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A fast low-rank inversion algorithm of dielectric matrix in GW approximation

Zhengbang Zhou, Huanhuan Ma, Wentiao Wu, Weiguo Gao, Jinlong Yang, Meiyue Shao, Wei Hu

TL;DR

This work targets the computational bottleneck in GW calculations—the inversion of the dielectric function matrix—by introducing a fast, cubic-scaling, low-rank inversion strategy that combines ISDF with the Sherman-Morrison-Woodbury formula and a Cauchy-integral refinement. Applied to G$_0$W$_0$ within the static COHSEX framework, the approach achieves $O(N_r N_e^2)$ cost with $N_r$ grid points in the range $100$--$1000 N_e$, and an accuracy around $0.03\,\mathrm{eV}$ for both molecular and solid systems. The method yields substantial speedups, up to roughly 50x faster than BerkeleyGW, due to reduced dimensionality and low-rank representations, enabling more scalable GW calculations. Key contributions include a cubic-scaling inversion in $N_{\text{vc}}$ space, rigorous error bounds for self-energies, and demonstrations of favorable low-rank properties in both $M_{\text{vc}}$ and $\chi$, which together support robust and efficient GW computations for large systems. The practical impact lies in enabling accurate excited-state predictions for larger molecules and extended solids with significantly reduced computational resources.

Abstract

The dielectric response function and its inverse are crucial physical quantities in materials science. We propose an accurate and efficient strategy to invert the dielectric function matrix. The GW approximation, a powerful approach to accurately describe many-body excited states, is taken as an application to demonstrate accuracy and efficiency. We incorporate the interpolative separable density fitting (ISDF) algorithm with Sherman--Morrison--Woodbury (SMW) formula to accelerate the inversion process by exploiting low-rank properties of dielectric function in plane-wave GW calculations. Our ISDF--SMW strategy produces accurate quasiparticle energies with $O(N_{\mathrm{r}}N_{\mathrm{e}}^2)$ computational cost $(N_{\mathrm{e}}$ is the number of electrons and $N_{\mathrm{r}}=100$--$1000N_{\mathrm{e}}$ is the number of grid points) with negligible small error of $0.03$ eV for both complex molecules and solids. This new strategy for inverting the dielectric matrix can be \(50\times\) faster than the current state-of-the-art implementation in BerkeleyGW, resulting in two orders of magnitude speedup for total GW calculations.

A fast low-rank inversion algorithm of dielectric matrix in GW approximation

TL;DR

This work targets the computational bottleneck in GW calculations—the inversion of the dielectric function matrix—by introducing a fast, cubic-scaling, low-rank inversion strategy that combines ISDF with the Sherman-Morrison-Woodbury formula and a Cauchy-integral refinement. Applied to GW within the static COHSEX framework, the approach achieves cost with grid points in the range --, and an accuracy around for both molecular and solid systems. The method yields substantial speedups, up to roughly 50x faster than BerkeleyGW, due to reduced dimensionality and low-rank representations, enabling more scalable GW calculations. Key contributions include a cubic-scaling inversion in space, rigorous error bounds for self-energies, and demonstrations of favorable low-rank properties in both and , which together support robust and efficient GW computations for large systems. The practical impact lies in enabling accurate excited-state predictions for larger molecules and extended solids with significantly reduced computational resources.

Abstract

The dielectric response function and its inverse are crucial physical quantities in materials science. We propose an accurate and efficient strategy to invert the dielectric function matrix. The GW approximation, a powerful approach to accurately describe many-body excited states, is taken as an application to demonstrate accuracy and efficiency. We incorporate the interpolative separable density fitting (ISDF) algorithm with Sherman--Morrison--Woodbury (SMW) formula to accelerate the inversion process by exploiting low-rank properties of dielectric function in plane-wave GW calculations. Our ISDF--SMW strategy produces accurate quasiparticle energies with computational cost is the number of electrons and -- is the number of grid points) with negligible small error of eV for both complex molecules and solids. This new strategy for inverting the dielectric matrix can be faster than the current state-of-the-art implementation in BerkeleyGW, resulting in two orders of magnitude speedup for total GW calculations.
Paper Structure (13 sections, 32 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 13 sections, 32 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Result and discussion
  3. Theoretical framework of fast inversion of dielectric matrix and its application in GW calculation
  4. Errors of self-energies
  5. Low-rank properties for functions and operators
  6. Computational complexity
  7. Methods
  8. GW approximation
  9. Interpolative separable density fitting
  10. Sherman--Morrison--Woodbury (SMW) formula combined with ISDF
  11. Cauchy integral representation
  12. Cubic scaling algorithm for G$_0$W$_0$ calculation
  13. Error analysis of implementing the ISDF algorithm on G$_0$W$_0$ calculation

Figures (4)

  • Figure 1: Flowchart difference between conventional and low-rank G$_0$W$_0$ calculations. The inversion step is highlighted.
  • Figure 2: Errors of self-energies for Si$_{64}$ in improved GW calculation strategy. (a) and (c): The ISDF coefficient $k_{\mathrm{vn}}$ and $k_{\mathrm{nn}}$ are fixed to $8.0$, and $k_{\mathrm{vc}}$ varies from $6.0$ to $10.0$. (b) and (d): The ISDF coefficient $k_{\mathrm{vc}} = 8.0$ is fixed, and $k_{\mathrm{vn}}=k_{\mathrm{nn}}$ varies from $6.0$ to $10.0$. The green, purple, and orange lines in (d) represent $k_{\mathrm{vn}} = k_{\mathrm{nn}} = 6.0$, $k_{\mathrm{vn}} = k_{\mathrm{nn}} = 8.0$, and $k_{\mathrm{vn}} = k_{\mathrm{nn}} = 10.0$, respectively. (a) to (d) do not involve Cauchy integral. (e) and (f): The ISDF coefficients $k_{\mathrm{vc}} = k_{\mathrm{vn}} = k_{\mathrm{nn}} = 8.0$ are fixed, and the upper bound of the relative error of Cauchy integral result in Algorithm \ref{['algo:COmegaC']} varies from $10^{-2}$ to $10^{-6}$. (g): Errors of Cauchy integral in terms of the number of quadrature nodes.
  • Figure 3: Singular values of conventional $M_{\mathrm{vc}}$ and $\chi$ in (a) Si$_{64}$ and (b) (SrTiO$_3$)$_8$. Here, $k_{\mathrm{vc}}=8.0$, $N_{\mu}=k_{\mathrm{vc}}\sqrt{N_{\mathrm{v}}N_{\mathrm{c}}}$, $N_{\mathrm{v}}$ and $N_{\mathrm{c}}$ are given in Table 1 of Supplementary information.
  • Figure 4: Total time for the conventional strategy, strategy from ma2021GWISDF and our improved strategy on Si with different bulk sizes. The ISDF coefficients are set to be $8.0$, and the threshold for Cauchy integral is set to be $10^{-7}$. (a) Execution time from calculating all operators to calculating self-energies in G$_0$W$_0$ calculation. The time for ISDF algorithm is included. (b) Execution time until obtaining the screened Coulomb interaction matrix $W$, excluding the time for ISDF. BerkeleyGW is not listed here as its implementation is a bit different so that it is not easy to make a fair comparison. (c) Total execution time for inverting $\epsilon$.