A fractional Landweber method for solving backward time-fractional diffusion problem

Han, Yaozong; Xiong, Xiangtuan; Xue, Xuemin

doi:10.1016/j.camwa.2019.02.017

Cited by 31 publications

(19 citation statements)

References 27 publications

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“…Theorem 4.1. Let f + be the best-approximate solution defined in (15), f m δ is the solution of the fractional Landweber regularization (21) or (22). Assume the noisy data satisfies (7) and the solution satisfies the a priori (25) under the choices for the a priori regularization parameter m and acceleration factor a:…”

Section: Under a Priori Parameter Choice Rulementioning

confidence: 99%

“…Wang and Wei [31] proposed an iterative method inspired by the work [7,8,5] to solve a backward problem, and gave convergence estimates under two kinds of regularization parameter choice rule. In [15], Han et al obtained a regularization solution by the fractional Landweber iterative regularization method for identifying the initial condition of the time-fractional diffusion equation.…”

mentioning

confidence: 99%

“…This idea was originally proposed by Klann et al [18] in a general framework based on the filter regularization technique for solving a linear inverse problem. It was then studied and applied in some extent by many authors with success, see [16,13,4,36,15] for examples. Our goal is to graft this approach to the time-fractional inverse problem considered in this paper.…”

mentioning

confidence: 99%

“…Let f + be the best-approximate solution defined in(15), f m δ is the solution of the fractional Landweber regularization (21) with 0 < a < ν+1 or (22) with a > 0. Assume the noisy data satisfies (7) and the solution satisfies the a priori condition f + D(L p 2 ) ≤ E for some p > 0.…”

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confidence: 99%

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An extension of the landweber regularization for a backward time fractional wave problem

Fan¹,

Azaïez²,

Xu³

2021

DCDS-S

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In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

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Section: Under a Priori Parameter Choice Rulementioning

confidence: 99%

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confidence: 99%

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confidence: 99%

mentioning

confidence: 99%

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An extension of the landweber regularization for a backward time fractional wave problem

Fan¹,

Azaïez²,

Xu³

2021

DCDS-S

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“…In recent years, the direct problems [13][14][15][16][17][18] corresponding to the time-fractional diffusion equations have been extensively studied, and inverse problems of fractional diffusion equations have been extensively studied; one can see previous works. [19][20][21][22][23][24][25][26] But there are few results for inverse problem of time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator. Therefore, in this paper, our purpose is to identify the initial value problem of time-fractional equation with Caputo-like counterpart hyper-Bessel operator.…”

Section: Introductionmentioning

confidence: 99%

Identifying initial value problem for time‐fractional diffusion equation with Caputo‐like counterpart hyper‐Bessel operator: Optimal error bound analysis and regularization method

Yang

Sun

2021

Math Methods in App Sciences

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In this paper, we consider the inverse problem for identifying the initial value problem for time-fractional diffusion equation with hyper-Bessel operator. Firstly, we show that this problem is ill-posed and analyze the optimal error bound. Next, we use the fractional Landweber iterative regularization method to solve this problem and give the error estimates between the regularization solution and the exact solution under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule. Finally, we use numerical examples to verify the effectiveness of this method. KEYWORDS Caputo-like counterpart hyper-Bessel operator, fractional Landweber regularization, ill-posed problem, initial value problem, time-fractional diffusion equation MSC CLASSIFICATION 35R25; 47A52; 35R30How to cite this article: Yang F, Sun Q-X, Li X-X. Identifying initial value problem for time-fractional diffusion equation with Caputo-like counterpart hyper-Bessel operator: Optimal error bound analysis and regularization method.

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Tikhonov regularization for simultaneous inversion of initial value and source term of a time‐fractional Black‐Scholes equation

Wu,

Yang

2024

Math Methods in App Sciences

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This paper studies simultaneous inversion of initial value and source term of a time‐fractional Black‐Scholes equation. This problem is ill‐posed, and we use Tikhonov regularization method to solve it. Under the selection rules of a priori and a posteriori regularization parameters, a priori and a posteriori Hölder type error estimates are derived. Finally, numerical experiments demonstrate the stability and effectiveness of the proposed regularization method.

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A fractional Landweber method for solving backward time-fractional diffusion problem

Cited by 31 publications

References 27 publications

An extension of the landweber regularization for a backward time fractional wave problem

An extension of the landweber regularization for a backward time fractional wave problem

Identifying initial value problem for time‐fractional diffusion equation with Caputo‐like counterpart hyper‐Bessel operator: Optimal error bound analysis and regularization method

Tikhonov regularization for simultaneous inversion of initial value and source term of a time‐fractional Black‐Scholes equation

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