Collaborative Research: Efficient Solvers for Nonlinear Eigenvalue Problems and Applications

这个项目包括发展先进的计算方法来解决真正的非线性特征值问题。在这个项目中，库布洛夫斯卡娅的非线性QR算法与现代的显示秩和结构保持技术巧妙地结合在一起，解决了中小型稠密问题，增强了新方法的能力。一种新的基于Pade有理逼近的修剪线性化扩展了对求解大型稀疏问题的增强。研究人员试图发展一个系统的和统一的处理相关的数学理论，并产生真正的非线性特征值问题的数值方法和软件工具。除了推进非线性特征值问题的研究，该项目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普遍存在，它们出现在结构动力学的研究、模拟纳米结构光伏转换材料以促进能源科学等许多场景中。本征值解释了一系列物理现象，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。许多本征值问题自然地以非线性形式出现。在这个项目中，研究人员不依赖线性化近似来研究潜在的非线性问题。极大地改进了计算非线性特征值问题解的方法，给广泛的实际应用带来了立竿见影的好处。

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.