Convergence results of Landweber iterations for linear systems

Qu, Gangrong; Ji, Ming; Yang, Yun

doi:10.1007/s10255-013-0299-y

Cited by 7 publications

(7 citation statements)

References 16 publications

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“…(2) can be transformed to the Landweber scheme without weights [3,21]. Generally, the linear systems (1) is inconsistent.…”

Section: Introductionmentioning

confidence: 99%

“…The necessary and sufficient conditions for the convergence of the iteration x (n) and for its convergence to a solution of the normal equation ( 6) are presented in [7]. A neat iterative representation formula was derived in [21] and the convergence conditions in [7] were refined in [21].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, with the neat iterative representation formula for the Landweber iteration in [21], we derive a new iterative representation (13) and find the optimal relaxation as λ n = 2 µ1+µm , by minimizing the spectral radius of the newly derived iterative matrix at each iteration, where µ 1 and µ m are the biggest and the smallest positive eigenvalues of matrix A * A. According to the optimal relaxation strategy and the convergence conditions, if only µ 1 is known, we prove that the spectral radius of the derived iterative matrix with a relaxation coefficient λ (1)…”

Section: Introductionmentioning

confidence: 99%

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Relaxation strategy for the Landweber method

Han

Jiang

2016

Signal Processing

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The Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the Landweber iteration is of both theoretical and practical importance. By the representation of the iterative formula and the convergence results of the Landweber iteration, we derive the optimal relaxation method under the minimization of the spectral radius of the newly derived iterative matrix. We also establish the iterative relaxation strategy to accelerate the convergence for the Landweber iteration when only the biggest singular value is available. As an immediate result, we derive the corresponding results for the Richardson's iteration for the symmetric nonnegative definite linear systems. Finally, numerical simulations are conducted to validate the theoretical results. The advantage of the proposed relaxation strategies is demonstrated by comparing with the existing strategies.

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“…(2) can be transformed to the Landweber scheme without weights [3,21]. Generally, the linear systems (1) is inconsistent.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Relaxation strategy for the Landweber method

Han

Jiang

2016

Signal Processing

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“…Convergence conditions of the method (2) have been established [7][8][9]. The semiconvergence behavior means that initially the iteration vector approaches a regularized solution, while continuing the iteration often leads to iteration vectors corrupted by noise [10].…”

mentioning

confidence: 99%

“…is divergent because of min = 0 [8]. In our previous work, based on minimizing the spectral radius of the new iterative matrix, a new relaxation coefficient…”

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confidence: 99%

Weighting Algorithm and Relaxation Strategies of the Landweber Method for Image Reconstruction

Han

Wang

2018

Mathematical Problems in Engineering

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The iterative approach is important for image reconstruction with ill-posed problem, especially for limited angle reconstruction. Most of iterative algorithms can be written in the general Landweber scheme. In this context, appropriate relaxation strategies and appropriately chosen weights are critical to yield reconstructed images of high quality. In this paper, based on reducing the condition number of matrix 푇 , we find one method of weighting matrices for the general Landweber method to improve the reconstructed results. For high resolution images, the approximate iterative matrix is derived. And the new weighting matrices and corresponding relaxation strategies are proposed for the general Landweber method with large dimensional number. Numerical simulations show that the proposed weighting methods are effective in improving the quality of reconstructed image for both complete projection data and limited angle projection data.

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The Convergence Properties for Regularized Landweber Method

Wang

2016

J Optim Theory Appl

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Convergence results of Landweber iterations for linear systems

Cited by 7 publications

References 16 publications

Relaxation strategy for the Landweber method

Relaxation strategy for the Landweber method

Weighting Algorithm and Relaxation Strategies of the Landweber Method for Image Reconstruction

The Convergence Properties for Regularized Landweber Method

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