Algorithms for the Inverse Problem of Matrix Construction

North Carolina State University, Raleigh NC

Investigators

Abstract

The inverse problem of matrix construction arises in many areas of important applications. Matrices under construction are supposed to satisfy certain specific constraints. The constraints could be inherited intrinsically from the physical feasibility of a certain mechanical structure or could be driven extrinsically by the desirable property of a certain design parameter. This proposal intends to extend the investigation that the PI has been conducting in the past years with emphasis on the the development of numerical algorithms for application to challenging inverse problems. Four specific inverse problems of matrix construction will be studied via three possible numerical approaches. Techniques to be used involves computer experiments, high resolution graphics and symbolic manipulation, in conjunction with mathematical analysis. This project is expected to find important applications ranging from new development of numerical algorithms to theoretic solution of difficult problems. Since matrix reconstruction with specified properties arises from a remarkably wide area of disciplines, the resulting technology would have substantial impact on the progress in scientific and engineering fields. In the era of information and digital technologies, massive data processing becomes an imperative task at almost every level of applications. In many situations the digitized information is gathered and stored as a data matrix. Nonetheless, because most of the information gathering devices or methods have only finite bandwidth, one cannot avoid the fact that the data collected often are not exact. Signals received by antenna arrays often are contaminated by instrumental noises; astronomical images acquired by telescopes often are blurred by atmospheric turbulence; and even empirical data obtained in laboratories often do not satisfy intrinsic physical constraints. Before any forward analysis technique can be applied, it is important to first reconstruct the data matrices so that the inexactness is reduced while certain feasibility conditions are satisfied. The general objective of this proposal is to develop numerical algorithms to carry out this kind of data reconstruction task. The work in this proposal concerns the mathematical theory and the numerical implementation of three algorithms for four specific inverse construction problems. This investigation could lead to improved techniques for use in several national strategic areas, including ground-based astro-imaging processing, medicine, communications, and laser technology.