Julia code for asymptotic estimators for quantites of interest in electronic structure. It can be used to reproduce the main results from our paper Practical error bounds for properties in plane-wave electronic structure calculations by Éric Cancès, Geneviève Dusson, Gaspard Kemlin and Antoine Levitt, available on arXiv. In particular, it reproduces the asymptotic error bound on the H^1-norm of the error from Figure 2 (middle) and the asymptotic estimator for the forces from Figures 8, 10 and 11.

This code is used with the refined reference solutions we introduced in the paper. A small example of the estimates for the forces is available here in the DFTK documentation.

Dependencies

Julia 1.7 with the following Julia libraries:

• DFTK.jl v0.5.0 for the simulation environment (this code might not work with more recent versions of DFTK);
• ForwardDiff.jl for the computation of dF with automatic differentiation;
• LinearAlgebra, IterativeSolvers, LinearMaps, FFTW for the linear algebra related computations;
• HDF5, CSV, Tables for saving and reading results.

Usage

To perform the computations, first open the Julia shell with julia --project from the location of this repository and then run

using Pkg
Pkg.instantiate()

to install the required dependencies.

Then, you can run all the computations at once and generates data with include("run_all_computations.jl"). This will go in every folder case/ where case is successively silicon, GaAs and TiO2 with the numerical setting described in our paper. Each folder contains:

• a file case_forces.jl which runs the computations with the Schur complement from our paper to improve the estimates on the forces and then saves the results in the file forces_case.h5;
• a file generate_data_case_forces.jl which generates various .csv files for plots;
• different .tex files that read the associated .csv files to generates .pdf files:
  • error_case files represent the estimation on the error Π(P-P_*) in the H^1-norm, as in Figure 2 (middle), without the bad error bound;
  • forces_case files represent the estimation of the error F-F* as in Figure 8 (left) and 10;
  • diff_forces_case files represent the improvement of this estimations, as in Figures 8 (right) and 11.

Note that, after running the computations, only .h5 and .csv files have been generated. You need to compile the .pdf files with the associated .tex files and your favorite LaTeX compiler.

Computational time

As we have chosen a quite precise reference grid, with many coarser grids, we ran all the computations on 32 Intel Xeon E5-2667 (3.30GHz) cores, which takes around half a day on a such a cluster.

Contact

This is research code, not necessarily user-friendly, actively maintened or extremely robust. For a first contact with the implementation, looking at the example mentionned above is maybe advised. If you have questions or encounter problems, get in touch!