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...autiful mathematical theoretical results to many successful applications in important real-world problems. Kaczmarz’s method is also a special case of several other algorithmic structures, see, e.g., =-=[4, 8, 12, 20]-=-. In a recent paper [24] (see also [23]) Strohmer and Vershynin propose a 2 randomized Kaczmarz’s algorithm and claim that “it outperforms all previously known methods on general extremely overdetermi...

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...ults in the sum being divided by the actual (and much smaller) number of nonzero summands appearing in it. Related to the CAV idea are the recent papers by Gordon and Gordon [34] and by Censor et al. =-=[20]-=-. In contrast with all the above which are related to the CFP, we use this idea here in conjunction with Algorithm 2 and Algorithm 5 for the BAP. We use it only heuristically and report on the success...

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...r, if we augment the iterative step (1.2) with a row scaling, x k+1 = x k + λkSA T M(b − Ax k ), k = 0, 1, 2, . . ., with S = diag(m/Nj), then we obtain the simultaneous version of the DROP algorithm =-=[6]-=-. The original proposals of some SIRT methods use weights, but for simplicity we do not include weights here. We now give a short summary of the contents of the paper. In Section 2 we study the semi-c...

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...a, many block parallel iterative methods suitable for parallel computing have been developed recently [7]. Among them Component Averaging (CAV) [6] and Diagonally-Relaxed Orthogonal Projection (DROP) =-=[5]-=- methods can be used on sensor networks [15] using fusion center. CAV [6] is a Cimmino-type method that projects current iterates simultaneously onto all the systems’ hyperplanes. In this method, they...

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...preceding Equation (4.2). The iterative step resulting from the first two features has the form xk+1 = xk + λk mX i=1 Gi PGiHi (x k)− xk ´ . (4.5) Recent work by Censor, Elfving, Herman and Nikazad =-=[36]-=- shows that component averaging is valid (i.e., generates convergent iterative sequences) even when orthogonal projections are used and not generalized oblique ones as described above. 4.1. Seminorm-i...

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...esultingssequence ofsis summable, as required for boundedsperturbations resilience.sThe projection operator is then applied to the newlysperturbed point. In this work we have chosen the DROPsoperator =-=[6]-=-, which belongs to the BIP algorithmic family.sIn DROP, the rows of the system matrix in (1) arespartitioned intosblocks, wheresis the set of rowsindices of thesblock and the total number of rows issT...

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...econstruction literature is named ordered subsets methods [4, 11, 19]), which lie between sequential and simultaneous cases, have been studied in several works with different applications, see, e.g., =-=[1, 4, 6, 7, 8, 13, 14, 15, 16, 25, 27, 29]-=-. Simultaneous block iterative methods have also been used to increase computational efficiency using parallel processors [3, 8, 21]. For an overview in a more general framework, see [9]. In the case ...

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...s, M is the maximum number of iterations, and wki are positive iteration dependent weights which must sum up (over i) to one, for every k ≥ 0 [CM99]. Algebraic Reconstruction Technique- ART Algorithm =-=[CEHN08]-=- Initialization: x0 ∈ Rn is arbitrary. Iterative Step: Given xk compute. for k = 1 to M do for i = 1 to m do xk+1j = x k j + λk n∑ i=1 bi − 〈 ai, xk 〉 ‖ai‖2 a i j end do 2.1.2 CIM Algorithm The Cimmin...

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...ation. Convergence of the CAV method was established by showing that no singular value of the matrix B exceeds one, where B has the entries Bij = Aij/ √√√√ J∑ t=1 st|Ait|2. In this paper we extend this result, to show that no eigenvalue of A†A exceeds the maximum of the numbers pi = J∑ t=1 st|Ait|2. Convergence of CAV then follows, as does convergence of several other methods, including the ART, Landweber’s method, the SART [1], the block-iterative CAV (BICAV) [9], the CARP1 method of Gordon and Gordon [12], and a block-iterative variant of CARP1 obtained from the DROP method of Censor et al. [7]. For a positive integer N with 1 ≤ N ≤ I, we let B1, ..., BN be not necessarily disjoint subsets of the set {i = 1, ..., I}; the subsets Bn are called blocks. We then let An be the matrix and b n the vector obtained from A and b, respectively, by removing all the rows except for those whose index i is in the set Bn. For each n, we let snt be the number of non-zero entries in the tth column of the matrix An, sn the maximum of the snt, s the maximum of the st, and Ln = ρ(A † nAn) be the spectral radius, or largest eigenvalue, of the matrix A†nAn, with L = ρ(A †A). We denote by Ai the ith row of...