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We present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image processing. The Kaczmarz method…

Numerical Analysis · Mathematics 2012-10-10 Deanna Needell , Rachel Ward

For solving large-scale consistent linear system, we combine two efficient row index selection strategies with Kaczmarz-type method with oblique projection, and propose a greedy randomized Kaczmarz method with oblique projection (GRKO) and…

Numerical Analysis · Mathematics 2021-06-28 Fang Wang , Weiguo Li , Wendi Bao , Li Liu

Kaczmarz's alternating projection method has been widely used for solving a consistent (mostly over-determined) linear system of equations Ax=b. Because of its simple iterative nature with light computation, this method was successfully…

Numerical Analysis · Computer Science 2014-07-22 Tim Wallace , Ali Sekmen

In this technical report we present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image…

Numerical Analysis · Mathematics 2012-04-03 Deanna Needell , Rachel Ward

The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax=b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin…

Numerical Analysis · Mathematics 2011-02-15 Yonina C. Eldar , Deanna Needell

The Randomized Kaczmarz Algorithm is a randomized method which aims at solving a consistent system of over determined linear equations. This note discusses how to find an optimized randomization scheme for this algorithm, which is related…

Systems and Control · Computer Science 2016-11-17 Liang Dai , Mojtaba Soltanalian , Kristiaan Pelckmans

The Kaczmarz method is a popular iterative scheme for solving large-scale linear systems. The randomized Kaczmarz method (RK) greatly improves the convergence rate of the Kaczmarz method, by using the rows of the coefficient matrix in…

Numerical Analysis · Mathematics 2020-12-01 Yutong Jiang , Gang Wu , Long Jiang

The Kaczmarz algorithm is a popular solver for overdetermined linear systems due to its simplicity and speed. In this paper, we propose a modification that speeds up the convergence of the randomized Kaczmarz algorithm for systems of linear…

Numerical Analysis · Computer Science 2013-05-17 Hassan Mansour , Ozgur Yilmaz

The Kaczmarz algorithm is an iterative method that solves linear systems of equations. It stands out among iterative algorithms when dealing with large systems for two reasons. First, at each iteration, the Kaczmarz algorithm uses a single…

Numerical Analysis · Mathematics 2024-04-10 Inês A. Ferreira , Juan A. Acebrón , José Monteiro

The Kaczmarz method is an iterative projection scheme for solving con-sistent system $Ax = b$. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the…

Numerical Analysis · Mathematics 2022-10-04 Chuan-gang Kang , Heng Zhou

Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend…

Numerical Analysis · Mathematics 2020-06-03 Anna Ma , Denali Molitor

The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution…

Numerical Analysis · Mathematics 2015-06-24 Deanna Needell , Ran Zhao , Anastasios Zouzias

The Kaczmarz algorithm is an iterative technique designed to solve consistent linear systems of equations. It falls within the category of row-action methods, focusing on handling one equation per iteration. This characteristic makes it…

Distributed, Parallel, and Cluster Computing · Computer Science 2024-02-01 Inês Ferreira , Juan A. Acebrón , José Monteiro

In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for large overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical…

Numerical Analysis · Mathematics 2020-04-07 Yanjun Zhang , Hanyu Li

The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical…

Numerical Analysis · Mathematics 2007-05-23 Thomas Strohmer , Roman Vershynin

Among recent developments centered around Randomized Kaczmarz (RK), a row-sampling iterative projection method for large-scale linear systems, several adaptions to the method have inspired faster convergence. Focusing solely on…

Numerical Analysis · Mathematics 2026-03-03 James Nguyen , Oleg Presnyakov , Adityakrishnan Radhakhrishnan

We propose a new method for preconditioning Kaczmarz method by sketching. Kaczmarz method is a stochastic method for solving overdetermined linear systems based on a sampling of matrix rows. The standard approach to speed up convergence of…

Numerical Analysis · Computer Science 2019-03-06 Alexandr Katrutsa , Ivan Oseledets

The Kaczmarz algorithm is a simple iterative scheme for solving consistent linear systems. At each step, the method projects the current iterate onto the solution space of a single constraint. Hence, it requires very low cost per iteration…

Optimization and Control · Mathematics 2019-02-27 Ion Necoara

The randomized version of the Kaczmarz method for the solution of linear systems is known to converge linearly in expectation. In this work we extend this result and show that the recently proposed Randomized Sparse Kaczmarz method for…

Optimization and Control · Mathematics 2016-10-11 Frank Schöpfer , Dirk A. Lorenz

We provide a complete characterization of the randomized Kaczmarz algorithm (RKA) for inconsistent linear systems. The Kaczmarz algorithm, known in some fields as the algebraic reconstruction technique, is a classical method for solving…

Numerical Analysis · Mathematics 2015-02-05 Chuang Wang , Ameya Agaskar , Yue M. Lu
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