[Paper Review] Convex Splitting Method for the Calculation of Transition States of Energy Functional

Shuting Gu, Xiang Zhou|arXiv (Cornell University)|Oct 23, 2016

Block Copolymer Self-Assembly26 references3 citations

TL;DR

This paper introduces a convex splitting method within the iterative minimization formulation (IMF) to efficiently compute index-1 saddle points (transition states) of energy functionals, such as those in Allen-Cahn and Cahn-Hilliard systems. By applying convex splitting to the auxiliary functional in each IMF cycle, the method ensures unconditional energy stability and enables large time step sizes, significantly improving computational efficiency for both 1D and 2D models including the Landau-Brazovskii functional.

ABSTRACT

Among numerical methods for partial differential equations arising from steepest descent dynamics of energy functionals (e.g., Allen-Cahn and Cahn-Hilliard equations), the convex splitting method is well-known to maintain unconditional energy stability for a large time step size. In this work, we show how to use the convex splitting idea to find transition states, i.e., index-1 saddle points of the same energy functionals. Based on the iterative minimization formulation (IMF) for saddle points (SIAM J. Numer. Anal., vol. 53, p1786, 2015), we introduce the convex splitting method to minimize the auxiliary functional at each cycle of the IMF. We present a general principle of constructing convex splitting forms for these auxiliary functionals and show how to avoid solving nonlinear equations. The new numerical scheme based on the convex splitting method allows for large time step sizes. The new methods are tested for the one dimensional Ginzburg-Landau energy functional in the search of the Allen-Cahn or Cahn-Hilliard types of transition states. We provide the numerical results of transition states for the two dimensional Landau-Brazovskii energy functional for diblock copolymers.

Motivation & Objective

• To develop an efficient numerical method for computing index-1 saddle points (transition states) of energy functionals in spatially extended systems.
• To adapt the convex splitting method—known for energy stability in gradient flow simulations—to the iterative minimization formulation (IMF) for saddle point computation.
• To enable large time step sizes in the solution of auxiliary minimization subproblems within IMF, overcoming stability limitations of standard schemes.
• To demonstrate the method's robustness and efficiency on both 1D Ginzburg-Landau and 2D Landau-Brazovskii energy functionals under $H^{-1}$ and $L^2$ metrics.
• To address practical challenges such as spurious convergence due to translation modes by enforcing orthogonality of the min-mode to the gradient of the state.

Proposed method

• The method integrates the convex splitting approach into the iterative minimization formulation (IMF), where each IMF cycle solves a subproblem via convex splitting of the auxiliary functional.
• A general principle is proposed to decompose the auxiliary functional $L$ into convex and concave parts, ensuring unconditional energy stability for large time steps.
• The convex splitting scheme avoids solving nonlinear equations at each time step by using a semi-implicit, linearized formulation, enhancing computational efficiency.
• The method ensures that the min-mode used in the auxiliary functional is orthogonal to the spatial gradient $\partial_x \phi$, preventing convergence to trivial translation modes.
• An adaptive stopping rule is applied to subproblems to balance accuracy and computational cost, preserving the quadratic convergence rate of IMF.
• The approach is validated on 1D Ginzburg-Landau and 2D Landau-Brazovskii models, with numerical results showing stable convergence even with large time steps.

Experimental results

Research questions

• RQ1 Can the convex splitting method be effectively adapted to the iterative minimization formulation (IMF) for computing index-1 saddle points?
• RQ2 Does the convex splitting approach for the auxiliary functional in IMF ensure unconditional energy stability and allow large time step sizes?
• RQ3 How does the method perform in avoiding convergence to spurious solutions caused by translation invariance in the system?
• RQ4 What is the computational efficiency gain of using convex splitting in IMF for transition state calculations compared to standard schemes?
• RQ5 Can the method be generalized to other energy functionals such as the Landau-Brazovskii model for diblock copolymers?

Key findings

• The convex splitting method applied within the IMF framework enables unconditionally energy-stable time integration, allowing large time step sizes without loss of stability.
• For the 1D Ginzburg-Landau energy, the method successfully computes both Allen-Cahn and Cahn-Hilliard type transition states under $L^2$ and $H^{-1}$ metrics, respectively.
• In the 2D Landau-Brazovskii model, the method converges to stable transition states even with large time steps, demonstrating robustness in higher dimensions.
• The method avoids convergence to trivial translation modes by enforcing orthogonality between the min-mode and the spatial gradient of the state, ensuring physically meaningful saddle point search.
• Extensive numerical experiments confirm that the method maintains quadratic convergence and significantly improves computational efficiency compared to standard schemes.
• The approach provides a general framework to repurpose existing energy-stable schemes for gradient flows into efficient tools for saddle point computation.