In this paper, we present a series of new preconditioners with parameters of strictly diagonally dominant Z-matrix, which contain properly two kinds of known preconditioners as special cases. Moreover, we prove the mo...In this paper, we present a series of new preconditioners with parameters of strictly diagonally dominant Z-matrix, which contain properly two kinds of known preconditioners as special cases. Moreover, we prove the monotonicity of spectral radiuses of iterative matrices with respect to the parameters and some comparison theorems. The results obtained show that the bigger the parameter k is（i.e., we select the more upper right diagonal elements to be the preconditioner）, the less the spectral radius of iterative matrix is. A numerical example generated randomly is provided to illustrate the theoretical results.展开更多
Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image...Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.展开更多
In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the m...In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives,which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices.The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system.Theoretically,we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval(12,32)and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size.Moreover,the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order.Our theoretical results fill in a vacancy in the literature.Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.展开更多
Discusses the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. Definition of the genuine-optimal circulant preconditioner; Use of the preconditioned conjugate gradient method; Numeric...Discusses the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. Definition of the genuine-optimal circulant preconditioner; Use of the preconditioned conjugate gradient method; Numerical treatments for high order quadrature rules.展开更多
A new favorable iterative algorithm named as PBiCGSTAB （preconditioned bi-conjugate gradient stabilized） algorithm is presented for solving large sparse complex systems. Based on the orthogonal list, the special tec...A new favorable iterative algorithm named as PBiCGSTAB （preconditioned bi-conjugate gradient stabilized） algorithm is presented for solving large sparse complex systems. Based on the orthogonal list, the special technique of only storing non-zero elements is carried out. The incomplete LU factorization without fill-ins is adopted to reduce the condition number of the coefficient matrix. The BiCGSTAB algorithm is extended from the real system to the complex system and it is used to solve the preconditioned complex linear equations. The locked-rotor state of a single-sided linear induction machine is simulated by the software programmed with the finite element method and the PBiCGSTAB algorithm. Then the results are compared with those from the commercial software ANSYS, showing the validation of the proposed software. The iterative steps required for the proposed algorithm are reduced to about one-third, when compared to the BiCG method, therefore the algorithm is fast.展开更多
基金Shaanxi Province Natural Science Foundation,2007A16,China
文摘In this paper, we present a series of new preconditioners with parameters of strictly diagonally dominant Z-matrix, which contain properly two kinds of known preconditioners as special cases. Moreover, we prove the monotonicity of spectral radiuses of iterative matrices with respect to the parameters and some comparison theorems. The results obtained show that the bigger the parameter k is（i.e., we select the more upper right diagonal elements to be the preconditioner）, the less the spectral radius of iterative matrix is. A numerical example generated randomly is provided to illustrate the theoretical results.
基金supported by the National Basic Research Program (No.2005CB321702)the National Outstanding Young Scientist Foundation(No. 10525102)the Specialized Research Grant for High Educational Doctoral Program(Nos. 20090211120011 and LZULL200909),Hong Kong RGC grants and HKBU FRGs
文摘Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.
基金supported in part by research grants of the Science and Technology Development Fund,Macao SAR(No.0122/2020/A3)University of Macao(No.MYRG2020-00224-FST)+1 种基金the HKRGC GRF(No.12306616,12200317,12300218,12300519,17201020))China Postdoctoral Science Foundation(Grant 2020M682897).
文摘In this paper,we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives,which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices.The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system.Theoretically,we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval(12,32)and thus the preconditioned conjugate gradient method converges linearly within an iteration number independent of the discretization step-size.Moreover,the proposed method can be extended to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by functions with zeros of fractional order.Our theoretical results fill in a vacancy in the literature.Numerical examples are presented to show the convergence performance of the proposed preconditioner that is better than other preconditioners.
基金Supported in part by the natural science foundation of China No. 19901017.
文摘Discusses the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. Definition of the genuine-optimal circulant preconditioner; Use of the preconditioned conjugate gradient method; Numerical treatments for high order quadrature rules.
文摘A new favorable iterative algorithm named as PBiCGSTAB （preconditioned bi-conjugate gradient stabilized） algorithm is presented for solving large sparse complex systems. Based on the orthogonal list, the special technique of only storing non-zero elements is carried out. The incomplete LU factorization without fill-ins is adopted to reduce the condition number of the coefficient matrix. The BiCGSTAB algorithm is extended from the real system to the complex system and it is used to solve the preconditioned complex linear equations. The locked-rotor state of a single-sided linear induction machine is simulated by the software programmed with the finite element method and the PBiCGSTAB algorithm. Then the results are compared with those from the commercial software ANSYS, showing the validation of the proposed software. The iterative steps required for the proposed algorithm are reduced to about one-third, when compared to the BiCG method, therefore the algorithm is fast.
