Algorithms for the Inverse Problem of Matrix Construction

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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.