| Issue |
ESAIM: M2AN
Volume 56, Number 6, November-December 2022
|
|
|---|---|---|
| Page(s) | 2021 - 2050 | |
| DOI | https://doi.org/10.1051/m2an/2022061 | |
| Published online | 14 September 2022 | |
The Morozov’s principle applied to data assimilation problems
1
Laboratoire POEMS, CNRS, INRIA, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France
2
Institut de Mathématiques de Toulouse, UMR5219, Univ. de Toulouse, CNRS, UPS, 118 Route de Narbonne, F-31062 Toulouse Cedex 9, France
* Corresponding author: laurent.bourgeois@ensta.fr
Received:
16
February
2022
Accepted:
6
July
2022
This paper is focused on the Morozov’s principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov’s choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov’s principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.
Mathematics Subject Classification: 35J05 / 35R25 / 35R30 / 65M60
Key words: Data assimilation / mixed formulation / Morozov’s principle / duality in optimization
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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