Optimization of Density Functional Methods for Atomic Structure Calculations

我们解决了建立在原子尺度物理和化学基础上的分子模拟所面临的数值挑战。模拟大规模分子系统的分子结构计算在量子化学、物理学和材料科学中具有重要意义。与该项目合作的人员正在使用它们来确定氧化物表面的电荷密度，确定纳米材料的原子结构，优化固体氧化物燃料电池中纳米材料的性能，并探索减少化学废物和提高催化剂效率的方法。在世界范围内，这些类型的计算正在耗尽世界的超级计算能力。S；这些计算背后的任何数值算法的任何进步都将通过扩大我们模拟越来越大的分子系统的能力而产生立竿见影的影响。这些类型的计算问题带来的数学挑战是数学研究的前沿；这些挑战是维度、非线性和模型不一致。我们从两个方向来解决这个问题。首先，我们将解决大规模优化问题的传统技术--所谓的有限记忆技术--与通过直接考虑模型中存在的多个尺度以及不一致的近似来稳定和加速这些技术的方法相结合。这涉及到对本地数值模型的敏感性的调查，以及针对有关系统真实状态的不完善或不完整信息的数学保障措施的发展。我们处理该问题的第二个方向是将模型划分为更小、更易于计算的部分--称为运算符拆分。运算符拆分方法打开了一系列算法选择的大门，允许人们以新颖的方式组合计算，从而避免更传统方法常见的问题；其中主要的是在糟糕的“局部解”处停滞不前的现象。由PI为与分子结构确定相关的问题开发的具有相似数学特征的新的算符分裂技术已被证明是有效的。这些算法背后的理论仍在由PI发展，并将当前的理论扩展到所谓的非凸和不一致问题的重要情况，这些问题在应用中普遍存在，远远超出了分子结构计算的范围。

We address numerical challenges facing simulations of molecules built upon atomic-scale physics and chemistry. Molecular structure calculations modeling large-scale molecular systems are of fundamental importance to quantum chemistry, physics, and materials science. They are being used by collaborators with this project for the determination of charge density at oxide surfaces, for atomic structure determination of nanoscale materials, for optimization of the performance of nanoscale materials in solid-oxide fuel cells, and for exploring ways to reduce chemical waste and improve the efficiency of catalysts. Worldwide, these types of calculations are exhausting the limits of the world''s supercomputing capacity; any advancements in the numerical algorithms behind these calculations would have immediate impact by expanding our capabilities for simulating ever-larger molecular systems. The mathematical challenges presented by these types of computational problems are at the frontiers of mathematical research; these challenges are dimensionality, nonlinearity, and model inconsistency.We approach the problem from two directions. In the first, we combine conventional techniques for solving large-scale optimization problems -- so-called limited memory techniques -- with a fresh look at ways to stabilize and accelerate the techniques by directly accounting for the presence of multiple scales as well as inconsistent approximations in the model. This involves an investigation into the sensitivity of the local numerical model and the development of a mathematical safeguard against imperfect or incomplete information about the true state of the system. Our second direction of approach to the problem is to divide the model into smaller, more computationally tractable pieces -- known as operator splitting. Operator-splitting methods open the door to a wide range of algorithmic options that allow one to combine computations in novel ways that avoid problems common for more conventional approaches; principal among these is the phenomenon of stagnation at a bad ''local solution''. New operator-splitting techniques developed by the PI for problems related to molecular structure determination and that share similar mathematical features have proven effective. The theory behind these algorithms is still being developed by the PI and extends current theory to the important cases of so-called nonconvex and inconsistent problems that are ubiquitous in applications far beyond molecular structure calculations.