### DATA ASSIMILATION FOR HYPERBOLIC CONSERVATION LAWS. A LUENBERGER OBSERVER APPROACH BASED ON A KINETIC DESCRIPTION

"... Abstract. Developing robust data assimilation methods for hyperbolic conservation laws is a chal-lenging subject. Those PDEs indeed show no dissipation effects and the input of additional infor-mation in the model equations may introduce errors that propagate and create shocks. We propose a new appr ..."

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Abstract. Developing robust data assimilation methods for hyperbolic conservation laws is a chal-lenging subject. Those PDEs indeed show no dissipation effects and the input of additional infor-mation 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. 1.

### Available online xxxx Keywords: Sensor optimisation

, 2014

"... are avoid locations with low impact, where impact is determined by a function such as one of the key phenomena may occur, such as slug initiation, gas entrainment, and slug propagation. Therefore, researchers often use correlations and closure equations to simplify the problem. Another example is a ..."

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are avoid locations with low impact, where impact is determined by a function such as one of the key phenomena may occur, such as slug initiation, gas entrainment, and slug propagation. Therefore, researchers often use correlations and closure equations to simplify the problem. Another example is a falling liquid film, which is a common phenomenon not only in industry but also in nature. Researchers have built many low-dimensional models to simplify the falling film problem, e.g., two to pursue higher photography in rtant that resources are used efficiently to capture the most importa tures. Therefore, it is necessary to define these features, o better, to develop a method to help researchers determi most critical features to simulate or to measure. Then it is possible to use these limited resources to accurately resolve the critical parts and try to eliminate costs due to unnecessary parts which have negligible effect. Once the important features are determined, researchers can speed up simulation by focusing only on the nec-essary features, and reduce experimental costs by measuring only the important information. ⇑ Corresponding author.

### unknown title

, 2015

"... www.atmos-chem-phys.net/15/10019/2015/ doi:10.5194/acp-15-10019-2015 © Author(s) 2015. CC Attribution 3.0 License. Ensemble data assimilation of total column ozone using a coupled meteorology–chemistry model and its impact on the structure of ..."

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www.atmos-chem-phys.net/15/10019/2015/ doi:10.5194/acp-15-10019-2015 © Author(s) 2015. CC Attribution 3.0 License. Ensemble data assimilation of total column ozone using a coupled meteorology–chemistry model and its impact on the structure of

### data assimilation scheme in OceanVar software

, 2012

"... Abstract: The most significant features ofDataAssimilation (DA)are that boththe models and the observations are very large and non-linear (of order at least O(10 8)). Further, DA is an ill-posed inverse problem. Such properties make the numerical solution of DA very difficult so that, as stated in [ ..."

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Abstract: The most significant features ofDataAssimilation (DA)are that boththe models and the observations are very large and non-linear (of order at least O(10 8)). Further, DA is an ill-posed inverse problem. Such properties make the numerical solution of DA very difficult so that, as stated in [19], ”solving this problem in ”real-time ” it is not always possible and many different approximations to the basic assimilation schemes are employed”. Thus, the exploitation of advanced computing environments is mandatory, reducing the computational cost to a suitable turnaround time. This activity should be done according to a co-design methodology where software requirements drive hardware design decisions andhardware design constraints motivate changes inthesoftware design tobetterfitwithin those constraints. In this paper, we address high performance computation issues of the three dimensional DA scheme underlying the oceanographic 3D-VAR assimilation scheme, named Ocean-VAR, developed at CMCC (Centro Euro Mediterraneo per i Cambiamenti Climatici), in Italy. The aim is to develop a parallel software architecture which is able to effectively take advantage of the available high performance computing resources.

### unknown title

"... What (if anything) can econometric forecasters learn from meteorologists (and vice versa)? ..."

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What (if anything) can econometric forecasters learn from meteorologists (and vice versa)?

### An Optimization Framework to Improve 4D-Var Data Assimilation System Performance

, 2014

"... ar ..."

### DATA ASSIMILATION OF TIME UNDER-SAMPLED MEASUREMENTS USING OBSERVERS, THE WAVE-LIKE EQUATION EXAMPLE

"... We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of th ..."

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We propose a sequential data assimilation scheme using Luenberger type observers when only some space restricted time under-sampled measurements are available. More precisely, we consider a wave-like equation for which we assume known the restriction of the solution to an open non-empty subset of the spatial domain and for some time samples (typically the sampling step in time is much larger than the time discretization step). To assimilate the available data, two strategies are proposed and analyzed. The first strategy consists in assimilating data only if they are available and the second one in assimilating interpolation of the available data at all the discretization times. In order to tackle the spurious high frequencies which appear when we discretize the wave equation, for both strategies, we introduce a numerical viscous term. In this case, we prove some error estimates between the exact solution and our observers. Numerical simulations illustrate the theoretical results in the case of the one dimensional wave equation.

### unknown title

, 2015

"... www.geosci-model-dev.net/8/1315/2015/ doi:10.5194/gmd-8-1315-2015 © Author(s) 2015. CC Attribution 3.0 License. Structure of forecast error covariance in coupled atmosphere–chemistry data assimilation ..."

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www.geosci-model-dev.net/8/1315/2015/ doi:10.5194/gmd-8-1315-2015 © Author(s) 2015. CC Attribution 3.0 License. Structure of forecast error covariance in coupled atmosphere–chemistry data assimilation