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: 1812695
• 负责人: Yousef Saad
• 金额: $ 13.9万
• 依托单位: University of Minnesota-Twin Cities
• 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 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812695&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.

特征值问题是由广泛的应用以及带来重大数值挑战引起的科学和工程学的核心主题。几十年来，这一直是许多理论研究活动的重点，用于开发各种有效的数值算法。这些努力导致了新软件的开发，这对于帮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其范围中仍然有限，并且到目前为止尚未产生有效的通用软件。该项目的主要目的是通过在数学和实际上比传统的线性特征值问题的非线性特征值问题的解决方案方法来填补这一空白。调查团队的综合专业知识非常适合探索该领域的新算法，分析它们，并开发新的有效软件，这些软件可以普遍影响广泛的学科（工程，物理，化学和生物学）。该项目的结果有望开放新的有效方法来解决非线性特征值问题。将开发，全面测试并公开发布一套新的最新数值例程。该项目的目的是推进针对非线性特征值问题的解决方案方法的最新方法。预计将设想的新方法对于使用并行性解决大规模问题特别有效。该项目的主要目的是基于非线性案例的Cauchy积分类型方法的概括的新型特征值算法的开发，并结合了Krylov和子空间迭代等投影方法。这项调查的起点是盛宴方法，该方法将适应非线性环境。由于所考虑的问题有望大且稀疏，因此团队将研究依赖域分解的方法，其中原始物理域被分配到许多子域中以利用并行性。除其他目标外，团队还将仔细研究在线性案例中所利用的工具的扩展，例如Spectrum切片（按部分计算特征值），块方法和迭代的线性求解，并将其求解到非线性案例。该奖项反映了NSF的法定任务，并通过评估了基金会的范围，并通过评估了基金会的范围。

项目成果

A rational approximation method for solving acoustic nonlinear eigenvalue problems

• DOI: 10.1016/j.enganabound.2019.10.006
• 发表时间: 2019-06
• 期刊: ArXiv
• 作者: Mohamed El-Guide;A. Miedlar;Y. Saad