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Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes

Haifan Chen, Guozhi Dong, José A. Iglesias, Wei Liu, Ziqing Xie

TL;DR

The paper develops and analyzes second-order dissipative hyperbolic PDEs (second-order flows) for nonconvex energies and introduces convex-splitting schemes that yield unconditional energy stability and unique solvability. It proves subsequential convergence of semi-discrete and second-order schemes to stationary points of the energy and establishes well-posedness of the time-continuous PDE through timestep-independent estimates. Numerical experiments on Ginzburg–Landau and Landau–de Gennes models show that second-order convex-splitting schemes achieve faster energy decay and robust stability across time steps, outperforming several traditional schemes. The results highlight the practical effectiveness of second-order flows for nonconvex variational problems and open avenues for complete-trajectory convergence analysis and alternative stabilized time-stepping methods.

Abstract

This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient flows to energy functionals defined on Sobolev spaces, and exhibiting significant performance particularly for the minimization of non-convex energies. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal (and spatial) discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the timestep tends to zero. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of representative non-convex variational models in scientific computing.

Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes

TL;DR

The paper develops and analyzes second-order dissipative hyperbolic PDEs (second-order flows) for nonconvex energies and introduces convex-splitting schemes that yield unconditional energy stability and unique solvability. It proves subsequential convergence of semi-discrete and second-order schemes to stationary points of the energy and establishes well-posedness of the time-continuous PDE through timestep-independent estimates. Numerical experiments on Ginzburg–Landau and Landau–de Gennes models show that second-order convex-splitting schemes achieve faster energy decay and robust stability across time steps, outperforming several traditional schemes. The results highlight the practical effectiveness of second-order flows for nonconvex variational problems and open avenues for complete-trajectory convergence analysis and alternative stabilized time-stepping methods.

Abstract

This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient flows to energy functionals defined on Sobolev spaces, and exhibiting significant performance particularly for the minimization of non-convex energies. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal (and spatial) discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the timestep tends to zero. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of representative non-convex variational models in scientific computing.
Paper Structure (17 sections, 9 theorems, 103 equations, 4 figures, 2 tables)

This paper contains 17 sections, 9 theorems, 103 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Convergence to stationary points via a convex-splitting scheme
  3. Timestep-independent estimates and well-posedness of the PDE
  4. A second-order semi-discrete convex-splitting scheme
  5. Applications and numerical results
  6. Ginzburg-Landau free energy
  7. Landau–de Gennes (LdG) model
  8. Conclusion
  9. Other numerical methods involved in the paper
  10. Other schemes for discretizing second-order flows
  11. Forward Euler Scheme
  12. Backward Euler Scheme
  13. Semi-implicit Scheme
  14. Crank-Nicolson Scheme
  15. Gradient flow and its convex-splitting schemes
  16. ...and 2 more sections

Key Result

Lemma 2.1

\newlabellemma3:energy-bound0 Let $\mathbf{u} \in H_0^1(\Omega, \mathbb{R}^N)$. Then where $C>0$ and $C^{\star}$ are constants depending only on $\Omega$, $\alpha$, $\beta$ and $\gamma$.

Figures (4)

  • Figure 1: Pseudo-energy decay comparison: Backward Euler scheme \ref{['eq:backward-euler-1']}-\ref{['eq:backward-euler-2']} vs. First-order Convex-splitting scheme \ref{['eq:1st-CS-FEM-1']}-\ref{['eq:1st-CS-FEM-2']} for $\tau=1$ in Example \ref{['eg:1']}.
  • Figure 2: Comparison of energy evolution (left column) and convergence to the minimum energy (right column) over iterations for Example \ref{['eg:3']} (an isotropic case) using Gradient Flow (GF-CS-1st $\&$ GF-CS-2nd) and Second-order Flow (SF-CS-1st $\&$ SF-CS-2nd) methodologies. Within the figures, $E$ denotes the calculated energy, while $E_{\star}$ signifies the minimum energy achieved throughout the iterations.
  • Figure 3: Comparison of energy evolution (left column) and convergence to the minimum energy (right column) over iterations for Example \ref{['eg:4']} (an anisotropic case) using Gradient Flow (GF-CS-2nd) and Second-order Flow (SF-CS-2nd) methodologies.
  • Figure 4: Stable liquid crystal configurations at varying $\vartheta$ values (5, 15, 50, 100, 200, 500), computed via the second-order flow method. The color gradient indicates the relative intensity of directional ordering, quantified as $|S(\mathbf{r})| / S_0$. The white bars depict the directional vectors of the nematic field, $\mathbf{n}(\mathbf{r})$.

Theorems & Definitions (24)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Theorem 2.3: Unconditional unique solvability
  • Proof 3
  • Theorem 2.4: Subsequential convergence to a stationary point
  • Proof 4
  • Lemma 3.1
  • Proof 5
  • ...and 14 more