 2009
[18]

Jiang, W

K. Jiang, W. Si, C. Chen, and C. Bao. Efficient numerical methods for computing the stationary states of phase field crystal models. SIAM J. Sci. Comput. , 42(6):B1350–B1377, 2020

work page 2020
[19]

Jiang, C

K. Jiang, C. Wang, Y. Huang, and P. Zhang. Discovery of new metastable patterns in diblock copolymers. Commun. Comput. Phys. , 14(2):443–460, 2013

work page 2013
[20]

C. Jin, R. Ge, P. Netrapalli, S. M. Kakade, and M. I. Jordan. How to escape saddle points efficiently. In International conference on machine learning , pages 1724–1732. PMLR, 2017

work page 2017
[21]

D Landau

L. D Landau. The theory of phase transitions. Nature, 138(3498):840–841, 1936

work page 1936
[22]

L. D. Landau and E. M. Lifshitz. Statistical physics, volume 5. Pergamon Press, 1958

work page 1958
[23]

J. D. Lee, M. Simchowitz, M. I. Jordan, and B. Recht. Gradient descent only converges to minimizers. In Conference on learning theory, pages 1246–1257. PMLR, 2016

work page 2016
[24]

L. Lin, X. Cheng, W. E, A.-C. Shi, and P. Zhang. A numerical method for the study of nucleation of ordered phases. J. Comput. Phys. , 229(5):1797–1809, 2010

work page 2010
[25]

Liu and J

C. Liu and J. Shen. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D, 179(3-4):211–228, 2003

work page 2003
[26]

McClenagan

D. McClenagan. Landau theory of complex ordered phases. Master’s thesis, McMaster Univer- sity, 2019

work page 2019
[27]

Nesterov and B

Y. Nesterov and B. T. Polyak. Cubic regularization of Newton method and its global perfor- mance. Math. Program., 108(1):177–205, 2006

work page 2006
[28]

J. W. Neuberger. Sobolev gradients and differential equations , volume 1670 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 2010

work page 2010
[29]

Nouiehed and M

M. Nouiehed and M. Razaviyayn. A trust region method for finding second-order stationarity in linearly constrained nonconvex optimization. SIAM J. Optim. , 30(3):2501–2529, 2020

work page 2020
[30]

C. W. Royer, M. O’Neill, and S. J. Wright. A Newton-CG algorithm with complexity guarantees for smooth unconstrained optimization. Math. Program., 180(1-2):451–488, 2020

work page 2020
[31]

Savitz, M

S. Savitz, M. Babadi, and R. Lifshitz. Multiple-scale structures: from faraday waves to soft- matter quasicrystals. IUCrJ, 5(3):247–268, 2018

work page 2018
[32]

J. Shen, J. Xu, and J. Yang. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev., 61(3):474–506, 2019

work page 2019
[33]

Shen and X

J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. , 28(4):1669–1691, 2010

work page 2010
[34]

S. Sial, J. Neuberger, T. Lookman, and A. Saxena. Energy minimization using Sobolev gra- dients: application to phase separation and ordering. J. Comput. Phys. , 189(1):88–97, 2003

work page 2003
[35]

Swift and P

J. Swift and P. C. Hohenberg. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15(1):319–328, 1977

work page 1977
[36]

S. M. Wise, C. Wang, and J. S. Lowengrub. An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal., 47(3):2269–2288, 2009

work page 2009
[37]

Yamada, M

K. Yamada, M. Nonomura, and T. Ohta. Fddd structure in AB-type diblock copolymers. Journal of Physics: Condensed Matter , 18(32):L421, 2006

work page 2006
[38]

X. Yang. Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. , 327:294–316, 2016

work page 2016
[39]

Zhang and X

P. Zhang and X. Zhang. An efficient numerical method of Landau-Brazovskii model.J. Comput. Phys., 227(11):5859–5870, 2008. 20 Energy -8.73e-04 -1.936e-02 -1.94e-02 -1.33e-01 initial value IMEX-TR First-order algorithms Switch to IMEX-TR SP-II SDP SDP Fig. 4.3: Trajectories of IMEX-TR activated at different stages of first-order meth- ods. Whether started ...

