arrow
Volume 18, Issue 6
Modified Tikhonov Regularization for Identifying Several Sources

Ole Iøseth Elvetun & Bjørn Fredrik Nielsen

Int. J. Numer. Anal. Mod., 18 (2021), pp. 740-757.

Published online: 2021-11

Export citation
  • Abstract

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.

  • AMS Subject Headings

35R30, 47A52, 65F22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-18-740, author = {Elvetun , Ole Iøseth and Nielsen , Bjørn Fredrik}, title = {Modified Tikhonov Regularization for Identifying Several Sources}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2021}, volume = {18}, number = {6}, pages = {740--757}, abstract = {

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/19948.html} }
TY - JOUR T1 - Modified Tikhonov Regularization for Identifying Several Sources AU - Elvetun , Ole Iøseth AU - Nielsen , Bjørn Fredrik JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 740 EP - 757 PY - 2021 DA - 2021/11 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/19948.html KW - Inverse source problems, PDE-constrained optimization, Tikhonov regularization, nullspace, numerical computations. AB -

We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.

Elvetun , Ole Iøseth and Nielsen , Bjørn Fredrik. (2021). Modified Tikhonov Regularization for Identifying Several Sources. International Journal of Numerical Analysis and Modeling. 18 (6). 740-757. doi:
Copy to clipboard
The citation has been copied to your clipboard