Convergence analysis of (statistical) inverse problems under conditional stability estimates

Werner, Frank; Hofmann, Bernd

doi:10.1088/1361-6420/ab4cd7

Cited by 19 publications

(9 citation statements)

References 37 publications

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“…In the following assumption, we briefly summarize the structural properties of the operator F and of its domain D(F ), in particular with respect to the the solution u † of equation (1.3). For examples of nonlinear inverse problems, which satisfy these assumptions (or at least substantial parts of it), we refer to [6,9] and to the appendices of the papers [11,28]. (f) Let a > 0, and let there exist finite constants 0 < c a ≤ C a such that the inequality chain…”

Section: Prerequisites and Assumptionsmentioning

confidence: 99%

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

Hofmann¹,

Plato²

2020

etna

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For the Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Hölder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence of regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies, without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

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Section: Prerequisites and Assumptionsmentioning

confidence: 99%

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

Hofmann¹,

Plato²

2020

etna

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“…The latter leads to the concept of conditional stability (cf. [57,58]) and corresponding stability estimates.…”

Section: Regularization Theorymentioning

confidence: 99%

Variational Regularization in Inverse Problems and Machine Learning

Burger¹

2021

Preprint

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This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances for convex functionals.In addition to the approach developed for inverse problems we will also discuss variational regularization in machine learning and work out some connections to the classical regularization theory. In particular we will discuss a reinterpretation of machine learning problems in the framework of regularization theory and a reinterpretation of variational methods for inverse problems in the framework of risk minimization. Moreover, we establish some previously unknown connections between error estimates in Bregman distances and generalization errors.

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“…In other work, the authors [4] consider a 2-step approach, however, again under the assumption of the norm in L 2 (X, ν; Y ) being known. The references [3] and [17,34] consider respectively a Gauss-Newton algorithm and the Tikhonov regularization for certain nonlinear inverse problem, but also in the idealized setting of Hilbertian white or colored noise with known covariance, which can only cover sampling effects when L 2 (X, ν; Y ) is known. Loubes et al [21] discussed the problem (1.1) under a fixed design and concentrate on the problem of model selection.…”

Section: Abhishake Rastogimentioning

confidence: 99%

“…This link condition is known as finitely smoothing. This condition is satisfied in various types of problems (for examples see [9, Example 10.2], [34,Example 4,5]).…”

Section: 2mentioning

confidence: 99%

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Rastogi¹

2020

Communications on Pure &Amp; Applied Analysis

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In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

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Convergence analysis of (statistical) inverse problems under conditional stability estimates

Cited by 19 publications

References 37 publications

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

Variational Regularization in Inverse Problems and Machine Learning

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

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