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A note on the behavior of the randomized Kaczmarz algorithm of Strohmer and Vershynin
 Journal of Fourier Analysis and Applications
, 2009
"... In a recent paper by T. Strohmer and R. Vershynin [“A Randomized Kaczmarz Algorithm with Exponential Convergence”, Journal of Fourier Analysis and Applications, published online on April 25, 2008] a “randomized Kaczmarz algorithm ” is proposed for solving systems of linear equations {ai, x ® = bi} ..."
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In a recent paper by T. Strohmer and R. Vershynin [“A Randomized Kaczmarz Algorithm with Exponential Convergence”, Journal of Fourier Analysis and Applications, published online on April 25, 2008] a “randomized Kaczmarz algorithm ” is proposed for solving systems of linear equations {ai, x ® = bi}mi=1. In that algorithm the next equation to be used in an iterative Kaczmarz process is selected with a 1 probability proportional to °°ai°°2. The paper illustrates the superiority of this selection method for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values. In this note we point out that the reported success of the algorithm of Strohmer and Vershynin in their numerical simulation depends on the specific choices that are made in translating the underlying problem, whose geometrical nature is “find a common point of a set of hyperplanes”, into a system of algebraic equations. If this translation is carefully done, as in the numerical simulation provided by Strohmer and Vershynin for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values, then indeed good performance may result. However, there will always be legitimate algebraic representations of the underlying problem (so that the set of solutions of the system of algebraic equations is exactly the set of points in the intersection of the hyperplanes), for which the selection method of Strohmer and Vershynin will perform in an inferior manner. Key words. Kaczmarz algorithm, projection method, rate of convergence.
Computational acceleration of projection algorithms for the linear best approximation problem
 Linear Algebra and Its Applications 416
, 2006
"... This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the HalpernLionsWittmannBauschke algorithm for the best approximation problem from a point to the int ..."
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This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the HalpernLionsWittmannBauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new componentaveraging variant that substantially accelerates their initial behavior for sparse systems. 1
SEMICONVERGENCE AND RELAXATION PARAMETERS FOR A CLASS OF SIRT ALGORITHMS ∗
"... Abstract. This paper is concerned with the Simultaneous Iterative Reconstruction Technique (SIRT) class of iterative methods for solving inverse problems. Based on a careful analysis of the semiconvergence behavior of these methods, we propose two new techniques to specify the relaxation parameters ..."
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Abstract. This paper is concerned with the Simultaneous Iterative Reconstruction Technique (SIRT) class of iterative methods for solving inverse problems. Based on a careful analysis of the semiconvergence behavior of these methods, we propose two new techniques to specify the relaxation parameters adaptively during the iterations, so as to control the propagated noise component of the error. The advantage of using this strategy for the choice of relaxation parameters on noisy and illconditioned problems is demonstrated with an example from tomography (image reconstruction from projections). Key words. SIRT methods, Cimmino and DROP iteration, semiconvergence, relaxation parameters, tomographic imaging AMS subject classifications. 65F10, 65R32 1. Introduction. Largescale discretizations of illposed problems (such as imaging problems in tomography) call for the use of iterative methods, because direct factorization methods are infeasible. In particular, there is an interest in regularizing iterations, where the iteration vector can be considered as a regularized solution with the iteration index playing the role of the regularizing parameter. Initially the iteration vector approaches a regularized solution,
Distributed Randomized Kaczmarz and Applications to Seismic Imaging in Sensor Network
 IN THE 11TH IEEE INTERNATIONAL CONFERENCE ON DISTRIBUTED COMPUTING IN SENSOR SYSTEMS (IEEE DCOSS
, 2015
"... Many realworld wireless sensor network applications such as environmental monitoring, structural health monitoring, and smart grid can be formulated as a leastsquares problem. In distributed CyberPhysical System (CPS), each sensor node observes partial phenomena due to spatial and temporal restr ..."
