Residual periodograms for choosing regularization parameters for ill-posed problems

Rust, Bert W.; O’Leary, Dianne P.

doi:10.1088/0266-5611/24/3/034005

Cited by 68 publications

(63 citation statements)

References 31 publications

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“…In accord with the comparison of statistically-based methods discussed in [23], we conclude that a statistically-based method should be used whenever such information is available. While we have not yet compared with new work described in [20] which is statistically-based but appears to be more computationally demanding, our results suggest that our new Newton χ 2 -curve method is competitive. These positive results are encouraging for the extension of the method, as suggested in [15], for a more general covariance structure of the data error.…”

Section: Discussionmentioning

confidence: 96%

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

Mead

Renaut

2008

Inverse Problems

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Abstract.We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a χ 2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of the system matrix is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for Generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted with standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This χ 2 -curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated. Tikhonov regularization, least squares, regularization parameterAMS classification scheme numbers: 15A09, 15A29, 65F22, 62F15, 62G08 Submitted to: Inverse Problems, 24 October 2008This is an author-produced, peer-reviewed version of this article. The final, definitive version of this document can be found online at Inverse Problems, published by Institute of Physics. Copyright restrictions may apply.

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Section: Discussionmentioning

confidence: 96%

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

Mead

Renaut

2008

Inverse Problems

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“…al. [74] developed a parameter choice method, called the normalized cumulative periodogram (NCP) method, and Rust and O'Leary [125] proposed a similar method. The basis of these methods is to make the residual as close as possible to white noise.…”

Section: Normalized Cumulative Periodogram Methodsmentioning

confidence: 99%

“…As shown in [74], the underlying assumption that the residual resembles white noise holds only approximately for a range of values of the Tikhonov regularization parameter α that are not too small, so it is likely that there are limitations for the method. However, the NCP method and its variant in [125] have been shown to perform well on several test problems [74,125]. It is clear from the basis of the NCP method, that the noise should be white.…”

Section: Normalized Cumulative Periodogram Methodsmentioning

confidence: 99%

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Comparingparameter choice methods for regularization of ill-posed problems

Bauer

Lukas

2011

Mathematics and Computers in Simulation

248

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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“…Most of the traditional approaches exposed in the literature for choosing the regularization parameter in discrete ill-posed problems are based on the norm of the residual error in one way or another, e.g., discrepancy principle, cross-validation, and the L-curve. Nevertheless, recent publications [32,33] suggest the use of a new parameter-choice method based on the residual spectrum. This technique is based on the evaluation of the shape of the Fourier transform of the residual error.…”

Section: Stopping Criterion: the Cumulative Spectrummentioning

confidence: 99%

Sparse covariance fitting for direction of arrival estimation

Blanco

Nájar

2012

EURASIP J. Adv. Signal Process.

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This article proposes a new algorithm for finding the angles of arrival of multiple uncorrelated sources impinging on a uniform linear array of sensors. The method is based on sparse signal representation and does not require either the knowledge of the number of the sources or a previous initialization. The proposed technique considers a covariance matrix model based on overcomplete basis representation and tries to fit the unknown signal powers to the sample covariance matrix. Sparsity is enforced by means of a l 1 -norm penalty. The final problem is reduced to an objective function with a non-negative constraint that can be solved efficiently using the LARS/homotopy algorithm. The method described herein is able to provide high resolution with a low computational burden. It proceeds in an iterative fashion solving at each iteration a small linear system of equations until a stopping condition is fulfilled. The proposed stopping criterion is based on the residual spectrum and arises in a natural way when the LARS/homotopy is applied to the considered objective function.

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Residual periodograms for choosing regularization parameters for ill-posed problems

Cited by 68 publications

References 31 publications

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

Comparingparameter choice methods for regularization of ill-posed problems

Sparse covariance fitting for direction of arrival estimation

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