On robust solutions to linear least squares problems affected by data uncertainty and implementation errors with application to stochastic signal modeling

Cataloged from PDF version of article.Engineering design problems, especially in signal and image processing, give rise to linear least squares problems arising from discretization of some inverse problem. The associated data are typically subject to error in these applications while the computed solution may only be implemented up to limited accuracy digits, i.e., quantized. In the present paper, we advocate the use of the robust counterpart approach of Ben-Tal and Nemirovski to address these issues simultaneously. Approximate robust counterpart problems are derived, which leads to semidefinite programming problems yielding stable solutions to overdetermined systems of linear equations affected by both data uncertainty and implementation errors, as evidenced by numerical examples from stochastic signal modeling. © 2003 Elsevier Inc. All rights reserved

Estimation and control with bounded data uncertainties

AbstractThe paper describes estimation and control strategies for models with bounded data uncertainties. We shall refer to them as BDU estimation and BDU control methods, for brevity. They are based on constrained game-type formulations that allow the designer to explicitly incorporate into the problem statement a priori information about bounds on the sizes of the uncertainties. In this way, the effect of uncertainties is not unnecessarily over-emphasized beyond what is implied by the a priori bounds; consequently, overly conservative designs, as well as overly sensitive designs, are avoided. A feature of these new formulations is that geometric insights and recursive techniques, which are widely known and appreciated for classical quadratic-cost designs, can also be pursued in this new framework. Also, algorithms for computing the optimal solutions with the same computational effort as standard least-squares solutions exist, thus making the new formulations attractive for practical use. Moreover, the framework is broad enough to encompass applications across several disciplines, not just estimation and control. Examples will be given of a quadratic control design, an H∞ control design, a total-least-square design, image restoration, image separation, and co-channel interference cancellation. A major theme in this paper is the emphasis on geometric and linear algebraic arguments, which lead to useful insights about the nature of the new formulations. Despite the interesting results that will be discussed, several issues remain open and indicate potential future developments; these will be briefly discussed

Methodik zur Integration von Vorwissen in die Modellbildung

This book describes how prior knowledge about dynamical systems and functions can be integrated in mathematical modelling. The first part comprises the transformation of the known properties into a mathematical model and the second part explains four approaches for solving the resulting constrained optimization problems. Numerous examples, tables and compilations complete the book

An Efficient Algorithm for a Bounded Errors-in-Variables Model

We pose and solve a parameter estimation problem in the presence of bounded data uncertainties. The problem involves a minimization step and admits a closed form solution in terms of the positive root of a secular equation

An Efficient Algorithm For A Bounded Errors-In-Variables Model

. We pose and solve a parameter estimation problem in the presence of bounded data uncertainties. The problem involves a minimization step and admits a closed form solution in terms of the positive root of a secular equation. Key words. Least-squares estimation, total least-squares, modeling errors, secular equation. AMS subject classifications. 15A06, 65F05, 65F10, 65F35, 65K10, 93C41, 93E10, 93E24 1. Introduction. Parameter estimation in the presence of data uncertainties is a problem of considerable practical importance, and many estimators have been proposed in the literature with the intent of handling modeling errors and measurement noise. Among the most notable is the total least-squares method [1, 2, 3, 4], also known as orthogonal regression or errors-in-variables method in statistics and system identification [5]. In contrast to the standard least-squares problem, the TLS formulation allows for errors in the data matrix. Its performance may degrade in some situations where ..