Data Errors and an Error Estimation for Ill-Posed Problems

Yagola, A. G.; Леонов, А. С.; Titarenko, Valeriy

doi:10.1080/10682760290031195

Cited by 42 publications

(27 citation statements)

References 9 publications

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“…We therefore skip detailed discussions about techniques in selecting regularization parameter, instead citing relevant references ( [26][27][28]) to keep the present text concise. Yagola et al [29] showed the inapplicability of error-free regularization techniques (such as L-curve, Generalized Cross Validation (GCV)) in a class of ill-posed problems. Recently, Sen and Roy [30] working on full waveform seismic inversion showed that among the methods, such as Engl's modified discrepancy principle (MDP), L-curve and GCV in computing a posteriori regularization parameter, Engl's MDP is computationally most efficient.…”

Section: Determination Of Regularization Parametermentioning

confidence: 99%

Iteratively adaptive regularization in inverse modeling with Bayesian outlook – application on geophysical data

Roy¹

2005

Inverse Problems in Science and Engineering

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An algorithm of model-based nonlinear inversion scheme with applications on geophysical examples is proposed. In the framework of classical least squares optimization, the algorithm uses adaptively controlled smoothness constraint in an iterative minimization scheme through an appropriate selection of regularization parameter. A new formula in computing regularization parameter, which is a variant of Engl's formalism of a posteriori regularization is proposed. A regularized Gauss-Newton type optimization scheme along with conjugate gradient algorithm is used for nonlinear error minimization. The applicability of the algorithm is demonstrated through numerical experiments inverting synthetically generated vertical electrical sounding, first arrival travel time refraction seismic data and travel time data of Rayleigh wave generated due to acoustic emission source over homogeneous and isotropic medium.

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Section: Determination Of Regularization Parametermentioning

confidence: 99%

Iteratively adaptive regularization in inverse modeling with Bayesian outlook – application on geophysical data

Roy¹

2005

Inverse Problems in Science and Engineering

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“…Response space is defined as the space containing response quantities of interest and its relationship with the parameter space is realized by the map Z(p) given in Equation (2). A geometric analysis of the domain R N Z and related quantities provides qualitative insight into the updating problem.…”

Section: Response Space: Z-spacementioning

confidence: 99%

“…Equation (4) represents the idealized case of model updating, which is, in general, an ill-posed problem [2]. In reality, the response is measured with only finite accuracy.…”

Section: Model Updating As An Inverse Problemmentioning

confidence: 99%

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Regularization in model updating

Titurus

Friswell

2007

Numerical Meth Engineering

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SUMMARYThis paper presents the theory of sensitivity-based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters. Model updating, together with the additional regularization criterion, is an optimization with two objective functions, and must be linearized to obtain the solution. Structured solutions are obtained, based on the generalized singular value decomposition (GSVD), and specific features of the parameter and response paths as the regularization parameter varies are explored. The four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families. These concepts are demonstrated on a simulated discrete example and on an experimental case study.

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“…For several (but not all) problems it has been observed to give a reasonably good and robust parameter choice, and it can cope with correlated errors [1,3,29,67,71,73,75]. However, it is known theoretically that the L-curve method (from [75]) has serious limitations [41,146]. First, the L-curve corner may not even exist [122].…”

Section: L-curve Methodsmentioning

confidence: 99%

Comparingparameter choice methods for regularization of ill-posed problems

Bauer

Lukas

2011

Mathematics and Computers in Simulation

252

111

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In the literature on regularization, many different parameter choice methods have been proposed in both deterministic and stochastic settings. However, based on the available information, it is not always easy to know how well a particular method will perform in a given situation and how it compares to other methods. This paper reviews most of the existing parameter choice methods, and evaluates and compares them in a large simulation study for spectral cut-off and Tikhonov regularization. The test cases cover a wide range of linear inverse problems with both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.

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Data Errors and an Error Estimation for Ill-Posed Problems

Cited by 42 publications

References 9 publications

Iteratively adaptive regularization in inverse modeling with Bayesian outlook – application on geophysical data

Iteratively adaptive regularization in inverse modeling with Bayesian outlook – application on geophysical data

Regularization in model updating

Comparingparameter choice methods for regularization of ill-posed problems

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