Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

Hohage, Thorsten; Miller, Philip

doi:10.1088/1361-6420/ab0b15

Cited by 17 publications

(54 citation statements)

References 44 publications

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“…For p > 1 only group sparsity in the levels is enforced, but not sparsity of the wavelet coefficients within each level. As a main result of this paper we demonstrate that the analysis in [24] as well as other works to be discussed below do not capture the full potential of estimators (1), i.e. the most commonly used case p = 1: Even though the error bounds in [24] are optimal in a minimax sense, more precisely in a worst case scenario in B s p,∞ -balls, we will derive faster rates of convergence for an important class of functions, which includes piecewise smooth functions.…”

Section: Introductionsupporting

confidence: 56%

“…As a main result of this paper we demonstrate that the analysis in [24] as well as other works to be discussed below do not capture the full potential of estimators (1), i.e. the most commonly used case p = 1: Even though the error bounds in [24] are optimal in a minimax sense, more precisely in a worst case scenario in B s p,∞ -balls, we will derive faster rates of convergence for an important class of functions, which includes piecewise smooth functions. The crucial point is that such functions also belong to Besov spaces with larger smoothness index s, but smaller integrability index p < 1.…”

Section: Introductionsupporting

confidence: 56%

“…In this setting Besov B r 1,1 -norms can be written in the form of the penalty term in (1). In a previous paper [24] we have already addressed sparsity promoting penalties in the form of Besov B 0 p,1 -norms with p ∈ [1,2]. For p > 1 only group sparsity in the levels is enforced, but not sparsity of the wavelet coefficients within each level.…”

Section: Introductionmentioning

confidence: 99%

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Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Miller

Hohage

2021

Numer. Math.

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We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $$L^2$$ L 2 -space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $$\ell ^2$$ ℓ 2 -norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces $$B^r_{1,1}$$ B 1 , 1 r . In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

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Section: Introductionsupporting

confidence: 56%

Section: Introductionsupporting

confidence: 56%

Section: Introductionmentioning

confidence: 99%

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Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Miller

Hohage

2021

Numer. Math.

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“…We believe that our upper bounds are too pessimistic, this can be seen e.g. by comparing with the case q = 1 treated in [HM18]. But note that in this case the VSC is not formulated with respect to the Bregman distance.…”

Section: Stability Estimates For Electrical Impedance Tomographymentioning

confidence: 89%

“…An analysis where the sparsity assumption is violated was developed in a series of papers starting with [BFH13] and leading to [FG18]. For an analysis in the spirit of the setup presented here that extends to the nonlinear case see [HM18]. Furthermore, the results have been extended to elastic net regularization in [CHZ17].…”

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confidence: 99%

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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Density matrix reconstructions in ultrafast transmission electron microscopy: uniqueness, stability, and convergence rates

2020

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In the recent paper [17] the first experimental determination of the density matrix of a free electron beam has been reported. The employed method leads to a linear inverse problem with a positive semidefinite operator as unknown. The purpose of this paper is to complement the experimental and algorithmic results in the work mentioned above by a mathematical analysis of the inverse problem concerning uniqueness, stability, and rates of convergence under different types of a-priori information.

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Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

Cited by 17 publications

References 44 publications

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

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

Density matrix reconstructions in ultrafast transmission electron microscopy: uniqueness, stability, and convergence rates

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