On Smoothness Concepts in Regularization for Nonlinear Inverse Problems in Banach Spaces

Hofmann, Bernd

doi:10.1002/9781118853887.ch8

Cited by 3 publications

(6 citation statements)

References 96 publications

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“…holds, one obtains (20). Since Φ is an index function, the same is valid also for Φ −1 , as well as for the compositions Φ −1 (δ 2 ) and ψ(Φ −1 (δ 2 )) = δ Φ −1 (δ 2 ) .…”

mentioning

confidence: 72%

“…Section 2). Then the convergence rate (22) can be derived from (20) for the choice (21). Due to (19), we have the inequality (18) with β = 0 and ψ(t) = t in the linear operator case) yields the convergence rate O(δ 1/3 ).…”

mentioning

confidence: 99%

“…Nevertheless, for completeness of exposition and since some of the estimates will be needed later on, in Section 2 we provide a short summary of the arguments leading to convergence of x δ α from (4) to x † (see also [1,6]). As in the case of Tikhonov's regularization method (cf., e.g., [35,Section 3.2] and [20]), also for Lavrentiev's regularization method a certain additional smoothness of x † with respect to the forward operator is required in order to derive convergence rates. At the origin of such studies, source conditions (range conditions) of Höldertype (cf., e.g., [25,28,37,38]) were under consideration, however attaining here the specific form…”

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confidence: 99%

“…Section 2). Then the convergence rate ( 22) can be derived from (20) for the choice (21). Due to (19), we have the inequality…”

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confidence: 99%

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Lavrentiev's regularization method in Hilbert spaces revisited

Hofmann¹,

Kaltenbacher²,

Resmerita³

2016

IPI

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In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

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“…holds, one obtains (20). Since Φ is an index function, the same is valid also for Φ −1 , as well as for the compositions Φ −1 (δ 2 ) and ψ(Φ −1 (δ 2 )) = δ Φ −1 (δ 2 ) .…”

mentioning

confidence: 72%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Section 2). Then the convergence rate ( 22) can be derived from (20) for the choice (21). Due to (19), we have the inequality…”

mentioning

confidence: 99%

See 2 more Smart Citations

Lavrentiev's regularization method in Hilbert spaces revisited

Hofmann¹,

Kaltenbacher²,

Resmerita³

2016

IPI

Self Cite

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“…with some initial guess f ∈ X. For obtaining convergence rates of the regularized solutions, an appropriate interplay of solution smoothness and structural conditions expressing the nonlinearity of F in a neighborhood of the solution is required (see, e.g., [17] for an overview). For the Tikhonov regularization of the form (25) and F from (6), the condition (23) acts as nonlinearity condition sufficiently well and allows proving the convergence rate…”

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confidence: 99%

Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization

et al. 2016

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Abstract. The SD-SPIDER method for the characterization of ultrashort laser pulses requires the solution of a nonlinear integral equation of autoconvolution type with a device-based kernel function. Taking into account the analytical background of a variational regularization approach for solving the corresponding ill-posed operator equation formulated in complex-valued L 2 -spaces over finite real intervals, we suggest and evaluate numerical procedures using NURBS and the TIGRA method for calculating the regularized solutions in a stable manner. In this context, besides the complex deautoconvolution problem with noisy but full data, a phase retrieval problem is introduced which adapts to the experimental state of the art in laser optics. For the treatment of this problem facet, which is formulated as a tensor product operator equation, we derive wellposedness of variational regularization methods. Case studies with synthetic and real optical data show the capability of the implemented approach as well as its limitation due to measurement deficits. MSC2010 subject classification: 47A52, 47J06, 78A60, 65R32, 45Q05, 65J15

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Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space

Boţ¹,

Hofmann²

2016

Inverse Problems

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In the literature on singular perturbation (Lavrentiev regularization) for the stable approximate solution of operator equations with monotone operators in the Hilbert space the phenomena of conditional stability and local well-posedness and ill-posedness are rarely investigated. Our goal is to present some studies which try to bridge this gap. So we discuss the impact of conditional stability on error estimates and convergence rates for the Lavrentiev regularization and distinguish for linear problems well-posedness and ill-posedness in a specific manner motivated by a saturation result. The role of the regularization error in the noise-free case, called bias, is a crucial point in the paper for nonlinear and linear problems. In particular, for linear operator equations general convergence rates, including logarithmic rates, are derived by means of the method of approximate source conditions. This allows us to extend well-known convergence rates results for the Lavrentiev regularization that were based on general source conditions to the case of non-selfadjoint linear monotone forward operators for which general source conditions fail. Examples presenting the self-adjoint multiplication operator as well as the non-selfadjoint fractional integral operator and Cesàro operator illustrate the theoretical results. Extensions to the nonlinear case under specific conditions on the nonlinearity structure complete the paper. MSC2010 subject classification: 47A52, 65F22, 47H05, 65J22, 65J15

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On Smoothness Concepts in Regularization for Nonlinear Inverse Problems in Banach Spaces

Cited by 3 publications

References 96 publications

Lavrentiev's regularization method in Hilbert spaces revisited

Lavrentiev's regularization method in Hilbert spaces revisited

Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization

Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space

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