CAREER: Extrapolation Methods for Matrix and Tensor Eigenvalue Problems

职业：矩阵和张量特征值问题的外推方法

• 批准号: 2045059
• 负责人: Sara Pollock
• 金额: $ 42.43万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2045059&HistoricalAwards=false
• 关键词: CAREER; Extrapolation; Methods; Matrix; Tensor

Eigenvalue problems arise naturally throughout many areas of mathematics and data science, and their numerical solution is crucial for understanding the behavior of many complex systems. Applications include structural mechanics, epidemiology, image processing, medical imaging, search-engine technology, modeling of population dynamics, and stability of numerical algorithms. Eigenvalue problems are often challenging to solve, as solutions can generally only be found by generating sequences of successive approximations. This work will develop efficient, robust and theoretically sound technologies that will accelerate convergence to solutions of matrix and tensor eigenvalue problems. In response to the increased prevalence of remote-learning, the integrated educational plan will develop stand-alone apps to aid in the delivery of standard and advanced topics in numerical analysis and linear algebra. The apps developed will include exposition of ideas and methods closely related to the research program.The technical aim of this work is the development and analysis of both novel and long-standing extrapolation techniques for eigenvalue problems. Extrapolation techniques are low-cost methods that combine a history of iterates and update steps to form the next approximation in a sequence. In this work they will be used to accelerate convergence of power-type iterations for challenging matrix and tensor eigenvalue problems. The main components of the research for matrix problems are development of novel methods that (1) damp multiple modes simultaneously; (2) resolve multiple modes simultaneously; and (3) extend the target problem class to indefinite matrices. For tensor problems, the main outcomes will be (1) development and convergence analysis of methods that accelerate robust but linearly converging power-type iterations; (2) extensions to accelerated versions of adaptively-shifted power-type iterations and generalized problems; and (3) studies on stability and clustering, and the development of fast techniques to capture complete sets of eigenvalues. This investigation of tensor methods is expected to advance the state of the art by introducing fast but low-complexity methods that are well suited to high-order problems.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.

特征值问题自然地出现在数学和数据科学的许多领域，而它们的数值解对于理解许多复杂系统的行为是至关重要的。应用领域包括结构力学、流行病学、图像处理、医学成像、搜索引擎技术、种群动态建模和数值算法的稳定性。特征值问题通常很难解决，因为解决方案通常只能通过生成逐次逼近序列来找到。这项工作将发展高效、稳健和理论上可靠的技术，以加速收敛到矩阵和张量特征值问题的解。为了应对日益普遍的远程学习，综合教育计划将开发独立的应用程序，以帮助教授数值分析和线性代数的标准和高级主题。开发的应用程序将包括与研究计划密切相关的思想和方法的阐述。这项工作的技术目标是开发和分析特征值问题的新的和长期存在的外推技术。外推技术是一种低成本的方法，它结合了迭代和更新步骤的历史，以形成序列中的下一个近似值。在这项工作中，它们将被用来加速挑战矩阵和张量特征值问题的幂类型迭代的收敛。矩阵问题研究的主要内容是发展新的方法：(1)同时抑制多个振型；(2)同时求解多个振型；(3)将目标问题类扩展到不定矩阵。对于张量问题，主要成果将是：(1)加速稳健但线性收敛的幂类型迭代的方法的发展和收敛分析；(2)对自适应移位的幂类型迭代和广义问题的加速版本的扩展；以及(3)关于稳定性和聚类性的研究，以及获取完整特征值集合的快速技术的发展。这项张量方法的研究有望通过引入快速但低复杂性的方法来推进最先进的技术，这些方法非常适合高阶问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。