Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

Anzengruber, Stephan W.; Hofmann, Bernd; Mathé, Peter

doi:10.1080/00036811.2013.833326

Cited by 50 publications

(124 citation statements)

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“…For the well-definedness of γ SDP from SDP, its properties and convergence of regularized solutions x δ γ SDP as δ → 0, we refer to [1]. In principle, one can say that γ SDP is uniquely determined for all 0 < q < 1 and y δ ∈ Y whenever γ 0 > 0 is large enough.…”

Section: Convergence Rates Under Variational Inequalitiesmentioning

confidence: 99%

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Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

2016

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We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between ℓ 2 and a Hilbert space Y , possessing the unique solution. For the cases that sparsity x † ∈ ℓ 0 is expected but often slightly violated in practice, we investigate in comparison with the ℓ 1 -regularization the elastic-net regularization, where the penalty is a weighted superposition of the ℓ 1 -norm and the ℓ 2 -norm square, under the assumption that x † ∈ ℓ 1 . There occur two positive parameters in this approach, the weight parameter η and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in ℓ 1 -regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x † k → 0 for k → ∞ and the classical smoothness properties of x † as an element in ℓ 2 .

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Section: Convergence Rates Under Variational Inequalitiesmentioning

confidence: 99%

“…We refer to [1,8,29] for further discussions. On the other hand, the application of Proposition 3.2 to the elastic-net regularization (1.5), where β is chosen by TDP and SDP, or LEP, requires us to construct an appropriate variational inequality.…”

Section: Application To Elastic-net Regularizationmentioning

confidence: 99%

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

2016

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“…For more details concerning the consequences of variational inequalities and the role of the choice of the regularization parameter for obtaining convergence rates in regularization we refer, for example, to [19] and [1,4,5,12,15,21]. Making use of Gelfand triples it was shown in [2] that, for a wide range of applied inverse problems, the forward operators A are such that link conditions of the form (3.1) apply for all e (k) , k ∈ N. On the other hand, the paper [13] gives counterexamples where (3.1) fails for specific operators A, but alternative link conditions presented there can compensate this deficit.…”

Section: Improved Convergence Ratesmentioning

confidence: 99%

“…Moreover, with focus on sparsity, the use of 1 -regularization can be motivated for specific classes of well-posed problems, too (cf., e.g., [10]). Based on the powerful tool of variational inequalities (also called variational source conditions), in [9] convergence rates results on 1 -regularization for linear ill-posed operator equations have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in 1 . In the present paper, we improve those results and illustrate the improvement level with respect to the associated convergence rates for the Cesáro operator equation in 2 and for specific denoising problems.…”

Section: Introductionmentioning

confidence: 99%

On ℓ¹-Regularization in Light of Nashed's Ill-Posedness Concept

Flemming

Hofmann

Veselić

2015

Computational Methods in Applied Mathematics

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Abstract. Based on the powerful tool of variational inequalities, in recent papers convergence rates results on 1 -regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in 1 . In the present paper we improve those convergence rates results and apply them to the Cesáro operator equation in 2 and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

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“…In this situation, we obtain for a choice α = α(δ, y δ ) of the regularization parameter by the sequential discrepancy principle (cf. [1,21]) convergence rates…”

Section: Introductionmentioning

confidence: 99%

About a deficit in low-order convergence rates on the example of autoconvolution

Burger

Hofmann

2014

Applicable Analysis

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Abstract. We revisit in L 2 -spaces the autoconvolution equation x * x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0, 1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F (x) = y with solutions x † in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x † ) * fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0, 2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L 2 -functions with a fixed and uniformly bounded modulus function.

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Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

Cited by 50 publications

References 20 publications

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

On ℓ¹-Regularization in Light of Nashed's Ill-Posedness Concept

About a deficit in low-order convergence rates on the example of autoconvolution

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Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

Cited by 50 publications

References 20 publications

Elastic-net regularization versusℓ1-regularization for linear inverse problems with quasi-sparse solutions

Elastic-net regularization versusℓ1-regularization for linear inverse problems with quasi-sparse solutions

On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept

About a deficit in low-order convergence rates on the example of autoconvolution

Contact Info

Product

Resources

About

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

Elastic-net regularization versusℓ¹-regularization for linear inverse problems with quasi-sparse solutions

On ℓ¹-Regularization in Light of Nashed's Ill-Posedness Concept