In a recent paper by Strohmer and Vershynin (J. Fourier Anal. Appl. 15:262-278, 2009) a "randomized kaczmarz algorithm" is proposed for solving consistent systems of linear equations {aOE (c) a (i) ,x >=b...
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In a recent paper by Strohmer and Vershynin (J. Fourier Anal. Appl. 15:262-278, 2009) a "randomized kaczmarz algorithm" is proposed for solving consistent systems of linear equations {aOE (c) a (i) ,x >=b (i) } (i=1) (m) . In that algorithm the next equation to be used in an iterative kaczmarz process is selected with a probability proportional to aEuro-a (i) aEuro-(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.
Projection adjustment is a technique that improves the rate of convergence, as measured by the number of iterations needed to achieve a given level of performance, of the kaczmarz algorithm (KA) for iteratively solvin...
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Projection adjustment is a technique that improves the rate of convergence, as measured by the number of iterations needed to achieve a given level of performance, of the kaczmarz algorithm (KA) for iteratively solving a system of consistent linear equations, however at the cost of requiring additional time per iteration and increased storage. This hinders the applicability of the previously published kaczmarz algorithm with projection adjustment (KAPA) to large-scale problems. An enhancement EKAPA of KAPA that uses projection adjustment only for a small subset of the equations is proposed for significantly reducing the time and storage requirements. An analysis of the behavior of EKAPA is provided. An illustration is given to show that EKAPA using a small subset of the equations for projection adjustment can achieve a speed-up over KA similar to that of KAPA in terms of the number of iterations, but requires much less computer time and storage;hence, it is more suitable for large-scale problems.
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 meth...
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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 estimates for its rate of convergence are still scarce. We introduce a randomized version of the kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling.
For a Banach space X and its dual X*, a sequence ((phi(n), psi(n))) subset of X* x X is effective if the kaczmarz algorithm provides a reconstruction for every vector in X. We give necessary and sufficient conditions ...
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For a Banach space X and its dual X*, a sequence ((phi(n), psi(n))) subset of X* x X is effective if the kaczmarz algorithm provides a reconstruction for every vector in X. We give necessary and sufficient conditions for a sequence to be effective. Starting with the mixed Gram matrix, we derive necessary matrix Eq.s for an effective sequence. When certain boundedness conditions are met, we show that these matrix Eq.s are also sufficient. We also give necessary conditions for related sequences to form a resolution of the identity. Finally, we provide examples of effective sequences in infinite dimensional Banach spaces.
We develop a stochastic approximation version of the classical kaczmarz algorithm that is incremental in nature and takes as input noisy real time data. Our analysis shows that with probability one it mimics the behav...
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We develop a stochastic approximation version of the classical kaczmarz algorithm that is incremental in nature and takes as input noisy real time data. Our analysis shows that with probability one it mimics the behavior of the original scheme: starting from the same initial point, our algorithm and the corresponding deterministic kaczmarz algorithm converge to precisely the same point. The motivation for this work comes from network tomography where network parameters are to be estimated based upon end-to-end measurements. Numerical examples via Matlab based simulations demonstrate the efficacy of the algorithm. (C) 2013 Elsevier Ltd. All rights reserved.
The kaczmarz algorithm is an iterative method for reconstructing a signal xaa"e (d) from an overcomplete collection of linear measurements y (n) =aOE (c) x,phi (n) >, na parts per thousand yen1. We prove quant...
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The kaczmarz algorithm is an iterative method for reconstructing a signal xaa"e (d) from an overcomplete collection of linear measurements y (n) =aOE (c) x,phi (n) >, na parts per thousand yen1. We prove quantitative bounds on the rate of almost sure exponential convergence in the kaczmarz algorithm for suitable classes of random measurement vectors . Refined convergence results are given for the special case when each phi (n) has i.i.d. Gaussian entries and, more generally, when each phi (n) /ayen phi (n) ayen is uniformly distributed on . This work on almost sure convergence complements the mean squared error analysis of Strohmer and Vershynin for randomized versions of the kaczmarz algorithm.
Sequences of unit vectors for which the kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and...
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Sequences of unit vectors for which the kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations Ax = b and characterize the (bounded) matrices A for which the kaczmarz algorithm always converges to a solution.
A modified version of kaczmarz algorithm for improving image reconstruction from projections in computerized tomography is proposed. Instead of taking projection onto hyperplanes sequentially in the given order in eac...
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ISBN:
(纸本)9781424441334
A modified version of kaczmarz algorithm for improving image reconstruction from projections in computerized tomography is proposed. Instead of taking projection onto hyperplanes sequentially in the given order in each iteration by classical kaczmarz algorithm, the proposed algorithm does projection in an adaptive order which is evaluated from pseudo measurement error of each hyperplane. Since this algorithm proceeds with a basic aspect of traditional kaczmarz algorithm, it can suit a variety of kaczmarz extended algorithm. Numerical experiments described in the last part of the paper indicate that the Modified kaczmarz algorithm gives much better results than the classical one.
To accelerate the convergence of kaczmarz algorithm (KA) in consistent case, an improved kaczmarz algorithm with projection adjustment (KAPA) has been designed by the author. In order to deal with inconsistent case, e...
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ISBN:
(纸本)9781479968114
To accelerate the convergence of kaczmarz algorithm (KA) in consistent case, an improved kaczmarz algorithm with projection adjustment (KAPA) has been designed by the author. In order to deal with inconsistent case, extended KAPA (EKAPA) and its randomized version (ERKAPA) have been proposed in this paper. Numerical simulations of solving inconsistent linear equations have been conducted. The efficiency of EKAPA and ERKAPA have been verified by the simulation results.
In this paper suggested methods for acceleration kaczmarz algorithm regularized modifications to solve the standard regularization problem of A.N. Tikhonov. As shown in numerical experiments, for certain classes of pr...
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In this paper suggested methods for acceleration kaczmarz algorithm regularized modifications to solve the standard regularization problem of A.N. Tikhonov. As shown in numerical experiments, for certain classes of problems, such methods allow reducing both the number of iterations and the time for finding solutions. For the two-dimensional problem of seismic tomography proposed greedy forms of kaczmarz algorithm regularized modifications can reduce the number of iterations up to 28 times. (C) 2019 The Authors. Published by Elsevier Ltd.
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