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

Didier Auroux; Maëlle Nodet

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 2, page 318-342
  • ISSN: 1292-8119

Abstract

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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 (Burgers’ 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.

How to cite

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Auroux, Didier, and Nodet, Maëlle. "The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations." ESAIM: Control, Optimisation and Calculus of Variations 18.2 (2012): 318-342. <http://eudml.org/doc/277813>.

@article{Auroux2012,
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 (Burgers’ 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. },
author = {Auroux, Didier, Nodet, Maëlle},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Data assimilation; inverse problems; linear transport equations; Burgers’ equation; data assimilation; linear transport equation; Burgers equation; back and forth nudging algorithm; convergence},
language = {eng},
month = {7},
number = {2},
pages = {318-342},
publisher = {EDP Sciences},
title = {The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations},
url = {http://eudml.org/doc/277813},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Auroux, Didier
AU - Nodet, Maëlle
TI - The Back and Forth Nudging algorithm for data assimilation problems : theoretical results on transport equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/7//
PB - EDP Sciences
VL - 18
IS - 2
SP - 318
EP - 342
AB - 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 (Burgers’ 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.
LA - eng
KW - Data assimilation; inverse problems; linear transport equations; Burgers’ equation; data assimilation; linear transport equation; Burgers equation; back and forth nudging algorithm; convergence
UR - http://eudml.org/doc/277813
ER -

References

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