The truncation method for identifying an unknown source in the Poisson equation

Yang, Fan

doi:10.1016/j.amc.2011.04.017

Cited by 20 publications

(10 citation statements)

References 16 publications

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“…The above inverse problem is particularly important for several applications, especially in the model case of the Laplace operator L = ∆ and the Poisson equation, see e.g. [1,8,14,20,28,29,31,33,34,35,36,37]. Herein we will assume that the noisy measurements on the solution take the form…”

Section: Introductionmentioning

confidence: 99%

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

Katzourakis¹

2019

SIAM J. Math. Anal.

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In this paper we utilise new methods of Calculus of Variations in L ∞ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in L 2 is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.

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Section: Introductionmentioning

confidence: 99%

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

Katzourakis¹

2019

SIAM J. Math. Anal.

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“…In [7], the authors identified the unknown source of the Poisson equation using the modified regularization method. In [8], the authors identified the unknown source of the Poisson equation using the truncation method. In [7,8], the regularization parameters which depend on the noise level and the a priori bound are selected by the a priori rule.…”

Section: Introductionmentioning

confidence: 99%

“…In [8], the authors identified the unknown source of the Poisson equation using the truncation method. In [7,8], the regularization parameters which depend on the noise level and the a priori bound are selected by the a priori rule. Generally speaking, there is a defect for any a priori method; that is, the a priori choice of the regularization parameter depends seriously on the a priori bound of the unknown solution.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation

2013

International Journal of Partial Differential Equations

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We consider the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with ana posterioriregularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method.

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“…The uniqueness and conditional stability results for these problems can be found in [2][3][4][5][6][7]. Some numerical reconstruction schemes can be found in [1,[8][9][10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales

Zhao¹,

Xie²,

Meng³

et al. 2014

JAMP

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In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.

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The truncation method for identifying an unknown source in the Poisson equation

Cited by 20 publications

References 16 publications

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation

Determination of an Unknown Source in the Heat Equation by the Method of Tikhonov Regularization in Hilbert Scales

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