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.
Keywords: Data assimilation, mixed formulation, Morozov’s principle, duality in optimization
@article{M2AN_2022__56_6_2021_0,
author = {Bourgeois, Laurent and Dard\'e, J\'er\'emi},
title = {The {Morozov{\textquoteright}s} principle applied to data assimilation problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2021--2050},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {6},
doi = {10.1051/m2an/2022061},
mrnumber = {4481123},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022061/}
}
TY - JOUR AU - Bourgeois, Laurent AU - Dardé, Jérémi TI - The Morozov’s principle applied to data assimilation problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 2021 EP - 2050 VL - 56 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022061/ DO - 10.1051/m2an/2022061 LA - en ID - M2AN_2022__56_6_2021_0 ER -
%0 Journal Article %A Bourgeois, Laurent %A Dardé, Jérémi %T The Morozov’s principle applied to data assimilation problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 2021-2050 %V 56 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022061/ %R 10.1051/m2an/2022061 %G en %F M2AN_2022__56_6_2021_0
Bourgeois, Laurent; Dardé, Jérémi. The Morozov’s principle applied to data assimilation problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2021-2050. doi: 10.1051/m2an/2022061
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