The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations

Didier Auroux; Maëlle Nodet

inria-00325305, version 1

The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations

Didier Auroux ()^a^, 1, 2, Maëlle Nodet ()^b^, 1

ESAIM: Control, Optimisation and Calculus of Variations 18, 2 (2012) 318-342

Abstract: In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Bürgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.

a – Université Paul Sabatier - Toulouse III
b – Université Joseph Fourier - Grenoble I
1: MOISE (INRIA Grenoble Rhône-Alpes / LJK Laboratoire Jean Kuntzmann)
CNRS : UMR5224 – INRIA – Laboratoire Jean Kuntzmann – Université Joseph Fourier - Grenoble I – Institut Polytechnique de Grenoble - Grenoble Institute of Technology
2: Mathématiques pour l'Industrie et la Physique (MIP)
CNRS : UMR5640 – Université des Sciences Sociales - Toulouse I – Université Paul Sabatier [UPS] - Toulouse III – Institut National des Sciences Appliquées (INSA) - Toulouse

inria-00325305, version 1
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From: Maëlle Nodet <>
Submitted on: Sunday, 28 September 2008 11:37:50
Updated on: Saturday, 23 February 2013 17:13:46

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arXiv: 0809.4838

DOI: 10.1051/cocv/2011004

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%% inria-00325305, version 1
%% http://hal.inria.fr/inria-00325305
@article{auroux:inria-00325305,
    hal_id = {inria-00325305},
    url = {http://hal.inria.fr/inria-00325305},
    title = {{The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations}},
    author = {Auroux, Didier and Nodet, Ma{\"e}lle},
    abstract = {{In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (B{\"u}rgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.}},
    keywords = {Data assimilation ; Inverse problems ; Linear transport equations ; Burgers' equation},
    language = {English},
    affiliation = {MOISE - INRIA Grenoble Rh{\^o}ne-Alpes / LJK Laboratoire Jean Kuntzmann , Math{\'e}matiques pour l'Industrie et la Physique - MIP},
    publisher = {EDP Sciences},
    pages = {318-342},
    journal = {ESAIM: Control, Optimisation and Calculus of Variations},
    volume = {18},
    number = {2 },
    note = {RR-6676 RR-6676 LEFE BFN },
    audience = {international },
    doi = {10.1051/cocv/2011004 },
    year = {2012},
    month = Apr,
    pdf = {http://hal.inria.fr/inria-00325305/PDF/BFN-burgers-thms-HAL.pdf},
}

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