New Preconditioned Solvers for Large and Complex Eigenvalue Problems

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与特征值相关的大规模建模和模拟，包括动力系统的线性稳定性分析、材料力学、声发射优化、超导研究、不确定条件下的振动等。某些方法对其他领域也有重要意义；例如，快速的指数矩阵向量积对于用传统方法难以处理的刚性含时微分方程组的指数积分器是必不可少的。新的教材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员，以便在未来产生进一步的影响。在应用数学和科学计算的许多领域，特征值和密切相关的数学工具(如伪谱)基本上是描述性的。本项目系统地开发和分析了几类重要的大规模和复杂特征值相关问题的创新预条件求解器。对于大型线性对称特征问题，预条件特征解算器的变种已经得到了深入的研究和广泛的应用，并取得了巨大的成功。我们的计划是证明，预条件和求解器框架都可以被显著地推广和集成，并具有极大的灵活性，以解决与特征值相关的更广泛类别的挑战性问题。待开发的方法将是可靠、高效和灵活的。具体的研究课题包括(I)矩阵函数的预处理(谱滤波)，(Ii)解决非线性特征问题，以及(Iii)计算大型结构矩阵的谱和伪谱。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.