On weakly bounded noise in ill-posed problems

Eggermont, P. P. B.; LaRiccia, V. N.; Nashed, M. Zuhair

doi:10.1088/0266-5611/25/11/115018

Cited by 22 publications

(17 citation statements)

References 45 publications

(101 reference statements)

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“…This paper aims to bridge the gap between them. Our results are closely related to earlier studies of Eggermont, LaRiccia, and Nashed [8,9,10], who studied weakly bounded noise. They assume that the noise is a L 2 -function and discuss regularization techniques when the noise tends to zero in the weak topology of L 2 .…”

Section: Above We Havesupporting

confidence: 89%

Analysis of regularized inversion of data corrupted by white Gaussian noise

2014

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Abstract. Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) iswhere δ > 0 is the noise magnitude. If ε was an L 2 -function, Tikhonov regularization gives an estimatefor u where α = α(δ) is the regularization parameter. Here penalization of the Sobolev norm u H r covers the cases of standard Tikhonov regularization (r = 0) and first derivative penalty (r = 1).Realizations of white Gaussian noise are almost never in L 2 , but do belong to H s with probability one if s < 0 is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.

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Section: Above We Havesupporting

confidence: 89%

Analysis of regularized inversion of data corrupted by white Gaussian noise

2014

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“…Surprisingly, this notion of stability with respect to weak perturbations has rarely been addressed. In fact, we are aware of only [19] and [21] 2 . A second property pertains to the existence of a class of clean simple signals which remain invariant under energy minimization.…”

Section: Introductionmentioning

confidence: 99%

A few remarks on variational models for denoising

Choksi

Fonseca

Zwicknagl

2014

Communications in Mathematical Sciences

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Abstract. Variational models for image and signal denoising are based on the minimization of energy functionals consisting of a fidelity term together with higher-order regularization. In addition to the choices of function spaces to measure fidelity and impose regularization, different scaling exponents appear. In this note we present a few simple, yet novel, remarks on (i) the stability with respect to deterministic noise perturbations, captured via oscillatory sequences converging weakly to zero, and (ii) exact reconstruction.

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“…The aforementioned references, besides [4], are restricted to the particular case in which p = 1 2 . Since T is compact and dim R(T ) = ∞, it follows that R(T ) is non-closed, which implies that the generalised inverse (see, e.g., [16]) T † is an unbounded operator.…”

Section: Introductionmentioning

confidence: 99%

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

Kindermann

Raik

2019

Numerical Functional Analysis and Optimization

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We study the choice of the regularisation parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyse several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyse and provide conditions for the convergence of the parameter choice by the generalised cross-validation and predictive mean-square error rules.

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On weakly bounded noise in ill-posed problems

Cited by 22 publications

References 45 publications

Analysis of regularized inversion of data corrupted by white Gaussian noise

Analysis of regularized inversion of data corrupted by white Gaussian noise

A few remarks on variational models for denoising

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

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