An accelerated Jacobi-gradient based iterative algorithm for solving sylvester matrix equations

Tian, Zhaolu; Mao, Tian; Gu, Changzhi; Hao, Xiaoning

doi:10.2298/fil1708381t

Cited by 35 publications

(28 citation statements)

References 25 publications

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“…Fixing , , the subproblem can be obtained by solving the following minimization problem. Recently, various acceleration techniques of iterative algorithms are proposed [55][56][57][58][59][60][61][62][63]. For this subproblem, in order to accelerate the convergence of the above iteration, we adopt the acceleration technique in [55] to solve it.…”

Section: Alternating Minimization Methodsmentioning

confidence: 99%

Image Restoration by a Mixed High-Order Total Variation and l1 Regularization Model

Zhu

Hao

2018

Mathematical Problems in Engineering

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Total variation regularization is well-known for recovering sharp edges; however, it usually produces staircase artifacts. In this paper, in order to overcome the shortcoming of total variation regularization, we propose a new variational model combining high-order total variation regularization and l1 regularization. The new model has separable structure which enables us to solve the involved subproblems more efficiently. We propose a fast alternating method by employing the fast iterative shrinkage-thresholding algorithm (FISTA) and the alternating direction method of multipliers (ADMM). Compared with some current state-of-the-art methods, numerical experiments show that our proposed model can significantly improve the quality of restored images and obtain higher SNR and SSIM values.

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Section: Alternating Minimization Methodsmentioning

confidence: 99%

Image Restoration by a Mixed High-Order Total Variation and l1 Regularization Model

Zhu

Hao

2018

Mathematical Problems in Engineering

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“…where X = (vec(X 1 ); vec(X 2 )) and B = (vec(C 1 ); vec(C 2 )). It can be verified that M is rank deficient and GB method is semi-convergent and its the optimum value is µ opt = 0.00109 which is computed by (19) Here we compare the performance of GB, DGB-version 1 and DGB-version 2 to solve (22). We used the following stopping criterion…”

Section: Numerical Experimentsmentioning

confidence: 99%

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Ameri¹,

Panjeh²

2018

Filomat

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Recently in [Journal of Computational Physics, 321 (2016), 829-907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper, we first prove semi-convergence of iterative methods with DORS to solve singular linear system of equations. In particular, we propose applying the DORS in the Modified HSS (MHSS) to solve singular complex symmetric systems and in the Richardson method to solve normal equations. Moreover, based on the obtained results, an algorithm is developed for solving coupled matrix equations. It is seen that the proposed method outperforms the relaxed gradient-based (RGB) method [Comput. Math. Appl. 74 (2017), no. 3, 597-604] for solving coupled matrix equations. Numerical results are examined to illustrate the validity of the established results and applicability of the presented algorithms.

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“…e Tikhonov regularisation (TR) method [21][22][23], truncated singular value method [24,25], kernel function-based regularisation method [26,27], and l 1 norm regularisation method [28] are often used to solve ill-posed problems. When estimating nonlinear parameters, iterative search methods such as the Gauss-Newton method, steepest gradient method, and LM method are commonly used [29][30][31][32][33][34]. In these methods, the Jacobian matrix has a strong influence on the computational efficiency of the algorithm.…”

Section: Introductionmentioning

confidence: 99%

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

Guo

2019

Complexity

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This paper considers the classical separable nonlinear least squares problem. Such problems can be expressed as a linear combination of nonlinear functions, and both linear and nonlinear parameters are to be estimated. Among the existing results, ill-conditioned problems are less often considered. Hence, this paper focuses on an algorithm for ill-conditioned problems. In the proposed linear parameter estimation process, the sensitivity of the model to disturbance is reduced using Tikhonov regularisation. The Levenberg–Marquardt algorithm is used to estimate the nonlinear parameters. The Jacobian matrix required by LM is calculated by the Golub and Pereyra, Kaufman, and Ruano methods. Combining the nonlinear and linear parameter estimation methods, three estimation models are obtained and the feasibility and stability of the model estimation are demonstrated. The model is validated by simulation data and real data. The experimental results also illustrate the feasibility and stability of the model.

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An accelerated Jacobi-gradient based iterative algorithm for solving sylvester matrix equations

Cited by 35 publications

References 25 publications

Image Restoration by a Mixed High-Order Total Variation and l1 Regularization Model

Image Restoration by a Mixed High-Order Total Variation and l1 Regularization Model

Delayed over-relaxation in iterative schemes to solve rank deficient linear system of (matrix) equations

Tikhonov Regularized Variable Projection Algorithms for Separable Nonlinear Least Squares Problems

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