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Many realworld wireless sensor network applications such as environmental monitoring, structural health monitoring, and smart grid can be formulated as a leastsquares problem. In distributed CyberPhysical System (CPS), each sensor node observes partial phenomena due to spatial and temporal restriction and is able to form only partial rows of leastsquares. Traditionally, these partial measurements were gathered at a centralized location. However, with the increase in sensors and their measurements, aggregation is becoming challenging and infeasible. In this paper, we propose distributed randomized kaczmarz that performs innetwork computation to solve leastsquares over the network by avoiding costly communication. As a case study, we present a volcano monitoring application on a distributed CORE emulator and use real data from Mt. St. Helens to evaluate our proposed method.
Iterative Projection Methods in Biomedical Inverse Problems
, 2007
"... The convex or quasiconvex feasibility problem and the split feasibility problem in the Euclidean space have many applications in various fields of science and technology, particularly in problems of image reconstruction from projections, in solving the fully discretized inverse problem in radiati ..."
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The convex or quasiconvex feasibility problem and the split feasibility problem in the Euclidean space have many applications in various fields of science and technology, particularly in problems of image reconstruction from projections, in solving the fully discretized inverse problem in radiation therapy treatment planning, and in other image processing problems. Solving systems of linear equalities and/or inequalities is one of them. The class of methods, generally called Projection Methods, has witnessed great progress in recent years and its member algorithms have been applied with success to fully discretized models of inverse problems in image reconstruction and image processing, and in intensitymodulated radiation therapy. We introduce the reader to this field by reviewing algorithmic structures and specific algorithms for the convex feasibility problem, the quasiconvex feasibility problem and the split feasibility problem.
1 Fast Superiorization Using a Dual Perturbation Scheme for Proton Computed Tomography
"... The purpose of proton Computed Tomography (pCT) is to provide an image that accurately captures the relative stopping power (RSP) needed for proton treatment ..."
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The purpose of proton Computed Tomography (pCT) is to provide an image that accurately captures the relative stopping power (RSP) needed for proton treatment
a,∗
"... In this paper we introduce a sequential block iterative method and its simultaneous version with optimal combination of weights (instead of convex combination) for solving convex feasibility problems. When the intersection of the given family of convex sets is nonempty, it is shown that any sequenc ..."
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In this paper we introduce a sequential block iterative method and its simultaneous version with optimal combination of weights (instead of convex combination) for solving convex feasibility problems. When the intersection of the given family of convex sets is nonempty, it is shown that any sequence generated by the given algorithms converges to a feasible point. Additionally for linear feasibility problems, we give equivalency of our algorithms with sequential and simultaneous block Kaczmarz methods explaining the optimal weights have been inherently used in Kaczmarz methods. In addition, a convergence result is presented for simultaneous block Kaczmarz for the case of inconsistent linear system of equations.
THE STUDY OF BINARY STEERING ALGORITHMS
, 2010
"... This thesis (open access) is brought to you for free and open access by the Jack N. Averitt College of Graduate Studies (COGS) at Digital ..."
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This thesis (open access) is brought to you for free and open access by the Jack N. Averitt College of Graduate Studies (COGS) at Digital
Bounds on the Largest Singular Value of a Matrix and the Convergence of Simultaneous and BlockIterative Algorithms for Sparse Linear Systems
"... Abstract We show that the eigenvalues of A † A do not exceed the maximum of the numbers for all i = 1, 2, ..., I, where s j denotes the number of nonzero elements in the jth column of the I by J matrix A. Using this result, we obtain convergence theorems for several iterative algorithms for solvin ..."
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Abstract We show that the eigenvalues of A † A do not exceed the maximum of the numbers for all i = 1, 2, ..., I, where s j denotes the number of nonzero elements in the jth column of the I by J matrix A. Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax = b, including the CAV, BICAV, CARP1 and the blockiterative DROP methods.