A Unified Theoryof Conditioning for Linear Least Squares and Tikhonov Regularization Solutions

Malyshev, Alexander

doi:10.1137/s0895479801389564

Cited by 25 publications

(18 citation statements)

References 11 publications

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“…In the above definition, we study the general case when both the coefficient matrix A and the right‐hand side b are perturbed, which is an extension to (6) and (7) by Malyshev 14, who only considered the perturbation on A or b .…”

Section: Normwise Mixed and Componentwise Condition Numbers For Tmentioning

confidence: 99%

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Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Chu

Lin

Tan

et al. 2010

Numer. Linear Algebra Appl.

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SUMMARYOne of the most successful methods for solving the least-squares problem min x Ax −b 2 with a highly ill-conditioned or rank deficient coefficient matrix A is the method of Tikhonov regularization. In this paper, we derive the normwise, mixed and componentwise condition numbers and componentwise perturbation bounds for the Tikhonov regularization. Our results are sharper than the known results. Some numerical examples are given to illustrate our results.

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Section: Normwise Mixed and Componentwise Condition Numbers For Tmentioning

confidence: 99%

“…For the Tikhonov regularization, Malyshev 14 studied the condition numbers and proved that where r = b − Ax λ . However, relative normwise condition numbers in (6) and (7) do not take into account the structures of A and b with respect to the scaling and/or sparsity.…”

Section: Introductionmentioning

confidence: 99%

Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Chu

Lin

Tan

et al. 2010

Numer. Linear Algebra Appl.

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“…[19,22]) and has been further studied in [6,9,15]. For the mixed and componentwise settings for linear least squares problems, the existing results consisted of bounds for both condition numbers(or first order perturbations bounds) and unrestricted perturbation bounds.…”

Section: A Brief Description Of Some Previous Workmentioning

confidence: 99%

Mixed and component wise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems

Zhao¹,

Sun

2009

Filomat

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Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure both input perturbations and output errors with norms. To take into account the relative scaling of data components or a possible sparseness, componentwise condition numbers have been increasingly considered. In this paper, we give explicit expressions for the mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix A, as well as for the solution and residue of a weighted linear least squares problem W

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“…For the related important references, TSVD is discussed by Hansen and co-workers [11][12][13], Chan and Hansen [14], and used by Chen et al [15,16]. TR was first proposed by Tikhonov [17] in 1963, introduced in Tikhonov and Arsenin [18] and analyzed in [19][20][21][22][23][24][25][26][27]. Recently, TSVD and TR are studied in [14,12,28], and Fierro et al [29] for the regularization by truncated total LS.…”

Section: Introductionmentioning

confidence: 99%

Ill‐conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations

Huang

Wei

2011

Numerical Linear Algebra App

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This paper explores some intrinsic characteristics of accuracy and stability for the truncated singular value decomposition (TSVD) and the Tikhonov regularization (TR), which can be applied to numerical solutions of partial differential equations (numerical PDE). The ill-conditioning is a severe issue for numerical methods, in particular when the minimal singular value min of the stiffness matrix is close to zero, and when the singular vector u min of min is highly oscillating. TSVD and TR can be used as numerical techniques for seeking stable solutions of linear algebraic equations. In this paper, new bounds are derived for the condition number and the effective condition number which can be used to improve ill-conditioning by TSVD and TR. A brief error analysis of TSVD and TR is also made, since both errors and condition number are essential for the numerical solution of PDE. Numerical experiments are reported for the discrete Laplace operator by the method of fundamental solutions.

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A Unified Theoryof Conditioning for Linear Least Squares and Tikhonov Regularization Solutions

Cited by 25 publications

References 11 publications

Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill-posed problems

Mixed and component wise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems

Ill‐conditioning of the truncated singular value decomposition, Tikhonov regularization and their applications to numerical partial differential equations

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