论文标题
直接最小化和自洽迭代的收敛分析
Convergence analysis of direct minimization and self-consistent iterations
论文作者
论文摘要
本文涉及子空间优化问题的数值解决方案，包括最大程度地限制固定等级的正交投影仪的平滑功能。在电子结构计算（Hartree-fock和Kohn-Sham密度功能理论-DFT-模型）中，尤其是遇到此类问题。我们从数值分析的角度比较了两个简单的代表，即阻尼自一致的领域（SCF）迭代和梯度下降算法，是在现场竞争的两类方法：SCF和直接最小化方法。我们得出这些算法的渐近收敛速率，并分析了它们对问题的光谱差距和其他特性的依赖。我们的理论结果通过数值模拟在各种示例上得到补充，从具有可调参数的玩具模型到现实的Kohn-Sham计算。我们还提供了一个非二次功能的简单SCF迭代的混乱行为的示例。
This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree-Fock and Kohn-Sham Density Functional Theory -DFT- models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn-Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.
