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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.

M. Thamban Nair & Samprita Das Roy. (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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