Error control and post-processing techniques in plane-wave density functional theory

Another important topic at the moment in electronic structure calculations is the estimation of errors, which can be of different sources, among which we can list:

1. modelling error, coming from the approximation of the $N$-body Schrödinger equation, fully known but way too complicated to be solved for real systems, by different models (e.g. Hartree-Fock, Kohn-Sham DFT, Gross-Pitaevskii, Coupled Cluster...);

2. discretization error, coming from the numerical approximation of the continuous equations given by the chosen model;

3. algorithmic error, coming from the resolution of the discretized equations.

Again, there is a lot to do in these topics for applied mathematicians. For instance, regarding the discretization error, there is a judgemental litterature on the topic for linear PDEs and eigenproblems, but there is much fewer results regarding nonlinear elliptic eigenproblems that are at the heart of electronic structure calculations. In my first works on these topics, we could propose efficient error estimates for nonlinear Kohn-Sham models. However, the strategy being based on the linearization of the underlying equations, such bounds are not guaranteed. We observed that they were still very useful in practice, with some mathematical justification in specific cases.

At the moment, with additional insights from works in progress, my understanding of the problem is that fully guaranted error estimates for nonlinear elliptic eigenproblems are either very expensive or too coarse to be used in practice. All hope is not gone as it is still possible to obtain efficient estimators if the guaranted constrainted is relaxed.

Note that getting an estimate of the error on some quantity is not the goal per se: such estimates can often be used in a post-processing calculations to enhance the accuracy too.

Related works:

• A. Bordignon, E. Cancès, G. Dusson, G. Kemlin, R.A. Lainez Reyes, B. Stamm. Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals. SIAM Journal on Scientific Computing, 47(5):A2881-A2905, 2025. doi - pdf

• E. Cancès, M. F. Herbst, G. Kemlin, A. Levitt, and B. Stamm. Numerical stability and efficiency of response property calculations in density functional theory. Letters in Mathematical Physics, 113(1):21, 2023. doi - pdf

• E. Cancès, G. Dusson, G. Kemlin, and A. Levitt. Practical error bounds for properties in plane-wave electronic structure calculations. SIAM Journal on Scientific Computing, 44(5):B1312-B1340, 2022. doi - pdf