Parallel Preconditioned Eigenvalue and Singular Value Solvers

该奖项来自数学科学部的计算数学计划，支持首席研究员的研究，该研究涵盖了理论研究，软件支持和开发以及现代应用的进展的平衡组合。本文的研究重点是设计新的高效的鲁棒并行预处理方法来计算内部特征值和奇异值。主要重点是以下具有挑战性的问题：调查的可用性和效率的预处理;避免折叠谱的方法，分析的可能性计算的奇异值的运营商使用预处理;和建立收敛理论的新的迭代求解器。对于后者，一个新的应用经典的控制理论允许块特征值和奇异值求解器的收敛行为的分析。这项工作也是相关的应用程序;奇异向量，对应于矩阵的最大奇异值用于执行由数据矩阵描述的数据的主成分分析。这里的重点是多维图像分割。本研究的主题是由这样一个事实，即所考虑的问题描述了许多物理和力学现象，并为实际计算可能需要极端的计算能力。计算方法的进步可以显著提高计算的准确性和效率。由于软件与广泛的应用程序相关，在公开提供的软件中实施开发工作可提高产生更广泛的重大影响的潜力。PI专门针对的应用是图像分割，例如在跟踪视频中的移动对象时发生的图像分割。该项目的其他影响是研究生培训。PI的研究提供了一个机会，培养学生在重要的计算研究动机的真实的实际应用。

This award, from the computational mathematics program of the Division of Mathematical Sciences, supports the research of the principal investigator, which covers a balanced mix of theoretical investigations, software support and development, and advances for modern applications. The research focuses on design of new efficient robust parallel preconditioned methods for interior eigenvalue and singular value computations. The main emphasis is on the following challenging issues: investigating availability and efficiency of preconditioning; avoiding the folded spectrum approach, analyzing the possibility of computing the singular values of an operator using preconditioning; and establishing convergence theory for novel iterative solvers. For the latter, a novel application of classical majorization theory permits analysis of the convergence behavior of block eigenvalue and singular value solvers. This work is also relevant for applications; the singular vectors, corresponding to the largest singular values of a matrix are used for performing the principal component analysis of data described by a data matrix. Here the focus is on multi-dimensional image segmentation.The topic of this research is motivated by the fact that the class of problems under consideration describes many phenomena in physics and mechanics, and for practical calculations may require extreme computational power. Advances in the computational approaches can provide noticeable improvements in the accuracy and efficiency of calculations. Implementing the developments in publicly available software enhances the potential for significant broader impact, due to the relevance of the software for a large range of applications. The application specifically targeted by the PI is image segmentation, such as occurs in tracking moving objects in videos. Additional impact of the project is on graduate student training. The research of the PI provides an opportunity to train students in important computational research motivated by real practical applications.