Comparingparameter choice methods for regularization of ill-posed problems

Bauer, Frank; Lukas, Mark A.

doi:10.1016/j.matcom.2011.01.016

Cited by 248 publications

(119 citation statements)

References 133 publications

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“…Note that not all results can be explained since some methods are still of rather heuristic nature and convergence analysis does not exist even for simpler regularizations such as ordinary Tikhonov regularization (see [18,19] for an overview). Often these methods are formulated and investigated without considering discretization and data distribution issues.…”

Section: Discussionmentioning

confidence: 99%

“…Table 1 shows the different parameter choice methods we tested and introduces their abbreviations. The tuning parameters are chosen in accordance to [18,19]. For the choice of the maximal indexK, see Section 4.5.…”

Section: Name Selection Criterion Specificationsmentioning

confidence: 99%

“…The test scenarios are chosen such that the parameter range lies between 1 and 10 −14 and includes the optimal parameter away from the boundaries λ 1 and λ 100 . For the choice of the parameters λ k , we refer to [18,19].…”

Section: Regularization Parametersmentioning

confidence: 99%

“…For some methods like the quasi-optimality criterion, the values of k have to be constrained by a suitable maximal indexK which must be chosen such that k opt <K. To increase computational efficiency, such a maximal index can be used for other methods as well without changing their performance. As in [18,19], we define this maximal index bŷ…”

Section: Maximal Indexmentioning

confidence: 99%

“…Note that, for stochastic noise, data-driven methods yield converging regularization methods with good performance in practice (cf. [18,19]). The outline of this paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

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Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Gutting

Kretz

Michel

et al. 2017

Front. Appl. Math. Stat.

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Recently, the regularized functional matching pursuit (RFMP) was introduced as a greedy algorithm for linear ill-posed inverse problems. This algorithm incorporates the Tikhonov-Phillips regularization which implies the necessity of a parameter choice. In this paper, some known parameter choice methods are evaluated with respect to their performance in the RFMP and its enhancement, the regularized orthogonal functional matching pursuit (ROFMP). As an example of a linear inverse problem, the downward continuation of gravitational field data from the satellite orbit to the Earth's surface is chosen, because it is exponentially ill-posed. For the test scenarios, different satellite heights with several noise-to-signal ratios and kinds of noise are combined. The performances of the parameter choice strategies in these scenarios are analyzed. For example, it is shown that a strongly scattered set of data points is an essentially harder challenge for the regularization than a regular grid. The obtained results yield, as a first orientation, that the generalized cross validation, the L-curve method and the residual method could be most appropriate for the RFMP and the ROFMP.

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Section: Discussionmentioning

confidence: 99%

Section: Name Selection Criterion Specificationsmentioning

confidence: 99%

Section: Regularization Parametersmentioning

confidence: 99%

Section: Maximal Indexmentioning

confidence: 99%

“…Note that, for stochastic noise, data-driven methods yield converging regularization methods with good performance in practice (cf. [18,19]). The outline of this paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

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Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Gutting

Kretz

Michel

et al. 2017

Front. Appl. Math. Stat.

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Using the L₁norm to select basis set vectors for multivariate calibration and calibration updating

Shahbazikhah

Kalivas

Andries

et al. 2016

Journal of Chemometrics

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dWith projection based calibration approaches, such as partial least squares (PLS) and principal component regression (PCR), the calibration space is spanned by respective basis vectors (latent vectors). Up to rank k basis vectors are formed where k ≤ min(m,n) with m and n denoting the number of calibration samples and measured variables. The user needs to decide how many and which respective basis vectors (tuning parameters). To avoid the second issue, basis vectors are selected top-down starting with the first and sequentially adding until model criteria are satisfied. Ridge regression (RR) avoids the issues by using the full set of basis vectors. Another approach is to select a subset from the total available. The presented work develops a process based on the L 1 vector norm to select basis vectors. Specifically, the L 1 norm is used to select singular value decomposition (SVD) basis set vectors for PCR (LPCR). Because PCR, PLS, RR, and others can be expressed as linear combination of the SVD basis vectors, the focus is on selection and comparison using the SVD basis set. Results based on respective tuning parameter selections and weights applied to the SVD basis vectors for LPCR, top-down PCR, correlation PCR (CPCR), PLS, and RR are compared for calibration and calibration updating using spectroscopic data sets. The methods are found to predict equivalently. In particular, the L 1 norm produces similar results to those obtained by the well-studied CPCR process. Thus, the new method provides a different theoretical framework than CPCR for selecting basis vectors.

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Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

Gazzola

Landman

2020

GAMM-Mitteilungen

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Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms. K E Y W O R D S hybrid methods, imaging problems, Krylov subspace methods, large-scale linear inverse problems, regularization parameter choice rules, Tikhonov regularization 1 INTRODUCTION This paper considers linear, large-scale, discrete ill-posed problems of the form Ax true + e = b true + e = b , (1) where the matrix A ∈ R m×n represents a forward mapping that is ill-conditioned with ill-determined rank (ie, the singular values of A decay and cluster at zero without an evident gap between two consecutive ones), x true ∈ R n is the desired solution, and e ∈ R m is some unknown noise that affects the data b ∈ R m. Systems like (1) typically stem from the discretization of first-kind Fredholm integral equations, and model inverse problems arising in a variety of applications, such This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Comparingparameter choice methods for regularization of ill-posed problems

Cited by 248 publications

References 133 publications

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Using the L₁norm to select basis set vectors for multivariate calibration and calibration updating

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

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Comparingparameter choice methods for regularization of ill-posed problems

Cited by 248 publications

References 133 publications

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Study on Parameter Choice Methods for the RFMP with Respect to Downward Continuation

Using the L1norm to select basis set vectors for multivariate calibration and calibration updating

Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches

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Using the L₁norm to select basis set vectors for multivariate calibration and calibration updating