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

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

Numerical methods and analysis for continuous data assimilation in fluid models

Modeling fluid flow arises in many applications of science and engineering, including the design of aircrafts, prediction of weather, and oceanography. It is vital that these models are both computationally efficient and accurate. In order to obtain good results from these models, one must have accurate and complete initial and boundary conditions. In many real-world applications, these conditions may be unknown, only partially known, or contain error. In order to overcome the issue of unknown or incomplete initial conditions, mathematicians and scientists have been studying different ways to incorporate data into fluid flow models to improve accuracy and/or speed up convergence to the true solution. In this thesis, we are studying one specific data assimilation technique to apply to finite element discretizations of fluid flow models, known as continuous data assimilation. Continuous data assimilation adds a penalty term to the differential equation to nudge coarse spatial scales of the algorithm solution to coarse spatial scales of the true solution (the data). We apply continuous data assimilation to different algorithms of fluid flow, and perform numerical analysis and tests of the algorithms

Nudging the particle filter

We investigate a new sampling scheme aimed at improving the performance of particle filters whenever (a) there is a significant mismatch between the assumed model dynamics and the actual system, or (b) the posterior probability tends to concentrate in relatively small regions of the state space. The proposed scheme pushes some particles towards specific regions where the likelihood is expected to be high, an operation known as nudging in the geophysics literature. We re-interpret nudging in a form applicable to any particle filtering scheme, as it does not involve any changes in the rest of the algorithm. Since the particles are modified, but the importance weights do not account for this modification, the use of nudging leads to additional bias in the resulting estimators. However, we prove analytically that nudged particle filters can still attain asymptotic convergence with the same error rates as conventional particle methods. Simple analysis also yields an alternative interpretation of the nudging operation that explains its robustness to model errors. Finally, we show numerical results that illustrate the improvements that can be attained using the proposed scheme. In particular, we present nonlinear tracking examples with synthetic data and a model inference example using real-world financial data

Final Report of the DAUFIN project

DAUFIN = Data Assimulation within Unifying Framework for Improved river basiN modeling (EC 5th framework Project