Analysis of Profile Functions for General Linear Regularization Methods

Hofmann, Bernd; Mathé, Peter

doi:10.1137/060654530

Cited by 77 publications

(106 citation statements)

References 17 publications

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“…Conditional stability estimates by using interpolation [24,25]. Such inequalities which extend the classical interpolation inequality became a powerful tool in the analysis of regularization under general smoothness conditions, see, e.g., [7,26,28,40,41,42,45,46,47,56,61]. Variable Hilbert scale interpolation is sometimes also called interpolation with a function parameter, see [6,44].…”

Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning

confidence: 99%

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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Abstract. The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

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Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning

confidence: 99%

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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“…Then the convergence rate is slower than (2.5) and the rate function depends on d A * x † (cf. [22,24], and also [4]). From Proposition 2.3 we have for all elements…”

Section: Problem Setting and Basic Assumptionsmentioning

confidence: 99%

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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“…For ill-posed operator equations Ax = y with bounded linear operators A : X → Y possessing a nonclosed range in Y such smoothness classes M yielding (4.1) are usually dense subsets M = {x ∈ X : x = Gv, v ∈ X} of the Hilbert space X characterized by ranges of some bounded linear operators G : X → X with unbounded Moore-Penrose inverse G † where G and A are connected by some link condition (cf. [20]). For nonlinear problems the smoothness classes have a much more complicated structure.…”

Section: Conclusion and Open Questionsmentioning

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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Analysis of Profile Functions for General Linear Regularization Methods

Cited by 77 publications

References 17 publications

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

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

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

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Analysis of Profile Functions for General Linear Regularization Methods

Cited by 77 publications

References 17 publications

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

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

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

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Product

Resources

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