Volume 34, Issue 3
On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis

Subhankar Mondal & M. Thamban Nair

J. Part. Diff. Eq., 34 (2021), pp. 240-257.

Published online: 2021-07

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  • Abstract

We consider the inverse problem of identifying a general source term, which is a function of both time variable and the spatial variable, in a parabolic PDE  from the knowledge of  boundary measurements of the solution on some portion of the lateral boundary. We transform this  inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data, we employ the standard  Tikhonov regularization, and its  finite dimensional realization is done using a discretization  procedure involving the space  $L^2(0,\tau;L^2(Ω))$. For illustrating the  specification of an a priori source condition,  we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.

  • Keywords

Ill-posed source identification Tikhonov regularization weak solution.

  • AMS Subject Headings

35R30, 65N21, 47A52

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

s.subhankar80@gmail.com (Subhankar Mondal)

mtnair@iitm.ac.in (M. Thamban Nair)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-34-240, author = {Mondal , Subhankar and Nair , M. Thamban}, title = {On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis}, journal = {Journal of Partial Differential Equations}, year = {2021}, volume = {34}, number = {3}, pages = {240--257}, abstract = {

We consider the inverse problem of identifying a general source term, which is a function of both time variable and the spatial variable, in a parabolic PDE  from the knowledge of  boundary measurements of the solution on some portion of the lateral boundary. We transform this  inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data, we employ the standard  Tikhonov regularization, and its  finite dimensional realization is done using a discretization  procedure involving the space  $L^2(0,\tau;L^2(Ω))$. For illustrating the  specification of an a priori source condition,  we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v34.n3.3}, url = {http://global-sci.org/intro/article_detail/jpde/19322.html} }
TY - JOUR T1 - On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis AU - Mondal , Subhankar AU - Nair , M. Thamban JO - Journal of Partial Differential Equations VL - 3 SP - 240 EP - 257 PY - 2021 DA - 2021/07 SN - 34 DO - http://doi.org/10.4208/jpde.v34.n3.3 UR - https://global-sci.org/intro/article_detail/jpde/19322.html KW - Ill-posed KW - source identification KW - Tikhonov regularization KW - weak solution. AB -

We consider the inverse problem of identifying a general source term, which is a function of both time variable and the spatial variable, in a parabolic PDE  from the knowledge of  boundary measurements of the solution on some portion of the lateral boundary. We transform this  inverse problem into a problem of solving a compact linear operator equation. For the regularization of the operator equation with noisy data, we employ the standard  Tikhonov regularization, and its  finite dimensional realization is done using a discretization  procedure involving the space  $L^2(0,\tau;L^2(Ω))$. For illustrating the  specification of an a priori source condition,  we have explicitly obtained the range space of the adjoint of the operator involved in the operator equation.

SubhankarMondal & M. ThambanNair. (2021). On Regularization of a Source Identification Problem in a Parabolic PDE and its Finite Dimensional Analysis. Journal of Partial Differential Equations. 34 (3). 240-257. doi:10.4208/jpde.v34.n3.3
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