Parallel Preconditioned Eigenvalue and Singular Value Solvers

该奖项来自数学科学部的计算数学项目，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进展的平衡组合。研究的重点是设计一种新的高效、稳健的并行预条件方法，用于内部特征值和奇异值的计算。主要集中在以下具有挑战性的问题上：研究预条件的有效性和效率；避免使用折叠谱方法；分析使用预条件计算算子奇异值的可能性；以及建立新的迭代求解器的收敛理论。对于后者，经典优化理论的新应用允许分析块本征值和奇异值求解器的收敛行为。这项工作也与应用相关；对应于矩阵的最大奇异值的奇异向量被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。这项研究的主题是因为所考虑的问题类描述了许多物理和力学中的现象，对于实际计算可能需要极大的计算能力。计算方法的进步可以显著提高计算的准确性和效率。由于软件与大范围的应用程序相关，在可公开获得的软件中实施开发增强了产生重大更广泛影响的可能性。PI专门针对的应用是图像分割，例如在跟踪视频中的运动对象时发生的图像分割。该项目还对研究生培训产生了影响。PI的研究提供了一个机会，以培养学生在实际应用中推动的重要计算研究。

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.