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NOISEREDUCING CASCADIC MULTILEVEL METHODS FOR LINEAR DISCRETE ILLPOSED PROBLEMS
"... Abstract. Cascadic multilevel methods for the solution of linear discrete illposed problems with noisereducing restriction and prolongation operators recently have been developed for the restoration of blur and noisecontaminated images. This is a particular illposed problem. The multilevel meth ..."
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Abstract. Cascadic multilevel methods for the solution of linear discrete illposed problems with noisereducing restriction and prolongation operators recently have been developed for the restoration of blur and noisecontaminated images. This is a particular illposed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noisereducing multilevel methods for the solution of general linear discrete illposed problems. Key words. illposed problem, multilevel method, noisereduction 1. Introduction. Many problems in science and engineering require the determination of the cause of an observed effect. These problems often can be formulated
Error estimates for largescale illposed problems
 NUMER. ALGORITHMS
"... The computation of an approximate solution of linear discrete illposed problems with contaminated data is delicate due to the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution to errors in the data by replacing the given illp ..."
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Cited by 7 (7 self)
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The computation of an approximate solution of linear discrete illposed problems with contaminated data is delicate due to the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution to errors in the data by replacing the given illposed problem by a nearby problem, whose solution is less sensitive to perturbation. This regularization method requires that a suitable value of the regularization parameter be chosen. Recently, Brezinski et al. [Numer. Algorithms (2008), in press] described new approaches to estimate the error in approximate solutions of linear systems of equations and applied these estimates to determine a suitable value of the regularization parameter in Tikhonov regularization when the approximate solution is computed with the aid of the singular value decomposition. This paper discusses applications of these and related error estimates to the solution of largescale illposed problems when approximate solutions are computed by Tikhonov regularization based on partial Lanczos bidiagonalization of the matrix. The connection between partial Lanczos bidiagonalization and Gauss quadrature is utilized to determine inexpensive bounds for a family of error estimates.
AN ITERATIVE METHOD FOR TIKHONOV REGULARIZATION WITH A GENERAL LINEAR REGULARIZATION OPERATOR
, 2010
"... Tikhonov regularization is one of the most popular approaches to solve discrete illposed problems with errorcontaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on GolubKahan bidiagonal ..."
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Cited by 5 (3 self)
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Tikhonov regularization is one of the most popular approaches to solve discrete illposed problems with errorcontaminated data. A regularization operator and a suitable value of a regularization parameter have to be chosen. This paper describes an iterative method, based on GolubKahan bidiagonalization, for solving largescale Tikhonov minimization problems with a linear regularization operator of general form. The regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method.
Tikhonov regularization based on generalized Krylov subspace methods
, 2010
"... We consider Tikhonov regularization of large linear discrete illposed 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 illposed problem an ..."
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Cited by 4 (3 self)
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We consider Tikhonov regularization of large linear discrete illposed 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 illposed 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.
On the reduction of Tikhonov minimization problems and the construction of regularization matrices
 NUMER. ALGORITHMS
"... Tikhonov regularization replaces a linear discrete illposed problem by a penalized leastsquares problem, whose solution is less sensitive to errors in the data and roundoff errors introduced during the solution process. The penalty term is defined by a regularization matrix and a regularization ..."
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Cited by 3 (3 self)
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Tikhonov regularization replaces a linear discrete illposed problem by a penalized leastsquares problem, whose solution is less sensitive to errors in the data and roundoff errors introduced during the solution process. The penalty term is defined by a regularization matrix and a regularization parameter. The latter generally has to be determined during the solution process. This requires repeated solution of the penalized leastsquares problem. It is therefore attractive to transform the leastsquares problem to simpler form before solution. The present paper describes a transformation of the penalized leastsquares problem to simpler form that is faster to compute than available transformations in the situation when the regularization matrix has linearly dependent columns and no exploitable structure. Properties of this kind of regularization matrices are discussed and their performance is illustrated.
Square regularization matrices for large linear discrete illposed problems
 Numer. Linear Algebra Appl
"... Abstract. Large linear discrete illposed problems with contaminated data are often solved with the aid of Tikhonov regularization. Commonly used regularization matrices are finite difference approximations of a suitable derivative and are rectangular. This paper discusses the design of square regul ..."
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Abstract. Large linear discrete illposed problems with contaminated data are often solved with the aid of Tikhonov regularization. Commonly used regularization matrices are finite difference approximations of a suitable derivative and are rectangular. This paper discusses the design of square regularization matrices that can be used in iterative methods based on the Arnoldi process for largescale Tikhonov regularization problems. Key words. Tikhonov regularization, regularization matrix, Arnoldi process
Simple square smoothing regularization operators
 Electron. Trans. Numer. Anal
"... Abstract. Tikhonov regularization of linear discrete illposed problems often is applied with a finite difference regularization operator that approximates a loworder derivative. These operators generally are represented by banded rectangular matrices with fewer rows than columns. They therefore ca ..."
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Cited by 2 (2 self)
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Abstract. Tikhonov regularization of linear discrete illposed problems often is applied with a finite difference regularization operator that approximates a loworder derivative. These operators generally are represented by banded rectangular matrices with fewer rows than columns. They therefore cannot be applied in iterative methods that are based on the Arnoldi process, which requires the regularization operator to be represented by a square matrix. This paper discusses two approaches to circumvent this difficulty: zeropadding of the rectangular matrices to make them square and extending the rectangular matrix to a square circulant. We also describe how to combine these operators by weighted averaging and with orthogonal projection. Applications to Arnoldi and Lanczos bidiagonalizationbased Tikhonov regularization, as well as to truncated iteration with a range restricted minimal residual method, are presented.
SIMPLIFIED GSVD COMPUTATIONS FOR THE SOLUTION OF LINEAR DISCRETE ILLPOSED PROBLEMS
"... Abstract. The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the co ..."
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Cited by 1 (0 self)
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Abstract. The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the context of Tikhonov regularization. Also, other regularization methods, including truncated GSVD, are considered. We compare the computational efforts required by the simplified GSVD method and the Aweighted generalized inverse introduced by Eldén.
FGMRES for linear discrete illposed problems
"... GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear illposed problems with errorcontaminated da ..."
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GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear illposed problems with errorcontaminated data represented by the righthand side. Such linear systems are commonly referred to as linear discrete illposed problems. The FGMRES method, proposed by Saad, is a generalization of GMRES that allows larger flexibility in the choice of solution subspace than GMRES. This paper explores application of FGMRES to the solution of linear discrete illposed problems. Numerical examples illustrate that FGMRES with a suitably chosen solution subspace may determine approximate solutions of higher quality than commonly applied iterative methods.