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Volume 36, Issue 2
A New Regularization Method for a Parameter Identification Problem in a Non-Linear Partial Differential Equation

M. Thamban Nair & Samprita Das Roy

DOI: 10.4208/jpde.v36.n2.3

J. Part. Diff. Eq., 36 (2023), pp. 147-190.

Published online: 2023-07

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  • Abstract

We consider a parameter identification problem associated with a quasilinear elliptic Neumann boundary value problem involving a parameter function $a(·)$ and the solution $u(·),$ where the problem is to identify $a(·)$ on an interval $I:=g(Γ)$ from the knowledge of the solution $u(·)$ as $g$ on $Γ,$ where Γ is a given curve on the boundary of the domain $Ω⊆\mathbb{R}^3$ of the problem and $g$ is a continuous function. The inverse problem is formulated as a problem of solving an operator equation involving a compact operator depending on the data, and for obtaining stable approximate solutions under noisy data, a new regularization method is considered. The derived error estimates are similar to, and in certain cases better than, the classical Tikhonov regularization considered in the literature in recent past.

  • Keywords

Ill-posed, regularization, parameter identification.

  • AMS Subject Headings

35R30, 65N30, 65J15, 65J20, 76S05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-36-147, author = {Nair , M. Thamban and Roy , Samprita Das}, title = {A New Regularization Method for a Parameter Identification Problem in a Non-Linear Partial Differential Equation}, journal = {Journal of Partial Differential Equations}, year = {2023}, volume = {36}, number = {2}, pages = {147--190}, abstract = {

We consider a parameter identification problem associated with a quasilinear elliptic Neumann boundary value problem involving a parameter function $a(·)$ and the solution $u(·),$ where the problem is to identify $a(·)$ on an interval $I:=g(Γ)$ from the knowledge of the solution $u(·)$ as $g$ on $Γ,$ where Γ is a given curve on the boundary of the domain $Ω⊆\mathbb{R}^3$ of the problem and $g$ is a continuous function. The inverse problem is formulated as a problem of solving an operator equation involving a compact operator depending on the data, and for obtaining stable approximate solutions under noisy data, a new regularization method is considered. The derived error estimates are similar to, and in certain cases better than, the classical Tikhonov regularization considered in the literature in recent past.

}, issn = {2079-732X}, doi = {https://doi.org/ 10.4208/jpde.v36.n2.3}, url = {http://global-sci.org/intro/article_detail/jpde/21839.html} }
TY - JOUR T1 - A New Regularization Method for a Parameter Identification Problem in a Non-Linear Partial Differential Equation AU - Nair , M. Thamban AU - Roy , Samprita Das JO - Journal of Partial Differential Equations VL - 2 SP - 147 EP - 190 PY - 2023 DA - 2023/07 SN - 36 DO - http://doi.org/ 10.4208/jpde.v36.n2.3 UR - https://global-sci.org/intro/article_detail/jpde/21839.html KW - Ill-posed, regularization, parameter identification. AB -

We consider a parameter identification problem associated with a quasilinear elliptic Neumann boundary value problem involving a parameter function $a(·)$ and the solution $u(·),$ where the problem is to identify $a(·)$ on an interval $I:=g(Γ)$ from the knowledge of the solution $u(·)$ as $g$ on $Γ,$ where Γ is a given curve on the boundary of the domain $Ω⊆\mathbb{R}^3$ of the problem and $g$ is a continuous function. The inverse problem is formulated as a problem of solving an operator equation involving a compact operator depending on the data, and for obtaining stable approximate solutions under noisy data, a new regularization method is considered. The derived error estimates are similar to, and in certain cases better than, the classical Tikhonov regularization considered in the literature in recent past.

Nair , M. Thamban and Roy , Samprita Das. (2023). A New Regularization Method for a Parameter Identification Problem in a Non-Linear Partial Differential Equation. Journal of Partial Differential Equations. 36 (2). 147-190. doi: 10.4208/jpde.v36.n2.3
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