Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion

Бакушинский, А. Б.

doi:10.1016/0041-5553(84)90253-2

Cited by 147 publications

(184 citation statements)

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“…parameter λ>0 depends on the amount of noise, as it should [3], and, as expected, it increases as the amount of noise increases. Also, the larger the portion of the data provided the less ill-posed the problem, hence a smaller regularization parameter is required.…”

Section: Example 2 (N=3-dimensions)mentioning

confidence: 99%

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

Marin

Karageorghis

Lesnic

et al. 2016

Inverse Problems in Science and Engineering

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Abstract. An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.

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Section: Example 2 (N=3-dimensions)mentioning

confidence: 99%

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

Marin

Karageorghis

Lesnic

et al. 2016

Inverse Problems in Science and Engineering

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“…The LSQR method is an implementation of the conjugate gradient method applied to the normal equations associated with (1). Let the initial iterate be x 0 = 0.…”

Section: Parameter Choice Rules For Large Problemsmentioning

confidence: 99%

“…We are interested in the situation when no such bound is known and, therefore, the discrepancy principle cannot be applied. It has been shown by Bakushinski [1] that regularization parameter choice rules that do not use a bound for e will fail to determine a suitable value of k for some problems. Such rules are therefore sometimes referred to as "heuristic."…”

Section: Introductionmentioning

confidence: 99%

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Old and new parameter choice rules for discrete ill-posed problems

2012

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“…We are concerned with the approximate solution of large-scale minimization problems (1.1) when no accurate estimate of e is available or for whichb ∈ R(A). We note that since our criterion for determining µ does not use e , it may fail for some problems; see [3] for a discussion. Nevertheless, numerous numerical experiments, some of which are reported in Section 5, show the proposed method to perform well for many problems and to be competitive with other schemes that do not use e for determining µ.…”

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confidence: 99%

Error estimates for large-scale ill-posed problems

2008

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Abstract. The computation of an approximate solution of linear discrete ill-posed 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 ill-posed 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 large-scale ill-posed 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.

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Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion

Cited by 147 publications

References 1 publication

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

Old and new parameter choice rules for discrete ill-posed problems

Error estimates for large-scale ill-posed problems

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