Predictive Mixing for Density Functional Theory (and other Fixed-Point Problems)

Abstract

Density functional theory calculations use a significant fraction of current supercomputing time. The resources required scale with the problem size, internal workings of the code and the number of iterations to convergence, the latter being controlled by what is called mixing. This note describes a new approach to handling trust-regions within these and other fixed-point problems. Rather than adjusting the trust-region based upon improvement, the prior steps are used to estimate what the parameters and trust-regions should be, effectively estimating the optimal Polyak step from the prior history. Detailed results are shown for eight structures using both the Good and Bad Multisecant versions as well as Anderson and a hybrid approach, all with the same predictive method. Additional comparisons are made for thirty-six cases with fixed algorithm Greed The predictive method works well independent of which method is used for the candidate step, and is capable of adapting to different problem types particularly when coupled with the hybrid approach. It would be premature to claim that it is the best possible approach, but the results suggest that it may be.