Collaborative Research: Efficient Solvers for Nonlinear Eigenvalue Problems and Applications

这个项目涉及到解决真正非线性特征值问题的先进计算方法的发展。在本项目中，Kublanovskaya的非线性QR算法与现代中小型密集问题的秩揭示和结构保持技术的巧妙结合，增强了新方法的能力。一种新颖的通过Pade有理逼近的裁剪线性化扩展了解决大型但稀疏问题的增强。研究人员寻求对相关数学理论进行系统和统一的处理，并为真正的非线性特征值问题提供数值方法和软件工具。除了推进非线性特征值问题的研究外，该项目还为研究生提供了计算数学和跨学科研究工具方面的培训。特征值问题在计算科学和工程中无处不在，它们出现在结构动力学研究，纳米结构光伏转换材料的模拟中，以推进能源科学，以及许多其他场景。特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。许多特征值问题通常以非线性形式出现。在这个项目中，研究人员研究潜在的非线性问题，而不依赖于线性化近似。非线性特征值问题的计算方法有了实质性的改进，为广泛的实际应用带来了直接的好处。

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.