Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Feng, Xiaoli; Eldén, Lars

doi:10.1088/0266-5611/30/1/015005

Cited by 27 publications

(16 citation statements)

References 48 publications

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“…There are many regularization methods for the study of inverse problems, such as Tikhonov regularization method, [29][30][31] quasi-boundary value method, [32][33][34][35] quasi-reversibility regularization method, 36,37 a mollification regularization method, 38 Fourier regularization method, [39][40][41][42][43][44] and Landweber iterative regularization method. [45][46][47][48] In this paper, we consider the following the potential-free field inverse time-fractional Schrödinger problem with boundary condition…”

Section: Introductionmentioning

confidence: 99%

“…There are many regularization methods for the study of inverse problems, such as Tikhonov regularization method, 29‐31 quasi‐boundary value method, 32‐35 quasi‐reversibility regularization method, 36,37 a mollification regularization method, 38 Fourier regularization method, 39‐44 and Landweber iterative regularization method 45‐48 . In this paper, we consider the following the potential‐free field inverse time‐fractional Schrödinger problem with boundary condition

({, \begin{matrix} left left \\ array i_{0}^{C} D_{t}^{α} u (x, t) + u_{x x} (x, t) = 0, & array x > 0, t > 0, \\ array u (x, 0) = 0, & array x \geq 0, \\ array u (1, t) = f (t), & array t \geq 0, \\ array u (x, t) |_{x \to \infty} b o u n d e d, & array t > 0, \end{matrix})

where

i = \sqrt{- 1}

is the imaginary unit and

_{0}^{C} D_{t}^{α}

is the Caputo time‐fractional derivative of order α defined as

_{0}^{C} D_{t}^{α} u (x, t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} {(t - τ)}^{- α}

…”

Section: Introductionmentioning

confidence: 99%

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A potential‐free field inverse time‐fractional Schrödinger problem: Optimal error bound analysis and regularization method

Yang

2020

Math Methods in App Sciences

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In this paper, an inverse time-fractional Schrödinger problem of potential-free field is studied. This problem is ill-posed; that is, the solution (if it exists) does not depend continuously on the data. Based on an a priori bound condition, the optimal error bound analysis is given. Moreover, a modified kernel method is introduced. The convergence error estimate obtained by this method under the a priori regularization parameter selection rule is optimal, and the convergence error estimate obtained under the a posteriori regularization parameter selection rule is order-optimal. Finally, some numerical examples are given to illustrate the effectiveness and stability of this method. KEYWORDS a modified kernel method, ill-posed problem, inverse problem, optimal error bound, time-fractional Schrödinger equation MSC CLASSIFICATION

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Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} left left \\ array i_{0}^{C} D_{t}^{α} u (x, t) + u_{x x} (x, t) = 0, & array x > 0, t > 0, \\ array u (x, 0) = 0, & array x \geq 0, \\ array u (1, t) = f (t), & array t \geq 0, \\ array u (x, t) |_{x \to \infty} b o u n d e d, & array t > 0, \end{matrix})

where

i = \sqrt{- 1}

is the imaginary unit and

_{0}^{C} D_{t}^{α}

is the Caputo time‐fractional derivative of order α defined as

_{0}^{C} D_{t}^{α} u (x, t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} {(t - τ)}^{- α}

…”

Section: Introductionmentioning

confidence: 99%

A potential‐free field inverse time‐fractional Schrödinger problem: Optimal error bound analysis and regularization method

Yang

2020

Math Methods in App Sciences

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“…Let us introduce some papers studying the deterministic problem (). In the case of the problem is homogeneous (where both f and σ are zero), the readers can refer to some related papers 8‐28 . Essentially optimal stability results were given by Elden and Simoncini, 9 in wide generality and under substantially minimal assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…Quian et al 19 applied two regularization methods including quasi‐reversibility and truncation method to deal with the homogeneous problem. Additionally, some useful numerical methods were developed to solve the homogeneous problem, see for example, other studies 9‐13,15 . In the inhomogeneous case, to the best of our knowledge, the literature for this problem is very limited.…”

Section: Introductionmentioning

confidence: 99%

Regularized solution of a Cauchy problem for stochastic elliptic equation

Tuan

Hoan²,

Zhou

et al. 2020

Math Methods in App Sciences

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“…For the study of inverse problems, there are many regularization methods, such as the truncation method [31][32][33], Tikhonov regularization method [34], quasi-boundary value method [35][36][37], quasi-reversibility regularization method [38,39], a mollification regularization method [40], Fourier regularization method [41][42][43][44], and Landweber iterative regularization method [45][46][47] which never appears saturation phenomenon. In this paper, firstly, we will prove that the inverse problem is ill-posed, that means, the solution does not depend continuously on the data.…”

Section: Introductionmentioning

confidence: 99%

A potential-free field inverse Schrödinger problem: optimal error bound analysis and regularization method

Yang

2019

Inverse Problems in Science and Engineering

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In this paper, an inverse Schrödinger problem of potential-free field is studied. This problem is ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. Based on an a priori assumption, the optimal errorbound analysis is given. Moreover, two different regularization methods are used to solve this problem, respectively. Under an a priori and an a posteriori regularization parameters choice rule, the convergent error estimates are all obtained. Compared with Landweber iterative regularization method, the convergent estimate between the exact solution and the regularization solution obtained by a modified kernel method is optimal for the priori regularization parameter choice rule, and the posteriori error estimate is orderoptimal. Finally, some numerical examples are given to illustrate the effectiveness, stability and superiority of these methods.

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Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

Cited by 27 publications

References 48 publications

A potential‐free field inverse time‐fractional Schrödinger problem: Optimal error bound analysis and regularization method

A potential‐free field inverse time‐fractional Schrödinger problem: Optimal error bound analysis and regularization method

Regularized solution of a Cauchy problem for stochastic elliptic equation

A potential-free field inverse Schrödinger problem: optimal error bound analysis and regularization method

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