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

特征值问题是科学和工程领域的一个中心课题，它有着广泛的应用，并带来了重大的数值挑战。几十年来，它一直是众多理论研究活动的焦点，以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上取得了进展，但用于处理这些问题的方法仍然局限于其范围，并且迄今为止还没有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求解方法来填补这一空白，这些问题在数学上和实际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这个竞技场中探索新算法，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）。该项目的成果有望为求解非线性特征值问题开辟新的有效途径。一套新的最先进的数值例程将被开发，充分测试，并公开发布。该项目的目标是推进非线性特征值问题的最先进的求解方法。所设想的新方法，预计将是特别有效的解决大规模的问题，使用并行。该项目的主要推力是开发新的特征值算法的基础上推广的柯西积分型方法的非线性情况下，结合投影方法，如Krylov和子空间迭代。在这项调查的出发点是FEAST的方法，这将是适应非线性背景。由于所考虑的问题预计将是大而稀疏的，该团队将研究依赖于域分解的方法，其中原始物理域被划分为许多子域，以利用并行性。除其他目标外，该团队将仔细研究在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，以非线性情况。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而被认为值得支持。

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.