Domain decomposition methods for electronic structure calculations

电子结构计算在理论化学、物理和材料科学中是至关重要的。事实上，在被引用次数最多的10篇科学论文中，有两篇与这个主题有关。尽管该领域涉及特征值问题的离散化，这是数值分析中一个成熟的主题，但应用数学的专业知识很少涉及。在提议的项目中，目标是为电子结构计算中出现的特征值问题开发新的域分解（DD）算法，如Kohn-Sham DFT（密度泛函理论）方程。本项目中的方法基于特征值问题的域分解与源问题的域分解没有本质上的不同。尽管如此，特征值问题的dd方法还是不如源问题流行。此外，最近在隐式溶剂化模型背景下的工作和对这些方法的分析表明，即使没有所谓的粗校正，dd方法也可以扩展到越来越多的固定大小子结构域，例如链状分子或蛋白质。该项目涉及的问题是多方面的，包括：i)非线性特征值问题，ii)大量待确定的特征值，iii)无界域上的特征值问题，iv)处理包含库仑奇点的势。该项目是一个更广泛的长期计划的第一步，该计划基于局部降阶建模推导出高效的局部基函数，作为广泛使用的经验收缩高斯基函数的替代方案。事实上，领域分解策略允许对方程进行局部化，并为在第二步中使用经过认证的后验误差估计的降阶建模的应用打开了大门。

Electronic structure calculations are paramount in theoretical chemistry, physics, and material science. Indeed, among the top ten most cited scientific articles, two papers are related to this topic. Despite the fact that this field deals with the discretization of eigenvalue problems, which is a well-established subject in numerical analysis, the expertise of applied mathematics is little involved.Within the proposed project, the aim is to develop novel domain decomposition (DD) algorithms for eigenvalue problems which arise in electronic structure calculation like the Kohn-Sham DFT (Density Functional Theory) equations. The approach within this project is based on the idea that domain decomposition for eigenvalue problems is not fundamentally different than for source problems. Despite this fact, DD-methods for eigenvalue problems are less popular than for source problems. Additionally, recent work in the context of implicit solvation models and the analysis of those methods show that the DD-method is scalable for domains of an increasing number of fixed-size sub-domains, like e.g. for chain-like molecules or proteins, even without a so-called coarse-correction.The problems to be embraced within this project are manifold and contain: i) non-linear eigenvalue problems, ii) a large number of eigenvalues to be determined, iii) eigenvalue problems on unbounded domains and iv) dealing with potentials that contain Coulomb-like singularities.The project is a first step within a broader long-term plan to derive efficient local basis functions based on local reduced order modeling as an alternative of the widely-used but empirical contracted Gaussian basis functions. In fact, the domain decomposition strategy allows to localise the equations and opens the door to the application of reduced order modeling with certified a posteriori error estimates in a second step.