Structure-Preserving Algorithms for Solving Large Scale Eigenvalue Problems

大规模特征值计算的优化是计算数学和科学计算界长期存在的问题。在这个项目中，PI和他的合作者将开发新的结构保持算法，以准确和高效地解决大规模特征值问题。具体地说，他们将集中在大规模的结构化矩阵笔的特征值问题，二次特征值问题和非线性特征值问题。他们将研究解决这些特征值问题的保结构Rayleight-Ritz子空间投影技术，包括Krylov子空间的多级正交化过程、二阶Arnoldi(SOAR)方法和非线性Arnoldi方法。该项目的目标是通过应用计算数学和模拟，在科学努力和工程设计的前沿领域取得重大进展。目标应用包括电路和MEMS器件的模拟、结构动力学、声学和电磁学的有限元分析所产生的这些类型的特征值问题。这些应用对下一代电子、汽车效率和安全以及能效监测设备等具有重要的技术意义。该项目的结果将是对这些问题的有效计算模拟策略的描述，也将是公开可用的软件。

Optimization of large scale eigenvalue computations is a long-standingproblem in computational mathematics and scientific computing community. In this project, the PI and his collaborators will develop novel structure-preserving algorithms for accurately and efficiently solvinglarge scale eigenvalue problems. Specifically, they will focus on large scale eigenvalue problems of structured matrix pencils, quadratic eigenvalue problems and nonlinear eigenvalue problems. They will study structure-preserving Rayleight-Ritz subspace projection techniques forsolving these eigenvalue problems, that include the multi-level orthogonalization process of Krylov subspaces, second-order Arnoldi (SOAR) method and the nonlinear Arnoldi method. The goal of the project represents a significant advance in a frontier area of scientific endeavor and engineering design through the application of computational mathematics and simulations. The target applications include these types of eigenvalue problems arising from simulations of electrical circuits and MEMS devices, finite element analysis of structure dynamics, acoustics and electromagnetics. These applications are of great technology importancefor next-generation electronics, automobile efficiency and safety and energy-efficiency monitoring devices and others. The results of this project will be both the description of effective computationalsimulation strategies for these problems and also software made publiclyavailable.