New Preconditioned Solvers for Large and Complex Eigenvalue Problems

该项目旨在开发新的算法，以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、声发射的优化、超导性的研究、不确定条件下的振动等等。某些方法对其他领域也有借鉴意义；例如，快速的指数矩阵向量积对于指数积分器求解刚性时变微分方程是必不可少的，而传统方法很难解决这些问题。新的教科书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，以产生进一步的影响。特征值和密切相关的数学工具（例如，伪谱）在应用数学和科学计算的许多领域都是基本描述性的。该项目涉及系统地开发和分析几个重要类别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，预条件特征解的变体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。我们的计划是表明，预处理和求解器框架都可以被显著地推广，并以极大的灵活性集成，以解决更广泛的具有挑战性的特征值相关问题。所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)矩阵函数预处理（谱滤波），（ii）求解非线性特征问题，（iii）计算大结构矩阵的谱和伪谱。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.