Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations

Abstract

Pulay's Direct Inversion in the Iterative Subspace (DIIS) method is one of the most widely used mixing schemes for accelerating the self-consistent solution of electronic structure problems. In this work, we propose a simple generalization of DIIS in which Pulay extrapolation is performed at periodic intervals rather than on every self-consistent field iteration, and linear mixing is performed on all other iterations. We demonstrate through numerical tests on a wide variety of materials systems in the framework of density functional theory that the proposed generalization of Pulay's method significantly improves its robustness and efficiency.

Uncertainty quantification through a model-based fuzzy set membership function

“…As the GRPulay algorithm failed to converge in the graphene fragment case, we proposed a modification, abbreviated as GR-Pulay-LM: the new charge density was generated using the standard linear mixing scheme, instead of the fixed α = 1.0 of the original GR-Pulay. This is similar to, but not the same as, the approach in (Banerjee et al, 2016). The last algorithm was our newly proposed adaptable hybrid scheme.…”

“…The GR-Pulay-LM scheme (our modification of the GR-Pulay algorithm) is competitive especially for the higher mixing parameter values in the complex test problem. In future we plan to compare the properties of the proposed scheme to other recently published schemes such as (Banerjee et al, 2016).…”

Effect of electron-lattice coupling on charge and magnetic order in rare-earth nickelates

“…The literature on nonlinear iterative equation solution techniques suggests that the choice of the mixing parameter α is typically problem-specific. 35 We find the smoothest and most reliable convergence over most of the parameter space of interest occurs for α = 0.3. This type of update can avoid the iteration being stuck in a loop, or helplessly hopping on either side of a "flat minimum".…”

“…The DIIS can quickly maximize the potential of the vectors within the subspace W N,n , but if the true solution is outside the subspace by more than the allowed residual 0 , then no matter how many times the Pulay mixing is carried out, it will not result in improved convergence. The solution 35 is to intersperse Pulay mixing with regular updates of the form (22), with a given periodicity k (typically k = 3). The algorithm thus alternates between expanding its iterative subspace, and finding the lowest residual vector within it, resulting in optimal convergence for most parameter values.…”

M-SPARC: Matlab-Simulation Package for Ab-initio Real-space Calculations

“…For the very first electronic ground state calculation, the superposition of isolated atom electron densities is used as initial guess for the electron density, whereas for every subsequent such calculation, extrapolation based on previous solutions is used [31]. The convergence of the SCF iteration is accelerated using the restarted variant of the Periodic Pulay mixing scheme [32,33], with the option of real-space preconditioning [34]. For spin-polarized calculations, mixing is performed simultaneously for both spin components, i.e., using a vector of twice the original length containing both spin-up and…”