A posteriori error estimates and adaptive strategies for nonlinear models in electronic structure calculations

本课题研究电子结构计算中密度泛函理论（DFT）中非线性特征值问题的证明后验误差估计。该项目建立在为更简单的Gross-Piteavskii特征值问题建立的初步工作的基础上。目标是在近似能量和精确能量之间提供有保证的误差界限。DFT模型主要使用所谓的自洽域（SCF）迭代来求解，其中在SCF迭代过程中的每一步都求解一个线性特征值问题。估计器的设计使总误差的每个误差分量都可以量化，即由于平面波离散化引起的离散误差，由于非收敛的SCF-迭代引起的迭代误差，以及由于线性特征值问题的迭代解算器引起的误差。这种分离允许在不同误差源之间设计具有误差平衡的自适应算法。估计量将是凸dft模型误差的保证上界，最后的工作包将处理扩展到非凸模型。开发的方法和估计器将在开源的DFTK软件包中实施和测试。

This project deals with certified a posteriori error estimations of nonlinear eigenvalue problems arising in Density-Functional Theory (DFT) of electronic structure calculations. The project builds upon preliminary work established for the simpler Gross-Piteavskii eigenvalue problem. The goal is to provide guaranteed bounds of the error between the approximate energy and the exact energy. DFT models are dominantly solved using the so-called Self-Consistent Field (SCF) iterations where a linear eigenvalue problem is solved at each step within the SCF-iterative procedure. The estimator will be designed such that each error component of the total error can be quantified, namely the discretization error due to the planewave discretization, the iteration error due to the non-converged SCF- iterations, and the error due to the iterative solver of the linear eigenvalue problems. This splitting allows the design of an adaptive algorithm with error balancing between the different error sources. The estimator will be a guaranteed upper bound of the error for convex DFT-models and the final work-package will treat the extension to non-convex models. The developed methods and estimators will be implemented and tested in the open-source DFTK software package.