New Preconditioned Solvers for Large and Complex Eigenvalue Problems

用于大型复杂特征值问题的新预处理求解器

基本信息

批准号：
1819097
负责人：
Fei Xue
金额：
$ 20万
依托单位：
Clemson University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819097&HistoricalAwards=false
关键词：
New Preconditioned Solvers Large Complex

项目摘要

This project aims to develop new algorithms that will help enable large-scale eigenvalue-related modeling and simulations in many scientific and engineering areas, including linear stability analysis of dynamical systems, mechanics of materials, optimization of acoustic emissions, study of superconductivity, vibrations under conditions of uncertainty, and many more. Certain methods are also of significance to other areas; for example, a fast exponential matrix-vector product is essential to the exponential integrator for solving stiff time-dependent differential equations that are difficult to tackle by traditional methods. New textbook writing and graduate student mentoring will help cultivate qualified researchers and industrial professionals to generate further impact in future.Eigenvalues and closely related mathematical tools (e.g., pseudospectra) are fundamentally descriptive in many areas of applied mathematics and scientific computing. This project concerns systematic development and analysis of innovative preconditioned solvers for several important classes of large-scale and complex eigenvalue-related problems. For large linear symmetric eigenproblems, variants of preconditioned eigensolvers have been thoroughly investigated and widely used with great success in many applications. The plan is to show that the preconditioning and the solver framework can both be generalized significantly and integrated with great flexibility to solve a much broader class of challenging eigenvalue-related problems. The methods to be developed will be reliable, efficient and flexible. The specific research topics include (i) preconditioning (spectral filtering) with matrix functions, (ii) solving nonlinear eigenproblems, and (iii) computing spectra and pseudospectra of large structured matrices.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.

该项目旨在开发新的算法，帮助在许多科学和工程领域实现大规模特征值相关的建模和模拟，包括动力系统的线性稳定性分析、材料力学、声发射优化、超导性研究、振动下的振动不确定性条件等等。某些方法对于其他领域也具有重要意义；例如，快速指数矩阵向量乘积对于指数积分器来说至关重要，用于求解传统方法难以解决的刚性瞬态微分方程。新教科书的编写和研究生指导将有助于培养合格的研究人员和工业专业人士，以在未来产生进一步的影响。特征值和密切相关的数学工具（例如伪谱）在应用数学和科学计算的许多领域中具有基本的描述性。该项目涉及针对几类重要的大规模和复杂特征值相关问题的创新预处理求解器的系统开发和分析。对于大型线性对称本征问题，预处理本征求解器的变体已被彻底研究并广泛使用，在许多应用中取得了巨大成功。该计划旨在表明预处理和求解器框架都可以显着推广并以极大的灵活性进行集成，以解决更广泛的具有挑战性的特征值相关问题。所开发的方法将是可靠、高效和灵活的。具体研究主题包括（i）使用矩阵函数进行预处理（谱滤波），（ii）解决非线性特征问题，以及（iii）计算大型结构化矩阵的谱和伪谱。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。