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

Sprung, Benjamin; Hohage, Thorsten

doi:10.1007/s00211-018-0987-x

Cited by 5 publications

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“…Recently, the authors have developed a method for the verification of variational source conditions in Hilbert spaces under standard smoothness conditions. This method has been successfully applied to a number of interesting inverse problems, both linear and nonlinear [25,26,35,40].…”

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Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Weidling¹,

Sprung²,

Hohage³

2020

SIAM J. Numer. Anal.

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This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.

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confidence: 99%

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Weidling¹,

Sprung²,

Hohage³

2020

SIAM J. Numer. Anal.

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“…The question arises whether faster rates up to O(δ 4/3 ) as seen in Theorem 2.5 are achievable with VSCs. For linear operators this question has been answered in [Gra13], see also [SH18] for even faster rates by using Bregman iteration as a regularization method. The idea is roughly the following: If we want to achieve faster rates than O(δ) then at least the condition that guarantees the rate of O(δ) should hold true and hence by Proposition 2.33 we should have f * = T * p for f * ∈ ∂R( f † ).…”

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Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs

Weidling¹

2019

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That this general procedure yields indeed a stable reconstruction and that f α converges to the true solution f † when g obs converges to the true data for some choice of α has been established e.g. in [Pös08,Fle11] (see Section 1.2.2 for more details). For the choice of α one roughly has to take the following effect into account: the larger we choose α the more stable is the problem we are solving but the less we try to match the data exactly. Hence one has to try to find an optimal balance of the two effects in the sense that the f α is as close as possible to the true solution f † . It is well known, however, in the regularization theory that any bound on the distance f α to f † requires a priori knowledge on f † , i.e. we cannot find such a bound which will hold globally for all possible choices of parameters f † . Such an a priori knowledge is typically formulated as a source condition. E.g. in [EHN96] for the case that F is a linear, compact operators acting between Hilbert spaces the spectral source condition f † = (TT) ν w, ν ∈ (0, 1] have been studied, where the right hand side is defined by the functional calculus.under mild assumptions on the involved quantities, where the observed data g obs has to satisfy g obs − g ≤ δ again. Recent results of [Fle18] illustrate that such a VSC is always fulfilled, but they do not tell us how the function ψ will look like. The subject of this thesis is to investigate conditions which allow to quantify ψ in the VSC and hence the convergence rate.If F is an operator mapping between function spaces F −1 is often not continuous due to the fact that F is a smoothing operator, that is F( f ) is a smoother function than f . It has been observed for such problems that the spectral source condition can be interpreted as a smoothness assumption on f relative to the smoothing properties of the operator. The smoothing properties of the operator are related to the degree of ill-posedness of the operator; e.g. in case of linear, compact operators acting between Hilbert spaces this is typically measured by the speed of the decay of the singular values of the operator.In this thesis we will use the two factors, smoothness of the solution and illposedness of the operator equation, in order to derive explicit forms of variationalfor all f 1 , f 2 in a (usually smooth) subset K. Such estimates are quite common for many interesting problems while results that show that for a given problem and conditions on the true solution f † a VSC for a specific function ψ is fulfilled are still rare in the literature (see Remark 2.28). As VSCs imply stability estimates this allows us to compare our new findings with existing results in the literature.This thesis is structured as follows:• In Chapter 1 we will give a more detailed introduction into inverse problems and Tikhonov regularization.• Chapter 2 will review the results on convergence rates known in the literature. We will discuss the advantages and limitations of various source conditions as well as their relation to each other. Further we w...

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Inverse Problems

Hohage¹,

Sprung²,

Weidling³

2020

Topics in Applied Physics

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Generally speaking, inverse problems typically consist in the reconstruction of causes for observed effects. In imaging applications the cause is usually a probe and the effect are observed data. The corresponding forward problems then consists in predicting experimental data given perfect knowledge of the probe. In some sense solving an inverse problems means “computing backwards”, which is usually more difficult then solving the forward problem.

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

Cited by 5 publications

References 36 publications

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs

Inverse Problems

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