Algorithms for the Inverse Problem of Matrix Construction

矩阵构造反问题的算法

• 批准号: 0073056
• 负责人: Moody Chu
• 金额: $ 11.6万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073056&HistoricalAwards=false
• 关键词: Algorithms; Inverse; Problem; Matrix; Construction

The inverse problem of matrix construction arises in manyareas of important applications. Matrices under constructionare supposed to satisfy certain specific constraints. Theconstraints could be inherited intrinsically from the physicalfeasibility of a certain mechanical structure or could bedriven extrinsically by the desirable property of a certaindesign parameter. This proposal intends to extend theinvestigation that the PI has been conducting in the pastyears with emphasis on the the development of numericalalgorithms for application to challenging inverse problems.Four specific inverse problems of matrix construction willbe studied via three possible numerical approaches. Techniquesto be used involves computer experiments, high resolutiongraphics and symbolic manipulation, in conjunction withmathematical analysis. This project is expected to findimportant applications ranging from new development ofnumerical algorithms to theoretic solution of difficultproblems. Since matrix reconstruction with specifiedproperties arises from a remarkably wide area of disciplines,the resulting technology would have substantial impact on theprogress in scientific and engineering fields.In the era of information and digital technologies,massive data processing becomes an imperative taskat almost every level of applications. In many situationsthe digitized information is gathered and stored as a datamatrix. Nonetheless, because most of the informationgathering devices or methods have only finite bandwidth, onecannot avoid the fact that the data collected often are notexact. Signals received by antenna arrays often arecontaminated by instrumental noises; astronomical imagesacquired by telescopes often are blurred by atmosphericturbulence; and even empirical data obtained in laboratoriesoften do not satisfy intrinsic physical constraints. Beforeany forward analysis technique can be applied, it is importantto first reconstruct the data matrices so that the inexactnessis reduced while certain feasibility conditions are satisfied.The general objective of this proposal is to develop numericalalgorithms to carry out this kind of data reconstruction task.The work in this proposal concerns the mathematical theory andthe numerical implementation of three algorithms for fourspecific inverse construction problems. This investigationcould lead to improved techniques for use in several nationalstrategic areas, including ground-based astro-imagingprocessing, medicine, communications, and laser technology.

矩阵构造逆问题在许多重要的应用领域中出现。构造中的矩阵应该满足某些特定的约束。这些约束可以从本质上继承自某种机械结构的物理可行性，也可以由某种设计参数的理想特性驱动。本提案旨在扩展PI在过去几年中一直在进行的调查，重点是开发应用于具有挑战性的逆问题的数值算法。通过三种可能的数值方法，我们将研究矩阵构造的四个特定逆问题。所使用的技术包括计算机实验、高分辨率图形和符号处理，以及数学分析。该项目有望在数值算法的新发展和难题的理论解决等方面找到重要的应用。由于具有特定性质的矩阵重构出现在一个非常广泛的学科领域，由此产生的技术将对科学和工程领域的进步产生重大影响。在信息和数字技术时代，海量数据处理几乎成为每个应用级别的必要任务。在许多情况下，数字化信息以数据矩阵的形式被收集和存储。然而，由于大多数信息收集设备或方法只有有限的带宽，人们无法避免这样一个事实，即收集的数据往往是不精确的。天线阵列接收到的信号经常受到仪器噪声的污染；望远镜获得的天文图像常常因大气湍流而模糊不清；甚至在实验室中获得的经验数据也常常不满足内在的物理约束。在应用任何正演分析技术之前，重要的是首先重构数据矩阵，以便在满足一定可行性条件的情况下减少不精确性。本提案的总体目标是开发数字算法来执行这种数据重建任务。本提案的工作涉及四个特定逆构造问题的三种算法的数学理论和数值实现。这项研究可能会导致一些国家战略领域的技术改进，包括地基天文成像处理、医学、通信和激光技术。