1,721,069 research outputs found
Inverse problems for regularization matrices
Full text linkDiscrete ill-posed problems are difficult to solve, because their solution is very sensitive to errors in the data and to round-off errors introduced during the solution process. Tikhonov regularization replaces the given discrete ill-posed problem by a nearby penalized least-squares problem whose solution is less sensitive to perturbations. The penalization term is defined by a regularization matrix, whose choice may affect the quality of the computed solution significantly. We describe several inverse matrix problems whose solution yields regularization matrices adapted to the desired solution. Numerical examples illustrate the performance of the regularization matrices determined
A modified truncated singular value decomposition method for discrete ill-posed problems
No full textTruncated singular value decomposition is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank-k approximant, which we denote by Ak. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L-curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as Ak. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by Ak. © 2014 John Wiley & Sons, Ltd
The structured distance to normality of banded Toeplitz matrices
No full textSpectral properties of normal (2k + 1)- banded Toeplitz matrices of order n, with k <= left perpendicularn/ 2right perpendicular, are described. Formulas for the distance of (2k + 1)-banded Toeplitz matrices to the algebraic variety of similarly structured normal matrices are presented
Generalized circulant Strang-type preconditioners
No full textStrang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {eif}-circulant Strang-type preconditioner. Copyright (C) 2011 John Wiley & Sons, Ltd
The structured distance to normality of Toeplitz matrices with application to preconditioning
No full textA formula for the distance of a Toeplitz matrix to the subspace of {e(i phi)}-circulant matrices is presented, and applications of {e(i phi)}-circulant matrices to preconditioning of linear systems of equations with a Toeplitz matrix are discussed. Copyright (C) 2010 John Wiley & Sons, Ltd
Tridiagonal Toeplitz matrices: Properties and novel applications
No full textThe eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the E-pseudospectrum are derived. The second part of the paper discusses applications of the theory to inverse eigenvalue problems, the construction of Chebyshev polynomial-based Krylov subspace bases, and Tikhonov regularization. Copyright (c) 2012 John Wiley & Sons, Ltd
Inverse subspace problems with applications
No full textGiven a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace-restricted SVD method for the solution of linear discrete ill-posed problems, also are considered. © 2013 John Wiley & Sons, Ltd
A note on superoptimal generalized circulant preconditioners
No full textCirculant matrices can be effective preconditioners for linear systems of equations with a Toeplitz matrix. Several approaches to construct such preconditioners have been described in the literature. This paper focuses on the superoptimal circulant preconditioners proposed by Tyrtyshnikov, and investigates a generalization obtained by allowing generalized circulant matrices. Numerical examples illustrate that the new preconditioners so obtained can give faster convergence than available preconditioners based on circulant and generalized circulant matrices. © 2013 IMACS
Lavrentiev-type regularization methods for Hermitian problems
Full text linkLavrentiev regularization is a popular approach to the solution of linear discrete ill-posed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods. © 2014 Springer-Verlag Italia
Tikhonov regularization based on generalized Krylov subspace methods
No full textAbstract
We consider Tikhonov regularization of large linear discrete ill-posed problems with a regularization operator of general form and present an iterative scheme based on a generalized Krylov subspace method. This method simultaneously reduces both the matrix of the linear discrete ill-posed problem and the regularization operator. The reduced problem so obtained may be solved, e.g., with the aid of the singular value decomposition. Also, Tikhonov regularization with several regularization operators is discussed.
Keywords
Ill-posed problem;
Regularization operator;
Tikhonov regularization;
Multiparameter regularizatio
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