AF: Small: Collaborative Research: Effective Numerical Algorithms and Software for Nonlinear Eigenvalue Problems

AF：小型：协作研究：非线性特征值问题的有效数值算法和软件

• 批准号: 1812927
• 负责人: Agnieszka Miedlar
• 金额: $ 14.09万
• 依托单位: University of Kansas Center for Research Inc
• 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: 2018-10-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812927&HistoricalAwards=false
• 关键词: AF; Small; Collaborative; Research; Effective

The eigenvalue problem is a central topic in science and engineering arising from a wide range of applications and posing major numerical challenges. For decades, it has been the focus of numerous theoretical research activities for developing various efficient numerical algorithms. These efforts have led to the development of new software that is essential to assist the everyday work of many engineers and scientists. In spite of progress made on solving the eigenvalue problem, methods available for handling these problems remain limited in their scope and they have not resulted in effective general-purpose software so far. The primary goal of this project is to fill this gap by advancing the state of the art in solution methods for nonlinear eigenvalue problems which are both mathematically and practically far more challenging than the traditional linear eigenvalue problems. The combined expertise of the investigating team is well suited for exploring new algorithms in this arena, analyzing them, and developing new effective software that can universally impact a wide range of disciplines (engineering, physics, chemistry, and biology). The outcome of the project are expected to open new and efficient ways to solve nonlinear eigenvalue problems. A new suite of state of the art numerical routines will be developed, fully tested, and publicly released.The goal of this project is to advance the state-of-the-art in solution methods for nonlinear eigenvalue problems. The new approaches that are envisioned are expected to be particularly effective for solving large-scale problems using parallelism. The main thrust of the project is the development of novel eigenvalue algorithms based on generalizations of Cauchy integral type methods for the nonlinear case, combined with projection methods such as Krylov and subspace iteration. A starting point in this investigation is the FEAST approach which will be adapted to the nonlinear context. Because the problems under consideration are expected to be large and sparse, the team will investigate methods that rely on domain decomposition where the original physical domain is partitioned into a number of subdomains in order to exploit parallelism. Among other goals, the team will carefully study the extension of the tools that are exploited in the linear case, such as spectrum slicing (computing eigenvalues by parts), block methods, and iterative linear solves, to the nonlinear case.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.

特征值问题是科学和工程领域的一个中心主题，具有广泛的应用范围，并提出了重大的数值挑战。几十年来，它一直是开发各种高效数值算法的众多理论研究活动的焦点。这些努力导致了新软件的开发，这对于协助许多工程师和科学家的日常工作至关重要。尽管在解决特征值问题方面取得了进展，但可用于处理这些问题的方法在范围上仍然有限，并且迄今为止尚未产生有效的通用软件。该项目的主要目标是通过推进非线性特征值问题的最新解决方法来填补这一空白，非线性特征值问题在数学上和实践上都比传统的线性特征值问题更具挑战性。调查团队的综合专业知识非常适合在该领域探索新算法、分析它们以及开发可以普遍影响广泛学科（工程、物理、化学和生物学）的新有效软件。该项目的成果预计将为解决非线性特征值问题开辟新的有效方法。将开发、全面测试并公开发布一套新的最先进的数值例程。该项目的目标是推进非线性特征值问题的最先进的解决方法。设想的新方法预计对于使用并行性解决大规模问题特别有效。该项目的主要目标是开发基于非线性情况柯西积分类型方法推广的新颖特征值算法，并结合 Krylov 和子空间迭代等投影方法。本研究的起点是 FEAST 方法，该方法将适应非线性环境。由于所考虑的问题预计会很大且稀疏，因此该团队将研究依赖于域分解的方法，其中将原始物理域划分为多个子域以利用并行性。除其他目标外，该团队将仔细研究线性情况下使用的工具的扩展，例如频谱切片（按部分计算特征值）、块方法和迭代线性求解。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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

FEAST eigensolver for nonlinear eigenvalue problems

用于非线性特征值问题的 FEAST 特征求解器

• DOI: 10.1016/j.jocs.2018.05.006
• 发表时间: 2018-07
• 期刊: Journal of Computational Science
• 影响因子: 3.3
• 作者: Gavin, Brendan;Międlar, Agnieszka;Polizzi, Eric
• 通讯作者: Polizzi, Eric

A rational approximation method for solving acoustic nonlinear eigenvalue problems

求解声学非线性特征值问题的有理逼近法

• DOI: 10.1016/j.enganabound.2019.10.006
• 发表时间: 2020-02
• 期刊: Engineering Analysis with Boundary Elements
• 影响因子: 3.3
• 作者: El;Miȩdlar, Agnieszka;Saad, Yousef
• 通讯作者: Saad, Yousef