标题： 迈向最佳标度DFT：任意温度下热力学和完备基组极限下的随机Hartree理论 显示英文标题

标题： Toward optimal-scaling DFT: stochastic Hartree theory in the thermodynamic and complete basis set limits at arbitrary temperature

摘要： 我们首次在Hartree近似下对随机密度泛函理论（DFT）进行了数学分析。 通过计算自由度数量 $n$ 的几乎最优或 $\tilde{O}(n)$ 标度的概念，我们提出了这一分析的动机，这种标度与电子数无关，在热力学极限和完全基组极限中均成立。确实，这种标度的潜力是随机DFT相对于传统轨道为基础的方法以及确定性轨道自由替代方法的主要动机。 我们强调了数学关注的三个关键目标，这些目标在我们的算法和分析中得以综合体现。首先，我们确定了一种特定的Hartree势能的随机估计器，其样本复杂度基本上独立于离散化大小。其次，我们将自洽场迭代重新表述为一种随机镜像下降法，其中Fermi-Dirac熵扮演Bregman势能的角色，并证明了几乎与离散化无关的迭代次数界限以达到固定精度。第三，受估计器的启发，我们引入了一种新的平方根Fermi-Dirac算符的极点展开方案，即使在完全基组极限下，每次镜像下降迭代的成本仍保持为 $\tilde{O}(n)$。 结合这些要素，我们在所选离散化基组的两个感兴趣的极限下建立了几乎最优的标度，假设条件合理。在多达 $10^{6}$ 自由度的问题上进行的大量数值实验验证了我们的算法，并支持了几乎最优标度的理论。 显示英文摘要

摘要： We present the first mathematical analysis of stochastic density functional theory (DFT) in the context of the Hartree approximation. We motivate our analysis via the notion of nearly-optimal or $\tilde{O}(n)$ scaling with respect to the number $n$ of computational degrees of freedom, independent of the number of electrons, in both the thermodynamic and complete basis set limits. Indeed, the promise of such scaling is the primary motivation for stochastic DFT relative to conventional orbital-based approaches, as well as deterministic orbital-free alternatives. We highlight three key targets for mathematical attention, which are synthesized in our algorithm and analysis. First, we identify a particular stochastic estimator for the Hartree potential whose sample complexity is essentially independent of the discretization size. Second, we reformulate the self-consistent field iteration as a stochastic mirror descent method where the Fermi-Dirac entropy plays the role of the Bregman potential, and we prove a nearly discretization-independent bound on the number of iterations needed to reach fixed accuracy. Third, motivated by the estimator, we introduce a novel pole expansion scheme for the square-root Fermi-Dirac operator, preserving $\tilde{O}(n)$ cost per mirror descent iteration even in the complete basis set limit. Combining these ingredients, we establish nearly-optimal scaling in both limits of interest under reasonable assumptions on the basis sets chosen for discretization. Extensive numerical experiments on problems with as many as $10^{6}$ degrees of freedom validate our algorithm and support the theory of nearly-optimal scaling.