• The Journal of Korean Institute of Communications and Information Sciences
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• v.31 no.8C
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• pp.786-794
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• 2006

We proposed a novel preconditioner based PCG(Preconditioned Conjugate Gradient) method for super resolution image reconstruction. Compared with the conventional block circulant type preconditioner, the proposed preconditioner can be more effectively applied for objective functions that include roughness penalty functions. The effectiveness of the proposed method was shown by simulations and experiments.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.213-222
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• 2008

Iterative methods are often suitable for solving least squares problems min$||Ax-b||_2$, where A $\epsilon\;\mathbb{R}^{m{\times}n}$ is large and sparse. The well known LSQR algorithm is among the iterative methods for solving these problems. A good preconditioner is often needed to speedup the LSQR convergence. In this paper we present the numerical experiments of applying a well known preconditioner for the LSQR algorithm. The preconditioner is based on the $A^T$ A-orthogonalization process which furnishes an incomplete upper-lower factorization of the inverse of the normal matrix $A^T$ A. The main advantage of this preconditioner is that we apply only one of the factors as a right preconditioner for the LSQR algorithm applied to the least squares problem min$||Ax-b||_2$. The preconditioner needs only the sparse matrix-vector product operations and significantly reduces the solution time compared to the unpreconditioned iteration. Finally, some numerical experiments on test matrices from Harwell-Boeing collection are presented to show the robustness and efficiency of this preconditioner.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.299-312
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• 2004

ILUS factorization has many desirable properties such as its amenability to the skyline format, the ease with which stability may be monitored, and the possibility of constructing a preconditioner with symmetric structure. In this paper we introduce a new preconditioning technique for general sparse linear systems based on the ILUS factorization strategy. The resulting preconditioner has the same properties as the ILUS preconditioner. Some theoretical properties of the new preconditioner are discussed and numerical experiments on test matrices from the Harwell-Boeing collection are tested. Our results indicate that the new preconditioner is cheaper to construct than the ILUS preconditioner.

• Journal of applied mathematics & informatics
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• v.34 no.1_2
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• pp.85-94
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• 2016

In this paper, we first propose a fast one-parameter relaxation (FOPR) method with a scaled preconditioner for solving the saddle point problems, and then we present a formula for finding its optimal parameter. To evaluate the effectiveness of the proposed FOPR method with a scaled preconditioner, numerical experiments are provided by comparing its performance with the existing one or two parameter relaxation methods with optimal parameters such as the SOR-like, the GSOR and the GSSOR methods.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.531-540
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• 2018

We first show how to construct the linear operator equations corresponding to Tikhonov regularization problems for solving image deblurring problems with nearly separable point spread functions. We next propose a Kronecker product preconditioner which is suitable for the global PCG method. Lastly, we provide numerical experiments of the global PCG method with the Kronecker product preconditioner for several image deblurring problems to evaluate its effectiveness.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.459-474
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• 2018

Cao et al. in (Numer. Linear. Algebra Appl. 18 (2011) 875-895) proposed a splitting method for saddle point problems which unconditionally converges to the solution of the system. It was shown that a Krylov subspace method like GMRES in conjunction with the induced preconditioner is very effective for the saddle point problems. In this paper we first modify the iterative method, discuss its convergence properties and apply the induced preconditioner to the problem. Numerical experiments of the corresponding preconditioner are compared to the primitive one to show the superiority of our method.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.25 no.11
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• pp.1581-1593
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• 2001

An efficient variable-reordering method for finite element meshes is used and the effect of variable-reordering is investigated. For the element renumbering of unstructured meshes, Cuthill-McKee ordering is adopted. The newsy reordered global matrix has a much narrower bandwidth than the original one, making the ILU preconditioner perform bolter. The effect of variable reordering on the convergence behaviour of saddle point type matrix it studied, which results from P2/P1 element discretization of the Navier-Stokes equations. We also propose and test 'level(0) preconditioner'and 'level(2) ILU preconditioner', which are another versions of the existing 'level(1) ILU preconditioner', for the global matrix generated by P2/P1 finite element method of incompressible Navier-Stokes equations. We show that 'level(2) ILU preconditioner'performs much better than the others only with a little extra computations.

• The KIPS Transactions:PartA
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• v.8A no.3
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• pp.227-234
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• 2001

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditiner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to $1025{\times}1024$. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications, The results show that Multi-Color Block SOR is scalabl and gives the best performances.

• Journal of IKEEE
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• v.23 no.2
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• pp.614-621
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• 2019

In this paper, we presents the integral equation-fast Fourier transform(IE-FFT) and block matrix preconditioner (BMP) to solve electromagnetic scattering problems of penetrable structures composed of dielectric or magnetic materials. IE-FFT can significantly improve the amount of calculation to solve the matrix equation constructed from the moment method(MoM). Moreover, the iterative method in conjunction with BMP can be significantly reduce the number of iterations required to solve the matrix equations which are constructed from electrically large structures. Numerical results show that IE-FFT and block matrix preconditioner can solve electromagnetic scattering problems for penetrable objects quickly and accurately.

• Journal of Digital Contents Society
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• v.10 no.4
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• pp.579-585
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• 2009

In [9], Tyrtshnikov proposed a preconditioned approach to derive a general solution from a Toeplitz linear system. Furthermore, the process of selecting a preconditioner matrix from symmetric Toeplitz matrix, which has been used in previous studies, is introduced. This research introduces a new method for finding the preconditioner in a Toeplitz system. Also, through analyzing these preconditioners, it is derived that eigenvalues of a symmetric Toeplitz are very close to eigenvalues of a new preconditioner for T. It is shown that if the spectrum of the preconditioned system $C_0^{-1}T$ is clustered around 1, then the convergence rate of the preconditioned system is superlinear. From these results, it is determined to get the superliner at the convergence rate by our good preconditioner $C_0$. Moreover, an advantage is driven by increasing various applications i. e. image processing, signal processing, etc. in this study from the proposed preconditioners for Toeplitz matrices. Another characteristic, which this research holds, is that the preconditioner retains the properties of the Toeplitz matrix.