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

Tautenhahn, Ulrich; Hämarik, Uno; Hofmann, Bernd; Shao, Y.

doi:10.1080/01630563.2013.819515

Cited by 23 publications

(12 citation statements)

References 77 publications

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“…In [42], under the assumption of the a-priori bound condition (14), the authors established the following condition stability by using the interpolation method:…”

Section: Conditional Stability and Regularization Methods 21 Conditiomentioning

confidence: 99%

Solving the Riesz–Feller space-fractional backward diffusion problem by a generalized Tikhonov method

Zhang

2020

Adv Differ Equ

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The article investigates a Riesz-Feller space-fractional backward diffusion problem. We develop a generalized Tikhonov regularization method to overcome the ill-posedness of this problem, and then based on the result of conditional stability, we derive the convergence estimates of logarithmic and double logarithmic types for the regularized method by adopting a-posteriori choice rules of regularization parameter. Finally, by using the finite difference method, we solve a direct problem to construct the data, and some corresponding results of numerical simulations are presented to verify the convergence and stability for this method.

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“…In [42], under the assumption of the a-priori bound condition (14), the authors established the following condition stability by using the interpolation method:…”

Section: Conditional Stability and Regularization Methods 21 Conditiomentioning

confidence: 99%

Solving the Riesz–Feller space-fractional backward diffusion problem by a generalized Tikhonov method

Zhang

2020

Adv Differ Equ

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“…subsequently, using the result of conditional stability in (15), there holds that Ultimately, we can prove the result of convergence (47) by combing (A20) with (A24).…”

Section: Proof Of Lemma 2 By Settingmentioning

confidence: 95%

“…It can be found that many scholars have researched this kind of problem in the aspect of theory and algorithm, such as [5][6][7][8][9][10][11][12][13][14][15][16][17], and so on. In [14], the author solved (4), (5) by constructing a generalized Tikhonov regularization method, meanwhile derived the a-priori convergence estimate of regularized method.…”

Section: Introductionmentioning

confidence: 99%

Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Zhang

2020

Mathematics

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This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

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“…A famous theorem by A. N. Tikhonov, see [11], asserts that the problem is conditionally well-posed whenever the set M ⊂ X is compact. For some further discussion and examples concerning conditional well-posedness we also refer to [4,5,14]. Proof.…”

Section: Modulus Functionsmentioning

confidence: 99%

“…Tight bounds for the modulus of continuity ω Mϕ can be obtained under additional geometric (convexity) assumptions by means of general interpolation results within the framework of variable Hilbert scales, and we mention [13] for an early work on this. More recently this is pursued within the framework of conditionally stability estimates, see the recent study [14].…”

Section: Smoothness Classes In Hilbert Spacementioning

confidence: 99%

Regularizability of Ill-Posed Problems and the Modulus of Continuity

Boţ¹,

Hofmann²,

Mathé³

2013

Z. Anal. Anwend.

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The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the well-posedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the additional feature of being star-shaped at zero, the authors develop a series of results with focus on continuity properties of the modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus of continuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space. This study concludes with a new perspective on a concavity problem for the modulus of continuity, recently addressed by two of the authors in A note on the modulus of continuity for ill-posed problems in Hilbert space, 2012.

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

Cited by 23 publications

References 77 publications

Solving the Riesz–Feller space-fractional backward diffusion problem by a generalized Tikhonov method

Solving the Riesz–Feller space-fractional backward diffusion problem by a generalized Tikhonov method

Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Regularizability of Ill-Posed Problems and the Modulus of Continuity

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