CAREER: Extrapolation Methods for Matrix and Tensor Eigenvalue Problems

特征值问题在数学和数据科学的许多领域中自然出现，它们的数值解对于理解许多复杂系统的行为至关重要。应用包括结构力学、流行病学、图像处理、医学成像、搜索引擎技术、人口动态建模和数值算法的稳定性。特征值问题通常很难解决，因为通常只能通过生成逐次逼近序列来找到解决方案。这项工作将开发高效，强大和理论上合理的技术，将加速收敛到矩阵和张量特征值问题的解决方案。针对远程学习日益普及的情况，综合教育计划将开发独立的应用程序，以帮助提供数值分析和线性代数方面的标准和高级专题。开发的应用程序将包括与研究项目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本征值问题的新颖和长期存在的外推技术。外推技术是低成本的方法，它结合了联合收割机的迭代历史和更新步骤，以形成序列中的下一个近似值。在这项工作中，他们将被用来加速收敛的幂型迭代具有挑战性的矩阵和张量特征值问题。矩阵问题研究的主要组成部分是开发新的方法，（1）同时阻尼多个模式;（2）同时解决多个模式;（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型迭代方法的开发和收敛分析;（2）自适应移位幂型迭代和广义问题的加速版本的扩展;（3）稳定性和聚类研究，以及捕获完整特征值集的快速技术的开发。这项张量方法的调查预计将通过引入快速但低复杂性的方法，非常适合高阶问题的先进水平。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.