Krylov subspace iterative methods for time domain equivalent sources method based nearfield acoustical holography

In this paper, we consider performance of relaxation iterative methods for four types of image deblurring problems with different regularization terms. We first study how to apply relaxation iterative methods efficiently to the Tikhonov regularization problems, and then we propose how to find good preconditioners and near optimal relaxation parameters which are essential factors for fast convergence rate and computational efficiency of relaxation iterative methods. We next study efficient applications of relaxation iterative methods to Split Bregman method and the fixed point method for solving the L1-norm or total variation regularization problems. Lastly, we provide numerical experiments for four types of image deblurring problems to evaluate the efficiency of relaxation iterative methods by comparing their performances with those of Krylov subspace iterative methods. Numerical experiments show that the proposed techniques for finding preconditioners and near optimal relaxation parameters of relaxation iterative methods work well for image deblurring problems. For the L1-norm and total variation regularization problems, Split Bregman and fixed point methods using relaxation iterative methods perform quite well in terms of both peak signal to noise ratio values and execution time as compared with those using Krylov subspace methods.

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A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

Journal of Applied Mathematics ◽

10.1155/2013/489295 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Wei-Hua Luo ◽

Ting-Zhu Huang

Keyword(s):

Saddle Point ◽

Unit Circle ◽

Iterative Methods ◽

Krylov Subspace ◽

Numerical Results ◽

Saddle Point Problems ◽

Good Convergence ◽

Stokes Problems ◽

Preconditioned Matrix ◽

Generalized Saddle Point Problems

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that whenαis big enough, it has an eigenvalue at 1 with multiplicity at leastn, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameterα→0. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner.

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