work page 2008

[1] [1]

Agarwal, Z

N. Agarwal, Z. Allen-Zhu, B. Bullins, E. Hazan, and T. Ma. Finding approximate local minima faster than gradient descent. In STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing , pages 1195–1199. ACM, New York, 2017

work page 2017

[2] [2]

Attouch, J

H. Attouch, J. Bolte, and B. F. Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss- Seidel methods. Math. Program., 137(1-2):91–129, 2013

work page 2013

[3] [3]

C. Bao, C. Chen, and K. Jiang. An adaptive block Bregman proximal gradient method for computing stationary states of multicomponent phase-field crystal model. CSIAM Trans. Appl. Math., 3(1):133–171, 2022

work page 2022

[4] [4]

C. Bao, C. Chen, K. Jiang, and L. Qiu. Convergence analysis for Bregman iterations in minimizing a class of Landau free energy functionals. SIAM J. Numer. Anal. , 62(1):476– 499, 2024

work page 2024

[5] [5]

Bolte, S

J. Bolte, S. Sabach, and M. Teboulle. Proximal alternating linearized minimization for non- convex and nonsmooth problems. Math. Program., 146(1-2):459–494, 2014

work page 2014

[6] [6]

S. A. Brazovskii. Phase transition of an isotropic system to a nonuniform state. Sov. Phys. JETP, 41:85, 1975

work page 1975

[7] [7]

S. A. Brazovski˘ ı. Phase transition of an isotropic system to a nonuniform state. In 30 Years Of The Landau Institute—Selected Papers , pages 109–113. World Scientific, 1996

work page 1996

[8] [8]

M. E. Caplan and C. J. Horowitz. Colloquium: Astromaterial science and nuclear pasta. Rev. Modern Phys., 89:041002, 2017

work page 2017

[9] [9]

Gradient descent finds the cubic-regularized nonconvex Newton step

Y. Carmon and J. C. Duchi. First-order methods for nonconvex quadratic minimization. SIAM Rev., 62(2):395–436, 2020. Revised reprint of “Gradient descent finds the cubic-regularized nonconvex Newton step” [4000224]

work page 2020

[10] [10]

Cartis, N

C. Cartis, N. I. M. Gould, and P. L. Toint. Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints. SIAM J. Optim., 30(1):513–541, 2020

work page 2020

[11] [11]

L. Q. Chen and J. Shen. Applications of semi-implicit fourier-spectral method to phase field equations. Computer Physics Communications , 108(2-3):147–158, 1998

work page 1998

[12] [12]

A. R. Conn, N. I. M. Gould, and P. L. Toint. Trust-region methods. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2000

work page 2000

[13] [13]

Cui and K

G. Cui and K. Jiang. Spring pair method of finding saddle points using the minimum energy 19 path as a compass. Phys. Rev. E , 110(6):Paper No. 064123, 11, 2024

work page 2024

[14] [14]

G. Cui, K. Jiang, and T. Zhou. An efficient saddle search method for ordered phase transitions involving translational invariance. Computer Physics Communications , 306:109381, 2025

work page 2025

[15] [15]

F. E. Curtis, D. P. Robinson, and M. Samadi. A trust region algorithm with a worst-case iteration complexity of O(ϵ−3/2) for nonconvex optimization. Math. Program., 162(1-2):1– 32, 2017

work page 2017

[16] [16]

Fu and J

Z. Fu and J. Yang. Energy-decreasing exponential time differencing Runge-Kutta methods for phase-field models. J. Comput. Phys. , 454:Paper No. 110943, 11, 2022

work page 2022

[17] [17]

V. L. Ginzburg and L. D. Landau. On the theory of superconductivity. In On superconductivity and superfluidity: a scientific autobiography , pages 113–137. Springer, 2009

work page 2009

[18] [18]

Jiang, W

K. Jiang, W. Si, C. Chen, and C. Bao. Efficient numerical methods for computing the stationary states of phase field crystal models. SIAM J. Sci. Comput. , 42(6):B1350–B1377, 2020

work page 2020

[19] [19]

