Collaborative Research: Efficient Solvers for Nonlinear Eigenvalue Problems and Applications

协作研究：非线性特征值问题的高效求解器及其应用

• 批准号: 1115817
• 负责人: Zhaojun Bai
• 金额: $ 15.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115817&HistoricalAwards=false
• 关键词: Collaborative; Research; Efficient; Solvers; Nonlinear

This project involves the development of advanced computational methods for solving genuine nonlinear eigenvalue problems. In this project, skillful combination of Kublanovskaya's nonlinear QR algorithm with modern rank-revealing and structure-preserving techniques for small and medium size dense problems enhances the capabilities of new methods. A novel trimmed linearization via Pade rational approximation extends the enhancements for solving large but sparse problems. The investigators seek to develop a systematic and unified treatment of the relevant mathematical theory, and produce numerical methods and software tools for the genuine nonlinear eigenvalue problems. In addition to advancing research in nonlinear eigenvalue problems, the project provides training for graduate students in computational mathematics and interdisciplinary research tools.Eigenvalue problems are ubiquitous in computational science and engineering, where they arise in the study of dynamics of structures, simulation of nanostructured photovoltaic conversion materials to advance energy science, and many other scenarios. Eigenvalues explain a wide range of physical phenomena such as vibrations and frequencies, (in)stabilities of dynamical systems, and energy excitation states of electrons and molecules. Many eigenvalue problems occur naturally in nonlinear form. In this project, the investigators study the underlying nonlinear problems without relying on linearization approximations. The promise of substantially improved methods for computing solutions of nonlinear eigenvalue problems, brings immediate benefits to a wide range of practical applications.

这个项目涉及解决真正的非线性特征值问题的先进计算方法的发展。 在这个项目中，巧妙地结合Kublanovskaya的非线性QR算法与现代的排名揭示和结构保持技术的小型和中型密集的问题，提高了新方法的能力。一种新的修剪线性化通过帕德有理逼近扩展的增强解决大，但稀疏的问题。 研究人员寻求发展一个系统的和统一的处理相关的数学理论，并产生数值方法和软件工具，真正的非线性特征值问题。 本项目除了推进非线性特征值问题的研究外，还为研究生提供计算数学和跨学科研究工具的培训。特征值问题在计算科学和工程中无处不在，在结构动力学研究中，在纳米结构光伏转换材料的模拟中，它们会促进能源科学的发展，以及许多其他场景。本征值解释了广泛的物理现象，如振动和频率，动力系统的稳定性，以及电子和分子的能量激发态。 许多特征值问题自然地以非线性形式出现。在这个项目中，研究人员在不依赖线性化近似的情况下研究潜在的非线性问题。非线性特征值问题的求解方法得到了很大的改进，这给实际应用带来了直接的好处。