Improving Numerical Methods for Large Eigenvalue Problems

特征值问题是计算科学和工程的基石。它们的应用范围从物理和化学中的电子结构计算，生物学中的超分子系统动力学到土木和机械工程中的结构和振动。新兴应用包括锂离子电解质和石墨烯等新材料的研究和设计，以及寨卡病毒和西尼罗河病毒等超分子系统病毒衣壳动力学的研究。这些应用需要大量的特征值。能够有效地计算大量特征值的能力不仅具有吸引力，而且对于下一代特征求解器也是强制性的。在这个项目中，我们将协同努力发展大特征值问题的数学理论和数值方法。该项目的结果将为科学家和工程师提供数学理论和计算工具，以在更短的时间内获得更精确的模拟输出，并允许他们追求更富有成效的模拟策略。该项目将把研究活动整合到跨学科的教学、教育和培养计算数学前沿的研究生中。该奖学金将资助一名研究生。为了解决大型线性特征值问题的现有算法和软件的挑战性问题，我们将重点关注两个核心技术。一种是明确的外部紧缩，用于可靠地移动计算出的特征对，以防止算法重新计算这些数量。第二种技术是在具有显式外部压缩的Krylov子空间解算器中用于快速稀疏加低秩矩阵向量积的避免通信的矩阵幂核。对于大型非线性特征值问题，我们开发了数学理论和算法模板来指导基于近似的非线性特征解算器的设计和实现。此外，我们将探讨一个新兴的大型非线性特征值问题的公式，其中潜在的非线性矩阵值函数不是显式可用的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Eigenvalue problems are the cornerstones of computational science and engineering. They arise in applications ranging from electronic structure calculations in physics and chemistry, dynamic of supramolecular systems in biology to structural and vibration in civil and mechanical engineering. Emerging applications include the investigation and design of new materials such as Lithium-ion electrolyte and graphene and the study of dynamics of viral capsids of supramolecular systems such as Zika and West Nile viruses. These applications require large number of eigenvalues. The capability of being able to efficiently compute large number of eigenvalues will not just be appealing, but also mandatory for the next generation of eigensolvers. In this project, we will undertake synergistic efforts to develop mathematical theory and numerical methods for large eigenvalue problems. The outcome of this project will provide mathematical theory and computational tools for scientists and engineers to obtain more precise simulation outputs in much less time, and to allow them to pursue more productive simulation strategies. This project will integrate research activities into interdisciplinary teaching, education and training of graduate students in the forefront of computational mathematics. One graduate student will be funded by this award.To address challenging issues of existing algorithms and software for large linear eigenvalue problems, we will focus on two core techniques. One is an explicit external deflation for reliably moving away the computed eigenpairs to prevent the algorithm from computing over again those quantities. The second technique is a communication-avoiding matrix powers kernel for fast sparse-plus-low-rank matrix-vector products in Krylov subspace solvers with the explicit external deflation. For large nonlinear eigenvalue problems, we develop mathematical theory and algorithm templates for guiding the design and implementation of approximation based nonlinear eigensolvers. In addition, we will explore an emerging formulation of large nonlinear eigenvalue problems where underlying nonlinear matrix-valued functions are not explicitly available.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.