@article{chu_plemmons_2003, title={Real-valued, low rank, circulant approximation}, volume={24}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0041663838&partnerID=MN8TOARS}, DOI={10.1137/S0895479801383166}, abstractNote={Partially due to the fact that the empirical data collected by devices with finite bandwidth often neither preserves the specified structure nor induces a certain desired rank, retrieving the nearest structured low rank approximation from a given data matrix becomes an imperative task in many applications. This paper investigates the case of approximating a given target matrix by a real-valued circulant matrix of a specified, fixed, and low rank. A fast Fourier transform (FFT)-based numerical procedure is proposed to speed up the computation. However, since a conjugate-even set of eigenvalues must be maintained to guarantee a real-valued matrix, it is shown by numerical examples that the nearest real-valued, low rank, and circulant approximation is sometimes surprisingly counterintuitive.}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Plemmons, RJ}, year={2003}, pages={645–659} }
@article{chu_funderlic_plemmons_2003, title={Structured low rank approximation}, volume={366}, ISSN={["0024-3795"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037409774&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(02)00505-0}, abstractNote={This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.}, number={SPEC. ISS.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Funderlic, RE and Plemmons, RJ}, year={2003}, month={Jun}, pages={157–172} }
@article{chu_pauca_plemmons_sun_2000, title={A mathematical framework for the linear reconstructor problem in adaptive optics}, volume={316}, ISSN={["0024-3795"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0034392470&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(00)00019-7}, abstractNote={The wave front field aberrations induced by atmospheric turbulence can severely degrade the performance of an optical imaging system. Adaptive optics refers to the process of removing unwanted wave front distortions in real time, i.e., before the image is formed, with the use of a phase corrector. The basic idea in adaptive optics is to control the position of the surface of a deformable mirror in such a way as to approximately cancel the atmospheric turbulence effects on the phase of the incoming light wave front. A phase computation system, referred to as a reconstructor, transforms the output of a wave front sensor into a set of drive signals that control the shape of a deformable mirror. The control of a deformable mirror is often based on a linear wave front reconstruction algorithm that is equivalent to a matrix–vector multiply. The matrix associated with the reconstruction algorithm is called the reconstructor matrix. Since the entire process, from the acquisition of wave front measurements to the positioning of the surface of the deformable mirror, must be performed at speeds commensurate with the atmospheric changes, the adaptive optics control imposes several challenging computational problems. The goal of this paper is twofold: (i) to describe a simplified yet feasible mathematical framework that accounts for the interactions among main components involved in an adaptive optics imaging system, and (ii) to present several ways to estimate the reconstructor matrix based on this framework. The performances of these various reconstruction techniques are illustrated using some simple computer simulations.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Pauca, VP and Plemmons, RJ and Sun, XB}, year={2000}, month={Sep}, pages={113–135} }
@book{davis_chu_mcconnell_dolan_norris_ortiz_plemmon_ridgeway_scaife_stewart_et al._1998, title={Cornelius Lanczos: Collected published papers with commentaries}, ISBN={0929493003}, publisher={Raleigh, NC: College of Physical and Mathematical Sciences, North Carolina State University}, author={Davis, W. R. and Chu, M. T. and McConnell, J. R. and Dolan, P. and Norris, L. K. and Ortiz, E. and Plemmon, R. J. and Ridgeway, D. and Scaife, B.K.P. and Stewart, W. J. and et al.}, year={1998} }