M. Z. Nashed

Bio: M. Z. Nashed is an academic researcher from University of Central Florida. The author has contributed to research in topics: Compact operator & Hilbert space. The author has an hindex of 9, co-authored 13 publications receiving 554 citations. Previous affiliations of M. Z. Nashed include Georgia Institute of Technology & University of Delaware.

Operator-theoretic and computational approaches to Ill-posed problems with applications to antenna theory

Abstract: A general framework for regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized: philosophy of resolution, regularization-approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted-norm least-squares solutions of first kind integral equations (and more generally of linear operator equations with nonclosed range). A common problem in all these methods is delineated: each method reduces the problem of resolution to a "nonstandard" minimization problem involving an unknown critical "parameter" whose "optimal" value is crucial to the numerical realization and amenability of the method. The "nonstandardness" results from the fact that one does not have explicitly, or a priori, the function to be minimized; it has to built up using additional information, convergence rate estimates, and robustness conditions, etc. Several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill-posed (identification and pattern synthesis) problems. An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The results will be applied specifically to problems of antenna synthesis and identification. However the thrust of the paper is devoted to the interdisciplinary character of operator-theoretic and numerical methods for ill-posed problems.

Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations

Abstract: In this paper a study of generalized inverses of linear operators in reproducing kernel Hilbert spaces (RKHS) is initiated. Explicit expressions for generalized inverses and minimal-norm solutions of linear operator equations in RKHS are obtained in several forms. The relation between the regularization operator of the equation $Af = g$ and the generalized inverse of the operator A in RKHS is demonstrated. In particular, it is shown that they are the same if the range of the operator is closed in an appropriate RKHS. Finally, properties of the regularized pseudosolutions in this setting are studied.

Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

Abstract: We consider approximations {xn } obtained by moment discretization to (i) the minimal ?2-norm solution of XCx = y where XC is a Hilbert-Schmidt integral operator on ?2, and to (ii) the least squares solution of minimal 22-norm of the same equation when y is not in the range 6R(3C) of XC. In case (i), if y EE R(NC), then x" -3* Cty, where Xct is the generalized inverse of XC, and I xII-* - otherwise. Rates of convergence are given in this case if further Xty EE 3*(29), where 3C* is the adjoint of XC, and the Hilbert-Schmidt kernel of XXQ* satisfies certain smoothness conditions. In case (ii), if y EE (R(K) ( R(X)-, then xn -* XCty, and I xn I-* c otherwise. If further Xty EE V *C(22), then rates of con- vergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of (XX3*)2. Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.

Generalized inverses: theory and applications

Abstract: * Glossary of notation * Introduction * Preliminaries * Existence and Construction of Generalized Inverses * Linear Systems and Characterization of Generalized Inverses * Minimal Properties of Generalized Inverses * Spectral Generalized Inverses * Generalized Inverses of Partitioned Matrices * A Spectral Theory for Rectangular Matrices * Computational Aspects of Generalized Inverses * Miscellaneous Applications * Generalized Inverses of Linear Operators between Hilbert Spaces * Appendix A: The Moore of the Moore-Penrose Inverse * Bibliography * Subject Index * Author Index

Spectrum estimation and harmonic analysis

Abstract: In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Digital Image Restoration

Abstract: The article introduces digital image restoration to the reader who is just beginning in this field, and provides a review and analysis for the reader who may already be well-versed in image restoration. The perspective on the topic is one that comes primarily from work done in the field of signal processing. Thus, many of the techniques and works cited relate to classical signal processing approaches to estimation theory, filtering, and numerical analysis. In particular, the emphasis is placed primarily on digital image restoration algorithms that grow out of an area known as "regularized least squares" methods. It should be noted, however, that digital image restoration is a very broad field, as we discuss, and thus contains many other successful approaches that have been developed from different perspectives, such as optics, astronomy, and medical imaging, just to name a few. In the process of reviewing this topic, we address a number of very important issues in this field that are not typically discussed in the technical literature.

Robust Solutions to Least-Squares Problems with Uncertain Data

Abstract: We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

Elastography: ultrasonic estimation and imaging of the elastic properties of tissues.

Abstract: The basic principles of using sonographic techniques for imaging the elastic properties of tissues are described, with particular emphasis on elastography. After some preliminaries that describe some basic tissue stiffness measurements and some contrast transfer limitations of strain images are presented, four types of elastograms are described, which include axial strain, lateral strain, modulus and Poisson's ratio elastograms. The strain filter formalism and its utility in understanding the noise performance of the elastographic process is then given, as well as its use for various image improvements. After discussing some main classes of elastographic artefacts, the paper concludes with recent results of tissue elastography in vitro and in vivo.