Advanced Eigensolvers for Science and Engineering Applications

大规模特征值计算是计算数学和计算科学与工程中的一个老大难问题。在模拟和数据分析中，它经常被视为一个关键的核心。在通用特征解析器和利用潜在的特定数学属性和数据结构的专用特征解析器方面都取得了重大进展。然而，科学和工程应用不断涌现出新的需求和挑战。这个项目涉及两类新兴的特征值问题的高级数学分析和稳健、高效的算法的发展：稀疏加低阶线性特征值问题和具有特征值非线性的特征值问题。此外，该项目对培养跨学科研究的研究生具有更广泛的影响。这两类特征值问题的新求解器的计算稳定性、效率和可靠性将通过巧妙地利用潜在的数学性质和矩阵结构而得到极大的提高。特别是，对于具有特征值非线性的特征值问题，新的求解器将非线性的有理近似用于高精度，剪裁线性化用于低维，投影子空间基的紧凑表示用于节省内存和通信效率。该项目的成果将是出版新的理论和算法以及开放源码软件。

Large-scale eigenvalue computation is a long-standing problem in computational mathematics and computational science and engineering. It is frequently encountered as a critical kernel in simulations and data analysis. Significant progress has been made both in general-purpose eigensolvers and also in specialized eigensolvers that exploit underlying particular mathematical properties and data structure. However, new needs and challenges continue to emerge from science and engineering applications. This project involves the development of advanced mathematical analysis and robust, efficient algorithms for two emerging classes of eigenvalue problems: sparse plus low rank linear eigenvalue problems and eigenvalue problems with eigenvalue nonlinearity. In addition, this project has broader impacts in training graduate students in interdisciplinary research. While much of the work involves significant technical expertise, other areas can be successfully understood and tackled by advanced undergraduates.The computational stability, efficiency and reliability of the new solvers for the two classes of eigenvalue problems will be greatly enhanced by skillful exploitation of underlying mathematical properties and matrix structure. In particular, for the eigenvalue problems with eigenvalue nonlinearity, new solver will combine rational approximations of nonlinearity for high accuracy, trimmed linearizations for low dimensionality, and compact representations of the projection subspace bases for memory-saving and communication efficiency. The outcomes of this project will be the publication of new theory and algorithms and open-source software.