CAREER: Fast and Accurate Computations of Applied Eigenproblems

特征问题在应用科学和工程领域中普遍存在，其有效计算已成为相关研究和教学的重要组成部分。该提案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。虽然现有的通用软件库，如LINPACK，EISPACK和LAPACK解决许多常见的特征值和奇异值问题相当有效，许多问题，从应用科学和工程，如主特征向量计算图像处理中的对象识别，可以更有效地解决，如果他们自己的固有特性，大多是被忽略的通用库适当地纳入。该项目展示了这一观点，初步的和有前途的研究，结合数值线性代数技术和图像数据的特点，攻击一个巨大的特征值（奇异值）问题的一个大的高分辨率图像集。研究目标是从应用背景中深入挖掘应用问题的结构特性，从而开发出准确有效的算法。近十年来，矩阵计算问题的相对摄动理论和高相对精度方法的研究十分活跃，并取得了令人振奋的进展。这些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系列想法和具体项目，包括利用新的信息技术进行更有效的教学，例如避免学生花太多时间做笔记，不断与有困难的学生保持联系，并将数学材料带到课堂上。 该项目还涉及修改现有的两个学期的数值线性代数课程的课程，以包括应用问题讲座，最终将导致一个新的课程解决应用计算问题。

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.