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- 1 décembre 2021
- Maria Cabrera Calvo (LJLL)
*Highly oscillatory integrators at low regularity for the Klein-Gordon equation*We propose a novel class of uniformly accurate integrators for the Klein–Gordon equation which capture classical $c = 1$ as well as highly-oscillatory non-relativistic regimes $c \gg 1$ and, at the same time, allow for low regularity approximations. In particular, our first- and second-order schemes require no step size restrictions and lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods. The new schemes in addition preserve the nonlinear Schrodinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c \to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.

This is joint work with Katharina Schratz (Sorbonne University). - 8 décembre 2021
- Elisabetta Brocchieri (Paris Saclay - UEVE)
*About entropy methods to a dietary diversity cross-diffusion system*Cross-diffusion systems are non-linear parabolic systems, modelling the evolution of densities or concentrations of multicomponent populations in interaction. They may be derived by random walk on lattices, in a microscopic scaling, or as limit of linear parabolic diffusion systems, at a mesoscopic level. In this talk, we propose the rigorous passage from a weak competitive reaction-diffusion system towards a reaction cross diffusion system, in the fast reaction limit. The resulting limit system shows a starvation driven cross-diffusion term. The main ingredients used to prove the existence of global solutions are an energy functional and a compactness argument. However, the analysis of an appropriate family of energy functionals allows to improve the regularity of the solution. Furthermore, we also investigate the linear stability of homogeneous steady states of those systems and rule out the possibility of Turing instability. Then, no pattern formations occur. To conclude, numerical simulations are included, proving the compatibility with the theoretical results.

Reference:

B., E., Corrias, L., Dietert, H. and Kim, Y-J. Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit. J. Math. Biol. 83, 58 (2021). https://doi.org/10.1007/s00285-021-01679-y - 5 janvier 2022
- Willy Haik (LJLL et LMT)
*A real-time variational data assimilation method with model bias identification and correction*Real-time monitoring on a physical system by means of a model-based digital twin may be difficult if occurring phenomena are multiphysics and multiscale. A main difficulty comes from the numerical complexity which is associated to an expensive computation hardly compatible with real-time. To overcome this issue, the high-fidelity parameterized physical model may be simplified which adds a model bias. Moreover, the parameter values can be inaccurate, and one part of the physics may be unknown. All those errors affect the effectiveness of numerical diagnostics and predictions and need to be corrected with assimilation techniques on observation data. Therefore, the monitoring of a process occurs in two stages: (1) the state estimation at the acquisition time which may be associated with an identification of the set of unknown parameters of the parametrized model and an update state which enriches the model; (2) a state prediction for future time steps with the updated model.

The present study is mainly denoted to perform the state estimation using an extension, for time-dependent problems, of the Parameterized Background Data-Weak (PBDW) method introduced in [1]. This method is a non-intrusive, reduced basis and in-situ data assimilation framework for physical systems modeled by parametrized Partial Differential Equations initially designed for steady-state problems. The key idea of the formulation is to seek an approximation to the true field employing projection-by-data, with a first contribution from a background estimate computed from a reduced-order method (ROM) enhanced on-the-fly, and a second contribution from an update state informed by the experimental observations (correction of model bias). Further research works [2,3] developed an extension to deal with noisy data and a nonlinear framework. Moreover, a priori error analysis was conducted by providing a bound on the state error and identifying individual contributions. In the present work, the state prediction for future time steps is also performed from an evaluation of the updated model and an extrapolation of the time function from the tensor-based decomposition (SVD) on prior updates.

Numerical experiments are conducted on a thermal conduction problem in the context of heating on a Printed Circuit Board (PCB) with different cases of model bias: a bias on heat source, a biased boundary condition and an error on the constitutive equation. These numerical experiments show that the method significantly reduces the online computational time while providing relevant state evaluations and predictions. We thus illustrate the considerable improvement in prediction provided by the hybrid integration of a best-knowledge model and experimental observations.

[1] Maday, Y., Patera, A. T., Penn, J. D., and Yano, M. (2015). A parameterized background dataweak approach to variational data assimilation: formulation, analysis, and application to acoustics. International Journal for Numerical Methods in Engineering, 102(5), 933-965.

[2] Yvon Maday and Tommaso Taddei (2017) Adaptive pbdw approach to state estimation: noisy observations; user-defined update spaces. arXiv preprint arXiv:1712.09594.

[3] Gong, H., Maday, Y., Mula, O., and Taddei, T. (2019). PBDW method for state estimation: error analysis for noisy data and nonlinear formulation. arXiv preprint arXiv:1906.00810.

- 16 février 2022
- Lucas Journel (LJLL) A valider
*TBA*TBA

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Pour tout renseignement sur le GTT, contacter: Matthieu Dolbeault (dolbeault[at]clipper.ens.psl.eu), Noemi David (noemi.david[at]ljll.math.upmc.fr), Pierre Le Bris (lebris[at]ljll.math.upmc.fr) et Maria Cabrera Calvo (cabreracalvo[at]ljll.math.upmc.fr)