Volume 51, Issue 2
Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform

Faker Ben Belgacem, M. Djalil Kateb & Vincent Robin

J. Math. Study, 51 (2018), pp. 115-130.

Published online: 2018-06

[An open-access article; the PDF is free to any online user.]

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  • Abstract

The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide new information about the observation of the initial data problem. It is actually exponentially ill-posed.

  • Keywords

Integral operators, regular kernels, Jacobi transform, separated variables approximation.

  • AMS Subject Headings

MASC 65N20, 65F22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

faker.ben-belgacem@utc.fr (Faker Ben Belgacem)

djalil.kateb@utc.fr (M. Djalil Kateb)

vincent.robin@utc.fr (Vincent Robin)

  • BibTex
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@Article{JMS-51-115, author = {Faker and Ben Belgacem and faker.ben-belgacem@utc.fr and 13293 and Alliance Sorbonne Université, UTC, EA 2222, Laboratoire de Mathématiques Appliquées de Compiègne, F-60205 Compiègne, France and Faker Ben Belgacem and M. Djalil and Kateb and djalil.kateb@utc.fr and 7067 and Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Universités, UTC, EA 2222, F-60205 Compiègne, France and M. Djalil Kateb and Vincent and Robin and vincent.robin@utc.fr and 7068 and Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Universités, UTC, EA 2222, F-60205 Compiègne, France and Vincent Robin}, title = {Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform}, journal = {Journal of Mathematical Study}, year = {2018}, volume = {51}, number = {2}, pages = {115--130}, abstract = {

The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide new information about the observation of the initial data problem. It is actually exponentially ill-posed.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v51n2.18.01}, url = {http://global-sci.org/intro/article_detail/jms/12458.html} }
TY - JOUR T1 - Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform AU - Ben Belgacem , Faker AU - Kateb , M. Djalil AU - Robin , Vincent JO - Journal of Mathematical Study VL - 2 SP - 115 EP - 130 PY - 2018 DA - 2018/06 SN - 51 DO - http://doi.org/10.4208/jms.v51n2.18.01 UR - https://global-sci.org/intro/article_detail/jms/12458.html KW - Integral operators, regular kernels, Jacobi transform, separated variables approximation. AB -

The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide new information about the observation of the initial data problem. It is actually exponentially ill-posed.

Faker Ben Belgacem, M. Djalil Kateb & Vincent Robin. (2020). Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform. Journal of Mathematical Study. 51 (2). 115-130. doi:10.4208/jms.v51n2.18.01
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