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A Nonnested Augmented Subspace Method for Kohn-Sham Equation

Guanghui Hu, Hehu Xie, Fei Xu, Gang Zhao

TL;DR

The paper addresses the high cost of solving the nonlinear Kohn-Sham equation in electronic-structure calculations by introducing a moving-mesh, nonnested augmented subspace approach. It transforms the KS problem on a fine, nonnested mesh into multiple linear boundary-value problems of the same scale plus a small-scale nonlinear eigenproblem in a low-dimensional augmented subspace, avoiding full-scale KS solves. A coercivity-based convergence analysis and an asymptotically optimal computational-work estimate are provided, and the method is validated on Helium, Hydrogen–Lithium, Methane, and Benzene, showing improved accuracy and substantial efficiency gains. This framework enables accurate KS computations on complex molecules with scalable performance on HPC platforms, thanks to adaptive mesh redistribution and efficient augmentation techniques.

Abstract

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.

A Nonnested Augmented Subspace Method for Kohn-Sham Equation

TL;DR

The paper addresses the high cost of solving the nonlinear Kohn-Sham equation in electronic-structure calculations by introducing a moving-mesh, nonnested augmented subspace approach. It transforms the KS problem on a fine, nonnested mesh into multiple linear boundary-value problems of the same scale plus a small-scale nonlinear eigenproblem in a low-dimensional augmented subspace, avoiding full-scale KS solves. A coercivity-based convergence analysis and an asymptotically optimal computational-work estimate are provided, and the method is validated on Helium, Hydrogen–Lithium, Methane, and Benzene, showing improved accuracy and substantial efficiency gains. This framework enables accurate KS computations on complex molecules with scalable performance on HPC platforms, thanks to adaptive mesh redistribution and efficient augmentation techniques.

Abstract

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.
Paper Structure (14 sections, 4 theorems, 79 equations, 9 figures, 5 tables, 3 algorithms)

This paper contains 14 sections, 4 theorems, 79 equations, 9 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Finite element method for Kohn-Sham equation
  3. Nonnested augmented subspace method for Kohn-Sham equation
  4. The calculation of exchange-correlation potential and Hartree potential
  5. Moving mesh adaptive technique
  6. Nonnested augmented subspace method for Kohn-Sham equation
  7. Convergence analysis and computational work estimate
  8. Numerical experiments
  9. Kohn-Sham equation for Helium
  10. Kohn-Sham equation for Hydrogen-Lithium
  11. Kohn-Sham equation for Methane
  12. Kohn-Sham equation for Benzene
  13. Concluding remarks
  14. Acknowledgement

Key Result

Lemma 3.1

Let $K$ be a mesh element (d-dimensional simplex) of mesh $\mathcal{T}_h$, $E_K$ is a set of all edges on $K$, and $u$ is a quadratic differentiable function. Then, we have the following $L^\infty-$interpolation error estimate where with $|\Lambda|:=diag \{ |\lambda_1|,|\lambda_2|,\cdots,|\lambda_n| \}$.

Figures (9)

  • Figure 1: Contour plot of the electron density (left) and the adaptive moving mesh (right) for Example 1
  • Figure 2: The ground state energy (left) and the corresponding error estimates (right) for Example 1
  • Figure 3: Contour plot of the electron density (left) and the adaptive moving mesh (right) for Example 2
  • Figure 4: The ground state energy (left) and the corresponding error estimates (right) for Example 2
  • Figure 5: Contour plot of the electron density (left) and the adaptive moving mesh (right) for Example 3
  • ...and 4 more figures

Theorems & Definitions (9)

  • Lemma 3.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.2