The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class. © 2014 World Scientific Publishing Company.
|Original language||English (US)|
|Number of pages||15|
|Journal||Mathematical Models and Methods in Applied Sciences|
|State||Published - Aug 15 2014|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The authors acknowledge Mourad Choulli for fruitful discussions and valuable bibliographical comments. This research was initiated when the first author was a visiting scholar at the Basque Center for Applied Mathematics, in the frame of the NUMERIWAVES AdvG of ERC. This research was supported by the Office of Naval Research Contracts ONR N00014-09-1-0107, ONR N00014-11-1-0712, ONR N00014-12-1-0561; and the National Science Foundation Grant DMS 12 22390. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). The work of the second author was supported by Grant MTM2011-29306-C02-00 of the MICINN (Spain), the Advanced Grant FP7-246775 of the European Research Council Executive Agency and the Grant PI2010-04 of the Basque Government. This work was finished while the second author was visiting the Laboratoire Jacques Louis Lions with the support of the Paris City Hall "Research in Paris" program and the CIMI - Toulouse on the Excellence Chair on "PDE, Control and Numerics".
This publication acknowledges KAUST support, but has no KAUST affiliated authors.