2020
DOI: 10.3390/math8030331
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The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Abstract: This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the re… Show more

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Cited by 7 publications
(10 citation statements)
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“…The basis for an analytical treatment of the balancing principle is formed by general error estimates for Tikhonov regularization in Hilbert scales with oversmoothing penalty. With the inequality chain (3) and for x † ∈ int(D(F )) such estimates have been developed recently in [9], [13], and [7] by using auxiliary elements as minimizers of (4). Introducing the injective linear operator G := B −(2a+2) , the corresponding minimizers xα can be expressed explicitly as…”
Section: General Error Estimate For Tikhonov Regularization In Hilber...mentioning
confidence: 99%
See 3 more Smart Citations
“…The basis for an analytical treatment of the balancing principle is formed by general error estimates for Tikhonov regularization in Hilbert scales with oversmoothing penalty. With the inequality chain (3) and for x † ∈ int(D(F )) such estimates have been developed recently in [9], [13], and [7] by using auxiliary elements as minimizers of (4). Introducing the injective linear operator G := B −(2a+2) , the corresponding minimizers xα can be expressed explicitly as…”
Section: General Error Estimate For Tikhonov Regularization In Hilber...mentioning
confidence: 99%
“…Proof. A substantial ingredient of the proof is the fact that due to [19] sub-linear index functions are qualifications for the classical Tikhonov regularization approach with norm square penalty, and the verification of formula ( 25) in [7], which in turn is based on the bounds ( 21)-( 23) ibid. As can be seen from there it is enough to show that under the above assumptions, and for 0 ≤ θ ≤ 2a+1 2a+2 , we have that ( 7)…”
Section: Error Decompositionmentioning
confidence: 99%
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“…In the case when L is based on a norm, such as L(x, y) = 1 2 x − y 2 2 , conditions on Φ to guarantee the existence of a minimum to (3.1) are given in [34,Thm. 4.1] and [22,Prop. 2].…”
mentioning
confidence: 99%