Preconditioned Iterative Methods for Weighted Toeplitz Least Squares Problems

Benzi, Michele; Ng, Michael K.

doi:10.1137/040616048

Cited by 64 publications

(47 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While many efficient algorithms have been developed for solving problems with Toeplitz structure, a few emerging applications lead to Toeplitz-related problems for which the available algorithms are not directly applicable [18,7]. In this paper, we consider the preconditioned iterative method for weighted Toeplitz regularized least squares problems min x∈R n Bx − b 2 2 , (1.…”

Section: Introductionmentioning

confidence: 99%

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Ng¹,

Pan²

2014

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

The main aim of this paper is to develop a fast algorithm for solving weighted Toeplitz regularized least squares problems arising from image restoration. Based on augmented system formulation, we develop new Hermitian and skew-Hermitian splitting (HSS) preconditioners for solving such linear systems. The advantage of the proposed preconditioner is that the blurring matrix, weighting matrix, and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use. We show that for a preconditioned system that is derived from a saddle point structure of size (m + n) × (m + n), the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 − λ with |λ| < 1. We also study how to choose the HSS parameter to minimize the magnitude of λ, and therefore the Krylov subspace method applied to solving the preconditioned system converges very quickly. Experimental results for image restoration problems are reported to demonstrate that the performance of the proposed preconditioner is better than the other testing preconditioners.

show abstract

Section: Introductionmentioning

confidence: 99%

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Ng¹,

Pan²

2014

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For example, in finite-difference or sinc discretizations of nonlinear partial differential equations [7][8][9][10], in collocation approximations of nonlinear integral equation [11] and in saddle point problems from image processing [12,13].…”

Section: Introductionmentioning

confidence: 99%

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

Zhu

2015

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H = 1 2 (A + A T ) is a centrosymmetric matrix and the skew-symmetric part H = 1 2 (A − A T ) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective.

show abstract

“…It is obvious that the coefficient matrix of (1) is Hermitian (non-Hermitian) when the matrix A is Hermitian (non-Hermitian). Linear systems like (1) appear in many different applications of scientific computing, such as the finite element approximation to solve the Navier-Stokes equation, constrained optimization, mixed finite element formulations for second-order elliptic problems, weighted Toeplitz least squares problems (see [1][2][3][4][5]). It is known that the above linear systems are generally indefinite and ill-conditioned, i.e., the eigenvalue of the coefficient matrix of (1) is with both positive and negative parts.…”

Section: Introductionmentioning

confidence: 99%

The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems

Huang

2009

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tIn this paper, we consider the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems with nonzero (2, 2) blocks. The spectral property of the preconditioned matrix is studied in detail. Under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will form two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter approaches to zero from above, so do all eigenvalues of the preconditioned matrix with the original system being Hermitian. Numerical experiments are given to demonstrate the results.

show abstract

Preconditioned Iterative Methods for Weighted Toeplitz Least Squares Problems

Cited by 64 publications

References 38 publications

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

Weighted Toeplitz Regularized Least Squares Computation for Image Restoration

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems

Contact Info

Product

Resources

About