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长期存在的问题。它作为模拟和数据分析中的关键内核经常遇到。通用特征求解器和利用底层特定数学属性和数据结构的专用特征求解器都取得了重大进展。然而，科学和工程应用不断出现新的需求和挑战。该项目涉及针对两类新兴特征值问题开发高级数学分析和稳健、高效的算法：稀疏加低秩线性特征值问题和具有特征值非线性的特征值问题。此外，该项目在培养研究生跨学科研究方面具有更广泛的影响。虽然大部分工作涉及重要的技术专业知识，但其他领域可以由高年级本科生成功理解和解决。通过巧妙地利用基础数学特性和矩阵结构，两类特征值问题的新求解器的计算稳定性、效率和可靠性将大大提高。特别是，对于具有特征值非线性的特征值问题，新的求解器将结合非线性有理近似以实现高精度、修剪线性化以实现低维，以及投影子空间基的紧凑表示以节省内存和提高通信效率。该项目的成果将是新理论和算法以及开源软件的发布。

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.