A robust and efficient line search for self-consistent field iterations

In an ongoing effort with Antoine Levitt our aim is to develop reliable density-functional theory (DFT) methods for computational materials design. Recently we looked into a strategy to automatically select the damping parameter for the self-consistent field iterations (SCF). Our adaptive damping approach is based on a theoretically sound quadratic model for the DFT energy, which is used to fix the step size (damping) adaptively along the search directions suggested by an underlying algorithm (such as Pulay mixing, Kerker mixing, etc.). Our algorithm is fully automatic, i.e. an a priori damping selection is no longer required. In our work we test our method successfully on a range of challenging systems including supercells, transition-metal alloys or metallic surfaces. Overall our study shows adaptive damping to provide superior robustness over the traditional fixed-damping approach.

As I have reported in previous blog articles and we also discussed in our previous publication on black-box mixing strategies for inhomogeneous systems the main motivation of our work is to design numerical methods, which are parameter-free and automatically self-adapt to the simulated material. In modern simulation scenarios where millions of DFT calculations are required in order to generate training data or screen over large design spaces, robustness and automation are the key requirements. Often it is in fact less the computational time of the individual calculations, which limits overall throughput. Much rather it is the human factor, i.e. the human time required to setup, check and verify computations.

Clearly at the level of millions of calculations computational parameters can no longer be selected manually. Instead elaborate heuristics are employed to select basis set size, k-point sampling, SCF algorithm or the damping parameter. In case a calculation fails heuristics are also employed for automatic restart. However, this approach is far from perfect and even an optimistic 1% failure rate easily equals thousands of calculations, which require human attention. With our work (both the previous paper as well as this one) we want to replace heuristic approaches to parameter selection by algorithms that employ a mixture of mathematical and physical insight to automatically adapt to the simulation at hand. As we demonstrate in this work, such algorithms might be associated with an increased effort compared to the best possible parameter setting, however it also makes calculations overall more robust. Therefore one saves (a) on the repeated effort to find a suitable parameter set by trial and error and (b) reduces the fraction of calculations, which need to be considered by a human. Overall the maximally attainable throughput can therefore be expected to increase from such a robust scheme despite the fact that an individual calculation might be more costly.

In this work in particular we considered the question of choosing the damping parameter. For this our adaptive damping approach is based on constructing an approximate quadratic model for the DFT energy and using this model within a line search procedure. Since this procedure is associated with an additional cost, we only employ it in case the proposed SCF step would either increase the DFT energy or SCF residual Notably our approach introduces no changes to the SCF in case each proposed SCF step by the mixing procedure is already perfect (i.e. energy or residual decreasing). Therefore adaptive damping can be considered a safeguared, which only comes into play if the proposed steps are noisy or erroneous. Adaptive damping is by construction orthogonal to any existing mixing and convergence acceleration technique for DFT methods and in our work we demonstrate it to integrate readily into an Anderson-accelerated SCF for various challenging systems. Overall we managed to increase performance and robustness at only a minor extra cost. The full abstract of our paper reads

We propose a novel adaptive damping algorithm for the self-consistent field (SCF) iterations of Kohn-Sham density-functional theory, using a backtracking line search to automatically adjust the damping in each SCF step. This line search is based on a theoretically sound, accurate and inexpensive model for the energy as a function of the damping parameter. In contrast to usual SCF schemes, the resulting algorithm is fully automatic and does not require the user to select a damping. We successfully apply it to a wide range of challenging systems, including elongated supercells, surfaces and transition-metal alloys.

Michael F. Herbst and Antoine Levitt. A robust and efficient line search for self-consistent field iterations. (submitted). arXiv:2109.14018 [code]