Fractional Tikhonov regularization with a nonlinear penalty term

Morigi, Serena; Reichel, Lothar; Sgallari, Fiorella

doi:10.1016/j.cam.2017.04.017

Cited by 23 publications

(7 citation statements)

References 21 publications

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“…Applying the fractional Tikhonov method given by Morigi et al, 23 we propose the following regularized solution with exact data (h, f):…”

Section: Fractional Tikhonov Methodsmentioning

confidence: 99%

“…Applying the fractional Tikhonov method given by Morigi et al, we propose the following regularized solution with exact data

false(sans-serifh, monospacef false)

{scriptG}_{μ, ϑ} (x) = \sum_{sans-serifq = 0}^{\infty} E_{β, 1}^{- 1} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ}) {[\frac{E_{β, 1}^{2} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ})}{E_{β, 1}^{2} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ}) + μ}]}^{ϑ} ⟨ Θ, {\hat{sans-serife}}_{sans-serifq} ⟩ {\hat{sans-serife}}_{sans-serifq} .

If the observation data

false(sans-serifh, monospacef false)

is noised by

false(h^{ε}, f^{ε} false)

, then we have

G_{μ, ϑ}^{ε} (x) = \sum_{sans-serifq = 0}^{\infty} E_{β, 1}^{- 1} (-)

…”

Section: Fractional Tikhonov Regularizationmentioning

confidence: 99%

See 1 more Smart Citation

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

Math Methods in App Sciences

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In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

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“…Applying the fractional Tikhonov method given by Morigi et al, 23 we propose the following regularized solution with exact data (h, f):…”

Section: Fractional Tikhonov Methodsmentioning

confidence: 99%

“…Applying the fractional Tikhonov method given by Morigi et al, we propose the following regularized solution with exact data

false(sans-serifh, monospacef false)

{scriptG}_{μ, ϑ} (x) = \sum_{sans-serifq = 0}^{\infty} E_{β, 1}^{- 1} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ}) {[\frac{E_{β, 1}^{2} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ})}{E_{β, 1}^{2} (- \frac{{sans-serifa}_{sans-serifq}}{{normalγ}^{normalβ}} {sans-serifT}_{o}^{β γ}) + μ}]}^{ϑ} ⟨ Θ, {\hat{sans-serife}}_{sans-serifq} ⟩ {\hat{sans-serife}}_{sans-serifq} .

If the observation data

false(sans-serifh, monospacef false)

is noised by

false(h^{ε}, f^{ε} false)

, then we have

G_{μ, ϑ}^{ε} (x) = \sum_{sans-serifq = 0}^{\infty} E_{β, 1}^{- 1} (-)

…”

Section: Fractional Tikhonov Regularizationmentioning

confidence: 99%

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

Tuan

Huynh

Băleanu

et al. 2019

Math Methods in App Sciences

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“…Tikhonov et al [21]. Related research on fractional regularization methods can refer to the literature [22][23][24][25][26][27][28][29]. The fractional Tikhonov method with the a priori parameter for the same analytic continuation problem has been researched by [30].…”

Section: Problemmentioning

confidence: 99%

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

Xue

Xiong

2021

Mathematics

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In this paper, the numerical analytic continuation problem is addressed and a fractional Tikhonov regularization method is proposed. The fractional Tikhonov regularization not only overcomes the difficulty of analyzing the ill-posedness of the continuation problem but also obtains a more accurate numerical result for the discontinuity of solution. This article mainly discusses the a posteriori parameter selection rules of the fractional Tikhonov regularization method, and an error estimate is given. Furthermore, numerical results show that the proposed method works effectively.

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“…The discrepancy principle is an example of the first type and the Cross-Validation and L-curve are examples from the second type (see [23]). Since it has been observed that x δ α oversmoothens the solution x [12], the Fractional Tikhonov regularization (FTR) method [1,2,7,[10][11][12]19] was considered, wherein the minimizer x δ α,β of the functional…”

Section: Introductionmentioning

confidence: 99%

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

Mekoth

George

Jidesh

2023

Hacettepe Journal of Mathematics and Statistics

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One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh. Fractional Tikhonov regularization method in Hilbert scales. Appl. Math. Comput.(2021), 392: 125701, 26 DOI:10.1016/j.amc.2020.125701] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

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Fractional Tikhonov regularization with a nonlinear penalty term

Cited by 23 publications

References 21 publications

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

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