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一个长期存在的问题。在模拟和数据分析中，它经常作为一个关键的内核出现。在通用特征求解器和利用潜在的特殊数学性质和数据结构的专门特征求解器方面都取得了重大进展。然而，科学和工程应用不断出现新的需求和挑战。本项目涉及发展先进的数学分析和鲁棒，有效的算法为两类新兴的特征值问题：稀疏加低秩线性特征值问题和特征值非线性特征值问题。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级本科生成功理解和解决。通过巧妙地利用基础数学性质和矩阵结构，将大大提高这两类特征值问题新解的计算稳定性、效率和可靠性。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的合理近似以获得高精度，裁剪线性化以获得低维数，以及投影子空间基的紧凑表示以节省内存和通信效率。该项目的成果将是发表新的理论和算法以及开源软件。

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.