CAREER: Fast and Accurate Computations of Applied Eigenproblems

特征问题普遍存在于应用科学和工程领域，其高效计算已成为相关研究和教学中不可或缺的一部分。这项建议的研究部分涉及对相关应用有重大影响的数值线性代数问题。虽然现有的LINPACK、EISPACK和LAPACK等通用软件库很有效地解决了许多常见的特征值和奇异值问题，但如果适当地结合通用软件库通常忽略的自身固有特性，则可以更有效地解决许多应用科学和工程问题，如图像处理中目标识别的主特征向量计算。这个项目通过一个初步的和有希望的研究证明了这一观点，该研究结合了数值线性代数技术和图像数据的特征来解决大量高分辨率图像的巨大特征值(奇异值)问题。研究的目标是从应用背景中深入挖掘应用问题的结构特性，从而开发出准确和高效的算法。在过去的十年里，对矩阵计算问题的相对摄动理论和高精度方法的研究非常活跃，并取得了令人振奋的进展。这一结果可能会对以前使用传统算法进行特征问题计算的其他学科的研究产生重大影响。该项目的教育部分是实施一套想法和具体项目，包括开发新的信息技术，以实现更有效的教学，如避免学生记笔记的时间过长，不断与有困难的学生保持联系，以及将数学材料带入课堂。该项目还涉及修改现有两学期数值线性代数课程的课程，以包括应用问题讲座，这最终将导致一门关于解决应用计算问题的新课程。

Eigenproblems appear ubiquitously all across applied science and engineering, and their efficient computations have become an integral part of related research and teaching. The research component of this proposal is concerned with numerical linear algebra problems that have significant impacts on related applications. Although the existing general purposed software libraries such as LINPACK, EISPACK, and LAPACK solve many common eigenvalue and singular value problems quite efficiently, many problems from applied science and engineering such as primary eigenvector computations for object recognition in image processing can be solved much more efficiently if their own intrinsic characteristics that are mostly ignored by general purpose libraries are incorporated properly. This projectdemonstrates that point of view with a preliminary and promising study that combines numerical linear algebra techniques and characteristics of image data to attack a huge eigenvalue (singular value) problem for a large set of high resolution images. The research objective is to exploit in depth the structural properties of applied problems from their application background and, consequently, will develop accurate and efficient algorithms. The research into relative perturbation theory and highly relative accurate methods for matrix computational problems has been extremely active in the last ten years, and exciting advances are being made. The results could have a significant impact on research in other disciplines that previously used conventional algorithms to carry out eigenproblem computations. The education component of this project is to implement a set of ideas and concrete projects, including the exploitation of new information technologies for more effective teaching such as avoiding too much time for students' note taking, keeping constantly in touch with struggling students, and bringing math material alive in the classroom. The project is also concerned with modifying the curriculum of the existing two-semester numerical linear algebra courses to include applied problem lectures that ultimately will lead to a new course on solving applied computational problems.