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的密度泛函理论(DFT)中出现的非线性特征值问题的后验误差估计。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。目标是提供近似能量和精确能量之间误差的有保证的界。DFT模型主要使用所谓的自洽场(SCF)迭代来求解，其中在SCF迭代过程中的每一步都要求解线性本征值问题。估计器将被设计成可以量化总误差的每个误差分量，即由平面波离散化引起的离散化误差，由非收敛SCF迭代引起的迭代误差，以及由线性特征值问题的迭代求解器引起的误差。这种分离允许设计具有不同误差源之间的误差平衡的自适应算法。对于凸DFT模型，估计器将是误差的保证上界，而最终的工作包将处理对非凸模型的扩展。所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.