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Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description

Anne-Céline Boulanger 1 Philippe Moireau 2 Benoît Perthame 3 Jacques Sainte-Marie 1
1 ANGE - Numerical Analysis, Geophysics and Ecology
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
2 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
LMS - Laboratoire de mécanique des solides, Inria Saclay - Ile de France
3 MAMBA - Modelling and Analysis for Medical and Biological Applications
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
Abstract : Developing robust data assimilation methods for hyperbolic conservation laws is a challenging subject. Those PDEs indeed show no dissipation effects and the input of additional information in the model equations may introduce errors that propagate and create shocks. We propose a new approach based on the kinetic description of the conservation law. A kinetic equation is a first order partial differential equation in which the advection velocity is a free variable. In certain cases, it is possible to prove that the nonlinear conservation law is equivalent to a linear kinetic equation. Hence, data assimilation is carried out at the kinetic level, using a Luenberger observer also known as the nudging strategy in data assimilation. Assimilation then resumes to the handling of a BGK type equation. The advantage of this framework is that we deal with a single "linear" equation instead of a nonlinear system and it is easy to recover the macroscopic variables. The study is divided into several steps and essentially based on functional analysis techniques. First we prove the convergence of the model towards the data in case of complete observations in space and time. Second, we analyze the case of partial and noisy observations. To conclude, we validate our method with numerical results on Burgers equation and emphasize the advantages of this method with the more complex Saint-Venant system.
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Submitted on : Monday, January 6, 2014 - 11:28:41 PM
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Anne-Céline Boulanger, Philippe Moireau, Benoît Perthame, Jacques Sainte-Marie. Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description. Communications in Mathematical Sciences, International Press, 2015, 13 (3), pp.587 - 622. ⟨10.4310/CMS.2015.v13.n3.a1⟩. ⟨hal-00924559⟩



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