Algorithms for the Inverse Problem of Matrix Construction

矩阵构造的逆问题出现在许多重要的应用领域。构造中的矩阵应该满足某些特定的约束条件。这些约束可以从某种机械结构的物理可行性中继承下来，也可以由某种设计参数的合意性质从外部驱动。这项建议旨在扩展PI在过去几年中所进行的研究，重点是开发应用于挑战逆问题的数值算法。将通过三种可能的数值方法来研究四个具体的矩阵构造的逆问题。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。从数值算法的新发展到困难问题的理论解决，这一项目有望找到重要的应用。由于具有特定性质的矩阵重构产生于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实质性的影响。在信息和数字技术时代，海量数据处理成为几乎每个层次的应用都必须完成的任务。在许多情况下，数字化信息被收集并存储为数据矩阵。尽管如此，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无文本的。天线阵接收到的信号往往受到仪器噪声的污染；望远镜获取的天文图像往往因大气湍流而变得模糊；甚至实验室获得的经验数据也往往不满足内在的物理约束。在应用任何正向分析技术之前，重要的是首先重构数据矩阵，以便在满足一定的可行性条件的同时减少不精确度。本方案的总体目标是开发实现这种数据重构任务的数值算法。本方案的工作涉及四个具体逆构造问题的数学理论和三种算法的数值实现。这项研究可能会改进用于几个国家战略领域的技术，包括地面天文成像处理、医学、通信和激光技术。

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.