Optimal a Posteriori Parameter Choice for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems

Scherzer, Otmar; Engl, Heinz W.; Kunisch, Karl

doi:10.1137/0730091

Cited by 127 publications

(158 citation statements)

References 16 publications

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“…Therefore it is helpful to recall the existing parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems. As we know, by generalizing the idea developed in [8], Scherzer, Engl and Kunisch [19] proposed a rule to choose the regularization parameter for Tikhonov regularization of nonlinear ill-posed problems in 1993, and used the root α := α(δ) of the equation…”

Section: The Stopping Rule and Main Resultsmentioning

confidence: 99%

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On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

Jin

2000

Math. Comp.

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Section: The Stopping Rule and Main Resultsmentioning

confidence: 99%

“…Tikhonov regularization is one of the best-known methods for solving nonlinear ill-posed problems, and it has received a lot of attention in recent years [20,7,19,13]. In this method, the solution x δ α of the minimization problem min…”

Section: Introductionmentioning

confidence: 99%

On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

Jin

2000

Math. Comp.

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“…If the benchmark source condition (1.7) fails, but the Fréchet derivative F (x) exists for all x ∈ B r (x † ) ⊂ D(F ) and some r > 0, by extending the ideas of [16,33,38] two further alternatives for obtaining convergence rates to (1.5) have been presented in the paper [26] with focus on low order Hölder source conditions (see also [23,38])…”

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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“…In order to obtain stable approximate solutions, regularization methods are often used. Tikhonov regularization is one of the classical regularization methods used in the literature (see [4,6,7,11,12,15]). Since F is monotone, one can also use the Lavrentiev regularization method (see [16,17]).…”

Section: Introductionmentioning

confidence: 99%

Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Problems

Mahale¹,

Nair²

2009

ANZIAM J.

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We consider an iterated form of Lavrentiev regularization, using a null sequence (α k ) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x) = y, where F : D(F) ⊆ X → X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova ["Iterative regularization and generalized discrepancy principle for monotone operator equations", Numer. Funct. Anal. Optim. 28 (2007) 13-25] considered an a posteriori strategy to find a stopping index k δ corresponding to inexact data y δ with y − y δ ≤ δ resulting in the convergence of the method as δ → 0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (α k ) is weaker than that considered by Bakushinsky and Smirnova.2000 Mathematics subject classification: primary 47A52; secondary 65F22, 65J15, 65J22, 65M30.

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Optimal a Posteriori Parameter Choice for Tikhonov Regularization for Solving Nonlinear Ill-Posed Problems

Cited by 127 publications

References 16 publications

On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems

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

Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Problems

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