New Preconditioned Solvers for Large and Complex Eigenvalue Problems

用于大型复杂特征值问题的新预处理求解器

• 批准号: 1819097
• 负责人: Fei Xue
• 金额: $ 20万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-15 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819097&HistoricalAwards=false
• 关键词: New; Preconditioned; Solvers; Large; Complex

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.

该项目旨在开发新的算法，这将有助于在许多科学和工程领域实现大规模特征值相关的建模和模拟，包括动力系统的线性稳定性分析，材料力学，声发射优化，超导性研究，不确定条件下的振动等等。某些方法对其他领域也有重要意义;例如，快速指数矩阵向量积对于求解传统方法难以处理的刚性时变微分方程的指数积分器是必不可少的。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，以在未来产生进一步的影响。伪谱）在应用数学和科学计算的许多领域中是基本描述性的。这个项目涉及系统的发展和分析创新的预处理解决方案的几个重要类别的大型和复杂的特征值相关的问题。对于大型线性对称特征值问题，预处理特征值求解器的变体已被深入研究并广泛应用，并在许多应用中取得了巨大成功。该计划是为了表明，预处理和求解器框架都可以显着推广和集成具有很大的灵活性，以解决更广泛的一类具有挑战性的特征值相关的问题。拟制定的方法将是可靠、有效和灵活的。具体的研究课题包括（i）矩阵函数的预处理（谱滤波），（ii）求解非线性特征问题，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。

项目成果

A Block Preconditioned Harmonic Projection Method for Large-Scale Nonlinear Eigenvalue Problems

大规模非线性特征值问题的分块预条件谐波投影方法

• DOI: 10.1137/17m112141x
• 发表时间: 2018
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Xue, Fei

Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation

• DOI: 10.4208/aamm.oa-2020-0270
• 发表时间: 2021-06
• 期刊: Advances in Applied Mathematics and Mechanics
• 影响因子: 1.4
• 作者: global sci

One-step convergence of inexact Anderson acceleration for contractive and non-contractive mappings

收缩和非收缩映射的不精确安德森加速的一步收敛

• DOI: 10.1553/etna_vol55s285
• 发表时间: 2021
• 期刊: ETNA - Electronic Transactions on Numerical Analysis
• 作者: Xue, Fei

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

• DOI: 10.1137/17m1157568
• 发表时间: 2017-11
• 期刊: ArXiv
• 作者: Lingfei Wu;Fei Xue;A. Stathopoulos

LOCALIZED STOCHASTIC GALERKIN METHODS FOR HELMHOLTZ PROBLEMS CLOSE TO RESONANCE

近共振亥姆霍兹问题的局部随机伽略金法

• DOI: 10.1615/int.j.uncertaintyquantification.2021034247
• 发表时间: 2021
• 期刊: International Journal for Uncertainty Quantification
• 影响因子: 1.7
• 作者: Wang, Guanjie;Xue, Fei;Liao, Qifeng
• 通讯作者: Liao, Qifeng