ITR/AP(DMS): Semidefinite Programming for Electronic Structure

这个项目的目的是在多电子系统性质的第一性原理计算的应用指导下，促进大规模半定规划和特征值优化的快速迭代算法的发展。半定优化代码将被重写，以适合于大型和稀疏问题的迭代求解过程来取代稠密线性代数和直接矩阵分解。将用不同的数学公式和稀疏优化程序的实现进行数值实验。其指导应用是电子结构基态问题的变分形式，其中未知数是多电子系统的一体和二体约化密度矩阵。这一应用产生了具有稀疏解矩阵和极稀疏约束集的大型半定程序。半定编程是许多科学和工程应用的有价值的框架，包括系统控制、结构分析、组合优化、统计估计和VLSI设计，仅举几个除了电子结构理论之外的一些成熟的领域。这个项目旨在提高我们计算问题实例的解决方案的能力，这些问题实例的大小超出了当前数值方法的范围。对于这个项目的具体应用，基于量子力学的多体系统性质的计算是理解分子和固态结构、化学和基本生化过程以及凝聚态物质的力学和电磁性质的基础。例如，多体量子力学计算精度和规模的进步可能给技术和社会带来好处，包括半导体、磁存储材料、高温超导体和化学催化剂的更好设计，以及合理药物设计和碳纳米结构设计等新兴领域的改进。

This project aims to advance the state of the art in fast iterative algorithms for large-scale semidefinite programming and eigenvalue optimization, guided by applications to first-principles computation of properties of systems of many electrons. Semidefinite optimization codes will be rewritten to replace dense linear algebra and direct matrix factorizations by iterative solution processes suited to large and sparse problems. Numerical experiments will be carried out with different mathematical formulations and implementations of the sparse optimization procedures. The guiding application is a variational formulation of the electronic structure ground state problem in which the unknowns are the one-body and two-body reduced density matrices of a many electron system. This application gives rise to a large semidefinite program having sparse solution matrices and an extremely sparse constraint set. The codes that are developed under this grant will be written and documented with the intent of wide dissemination.Semidefinite programming is a valuable framework for many scientific and engineering applications, including systems control, structural analysis, combinatorial optimization, statistical estimation and VLSI design, just to name a few well-established ones besides electronic structure theory. This project aims to increase our ability to compute solutions to problem instances whose size places them beyond the reach of current numerical methods. As to the specific application in this project, the computation of properties of many-body systems on the basis of quantum mechanics is fundamental to the understanding of molecular and solid state structure, chemical and elementary biochemical processes, and mechanical and electromagnetic properties of condensed matter. Potential benefits to technology and society from advances in the accuracy and scale of many-body quantum mechanical computations include, for example, better design of semiconductors, magnetic storage materials, high-temperature superconductors and chemical catalysts, as well as improvements in the emerging areas of rational drug design and the design of carbon nanostructures.