Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

Albani, Vinicius; Elbau, Peter; Hoop, Maarten V. de; Scherzer, Otmar

doi:10.1080/01630563.2016.1144070

Cited by 27 publications

(89 citation statements)

References 12 publications

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“…In this case, which has been comprehensively discussed in the literature (cf., e.g., [7,Chap. 5] and [1,2,13]), we can explicitly calculate the terms under consideration.…”

Section: 1mentioning

confidence: 99%

“…Unlesss specified otherwise, the norm · in this study always refers to that in H. We assume that instead of the exact right-hand side y ∈ R(A) only noisy data y δ ∈ H satisfying (2) y δ − y ≤ δ Date: September 13, 2018. 1 with noise level δ ≥ 0 are available. Based on y δ we try to recover x † in a stable approximate manner by using variational regularization with general convex penalty functionals J.…”

Section: Introductionmentioning

confidence: 99%

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Penalty-based smoothness conditions in convex variational regularization

Hofmann

Kindermann

Mathé

2019

Journal of Inverse and Ill-Posed Problems

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The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penaltyminimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothnessdependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

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“…In this case, which has been comprehensively discussed in the literature (cf., e.g., [7,Chap. 5] and [1,2,13]), we can explicitly calculate the terms under consideration.…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Penalty-based smoothness conditions in convex variational regularization

Hofmann

Kindermann

Mathé

2019

Journal of Inverse and Ill-Posed Problems

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“…Suppose Assumption 2.1 holds true. Letf (1) α :=f α be the solution to (P 1 ) := (P 1 ), and set R 1 := R. Then for n = 1, 2, . .…”

Section: Bregman Iterationsmentioning

confidence: 99%

“…Again the limiting case of this dual source condition, which we tag second order source condition, is equivalent to a simpler condition, T ω ∈ ∂S * (p), which was studied earlier in [34,36,37]. Hence, Grasmair's second order condition corresponds to the indices ν ∈ (1,2] in (4). The aim of this paper is to derive rates of convergence corresponding to indices ν > 2, i.e.…”

Section: Introductionmentioning

confidence: 97%

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Higher order convergence rates for Bregman iterated variational regularization of inverse problems

Sprung¹,

Hohage²

2018

Numer. Math.

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We study the convergence of variationally regularized solutions to linear ill-posed operator equations in Banach spaces as the noise in the right hand side tends to 0. The rate of this convergence is determined by abstract smoothness conditions on the solution called source conditions. For non-quadratic data fidelity or penalty terms such source conditions are often formulated in the form of variational inequalities. Such variational source conditions (VSCs) as well as other formulations of such conditions in Banach spaces have the disadvantage of yielding only low-order convergence rates. A first step towards higher order VSCs has been taken by Grasmair (2013) who obtained convergence rates up to the saturation of Tikhonov regularization. For even higher order convergence rates, iterated versions of variational regularization have to be considered. In this paper we introduce VSCs of arbitrarily high order which lead to optimal convergence rates in Hilbert spaces. For Bregman iterated variational regularization in Banach spaces with general data fidelity and penalty terms, we derive convergence rates under third order VSC. These results are further discussed for entropy regularization with elliptic pseudodifferential operators where the VSCs are interpreted in terms of Besov spaces and the optimality of the rates can be demonstrated. Our theoretical results are confirmed in numerical experiments.

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On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

Fabian

Scherzer²

2022

Numer. Math.

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In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nyström method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.

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Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

Cited by 27 publications

References 12 publications

Penalty-based smoothness conditions in convex variational regularization

Penalty-based smoothness conditions in convex variational regularization

Higher order convergence rates for Bregman iterated variational regularization of inverse problems

On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

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