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

Density functional theory calculations
use a significant fraction
of current supercomputing time. The resources required scale with
the problem size, the internal workings of the code, and the number
of iterations to convergence, with the latter being controlled by
what is called “mixing”. This paper 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 the Anderson method and a hybrid approach, all
with the same predictive method. Additional comparisons are made for
36 cases with a fixed algorithm greed. The predictive method works
well independent of which method is used for the candidate step, and
it is capable of adapting to different problem types particularly
when coupled with the hybrid approach.

密度泛函理论（Density Functional Theory）计算占据了当前超级计算算力的相当一部分份额。其所需资源随问题规模、程序内部运行机制以及收敛迭代次数而变化，其中收敛迭代次数由所谓的‘混合（mixing）’策略调控。本文提出了一种用于处理上述及其他不动点问题（fixed-point problems）中信赖域（trust regions）的全新方法。该方法并非依据迭代改进效果调整信赖域，而是利用先前迭代步估算参数与信赖域的合理取值，从过往迭代历史中有效推导出最优波利亚（Polyak）步。针对8种结构，本文分别采用"good"与"bad"两类多割线（multisecant）变体、安德森（Anderson）方法以及混合方法展开测试，所有测试均采用统一的预测方法，最终给出了详细的测试结果。此外，本文还针对36个测试案例，采用固定贪心算法开展了额外对比实验。该预测方法适配性优异，不受候选步选取方法的限制，且尤其在与混合方法结合时，能够更好地适配不同类型的问题。