Jiang, C

K. Jiang, C. Wang, Y. Huang, and P. Zhang. Discovery of new metastable patterns in diblock copolymers. Commun. Comput. Phys. , 14(2):443–460, 2013

work page 2013

[20] [20]

C. Jin, R. Ge, P. Netrapalli, S. M. Kakade, and M. I. Jordan. How to escape saddle points efficiently. In International conference on machine learning , pages 1724–1732. PMLR, 2017

work page 2017

[21] [21]

D Landau

L. D Landau. The theory of phase transitions. Nature, 138(3498):840–841, 1936

work page 1936

[22] [22]

L. D. Landau and E. M. Lifshitz. Statistical physics, volume 5. Pergamon Press, 1958

work page 1958

[23] [23]

J. D. Lee, M. Simchowitz, M. I. Jordan, and B. Recht. Gradient descent only converges to minimizers. In Conference on learning theory, pages 1246–1257. PMLR, 2016

work page 2016

[24] [24]

L. Lin, X. Cheng, W. E, A.-C. Shi, and P. Zhang. A numerical method for the study of nucleation of ordered phases. J. Comput. Phys. , 229(5):1797–1809, 2010

work page 2010

[25] [25]

Liu and J

C. Liu and J. Shen. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D, 179(3-4):211–228, 2003

work page 2003

[26] [26]

McClenagan

D. McClenagan. Landau theory of complex ordered phases. Master’s thesis, McMaster Univer- sity, 2019

work page 2019

[27] [27]

Nesterov and B

Y. Nesterov and B. T. Polyak. Cubic regularization of Newton method and its global perfor- mance. Math. Program., 108(1):177–205, 2006

work page 2006

[28] [28]

J. W. Neuberger. Sobolev gradients and differential equations , volume 1670 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 2010

work page 2010

[29] [29]

Nouiehed and M

M. Nouiehed and M. Razaviyayn. A trust region method for finding second-order stationarity in linearly constrained nonconvex optimization. SIAM J. Optim. , 30(3):2501–2529, 2020

work page 2020

[30] [30]

C. W. Royer, M. O’Neill, and S. J. Wright. A Newton-CG algorithm with complexity guarantees for smooth unconstrained optimization. Math. Program., 180(1-2):451–488, 2020

work page 2020

[31] [31]

Savitz, M

S. Savitz, M. Babadi, and R. Lifshitz. Multiple-scale structures: from faraday waves to soft- matter quasicrystals. IUCrJ, 5(3):247–268, 2018

work page 2018

[32] [32]

J. Shen, J. Xu, and J. Yang. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev., 61(3):474–506, 2019

work page 2019

[33] [33]

Shen and X

J. Shen and X. Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. , 28(4):1669–1691, 2010

work page 2010

[34] [34]

S. Sial, J. Neuberger, T. Lookman, and A. Saxena. Energy minimization using Sobolev gra- dients: application to phase separation and ordering. J. Comput. Phys. , 189(1):88–97, 2003

work page 2003

[35] [35]

Swift and P

J. Swift and P. C. Hohenberg. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A, 15(1):319–328, 1977

work page 1977

[36] [36]

S. M. Wise, C. Wang, and J. S. Lowengrub. An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal., 47(3):2269–2288, 2009

work page 2009

[37] [37]

Yamada, M

K. Yamada, M. Nonomura, and T. Ohta. Fddd structure in AB-type diblock copolymers. Journal of Physics: Condensed Matter , 18(32):L421, 2006

work page 2006

[38] [38]

X. Yang. Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. , 327:294–316, 2016

work page 2016

[39] [39]

Zhang and X

P. Zhang and X. Zhang. An efficient numerical method of Landau-Brazovskii model.J. Comput. Phys., 227(11):5859–5870, 2008. 20 Energy -8.73e-04 -1.936e-02 -1.94e-02 -1.33e-01 initial value IMEX-TR First-order algorithms Switch to IMEX-TR SP-II SDP SDP Fig. 4.3: Trajectories of IMEX-TR activated at different stages of first-order meth- ods. Whether started ...

work page 2008