Section: New Results
Homogenization
Participants : Ronan Costaouec, Claude Le Bris, Frédéric Legoll, Francis Nier, Florian Thomines.
The project-team has pursued its efforts in the field of stochastic homogenization of elliptic equations. The various contributions of the team, which aim at designing numerical approaches that both are pratically relevant and keep the computational workload limited, have been presented from a unified perspective in [59] .
An interesting case in that context is when
the randomness comes as a small perturbation of the deterministic
case. As previously shown by former works of the project-team, this situation can indeed be
handled with a dedicated approach, which turns out to be far more
efficient than the standard approach of stochastic homogenization. This
previous analysis was performed manipulating the exact correctors,
solutions to PDEs posed on
In the work mentioned above, the perturbation to the deterministic case
is supposed to be small in the
The team has also proceeded to address, from a numerical
viewpoint, the case when
the randomness is not small. In that case, using the standard
homogenization theory, one knows that the homogenized tensor, which a
deterministic matrix, depends on the solution of a stochastic equation,
the so-called corrector problem, which is posed on the whole space
From a numerical perspective, the Multiscale Finite Element Method is a classical strategy to address the situation when the homogenized problem is not known, or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as a sufficiently accurate model). The extension of this strategy to the stochastic case, when the tensor describing the properties of the material is the sum of a periodic term and a small random term, has been studied by C. Le Bris, F. Legoll and F. Thomines [46] . A method with a much smaller computational cost than the original MsFEM in the stochastic setting has been proposed. Provided the stochastic perturbation is indeed small, the proposed method is as accurate as the original one. The work [46] also provides a complete analysis of the approach, extending that available for the deterministic setting. Such an analysis often relies on the rate of convergence of the two scale expansion (in the sense of homogenization theory) of the solution to the highly oscillatory elliptic partial differential equation. The result is classic for periodic homogenization. In generic stochastic homogenization, the rate can be arbitrary small, depending on the rate with which the correlations of the random coefficient vanish. In [47] , such a result has been established for weakly stochastic homogenization. This result is a key ingredient for the numerical analysis of the MsFEM approach proposed in [46] .
Still in the framework of the Multiscale Finite Element approach, F. Thomines has further investigated, in collaboration with Y. Efendiev and J. Galvis (Texas A&M University), the use of Reduced Basis methods. They have considered an extension of the MsFEM approach, well suited to the high contrast case, i.e. the case when the ratio between the maximum and the minimum values of the heterogeneous coefficient is large. The main idea of this extension is to complement the standard MsFEM basis functions with the eigenfunctions (associated to the first small eigenvalues) of a local eigenproblem. In [39] , Y. Efendiev, J. Galvis and F. Thomines have considered the case when the problem depends on an additional parameter, and shown how to use the Reduced Basis approach to more efficiently compute the eigenfunctions mentioned above.
The theoretical results obtained by the team on variance
reduction [19] and on the rate of
convergence of the two scale expansion of the solution to a highly
oscillatory, weakly random PDE [47] , both rely on asymptotic
properties of the Green function of the elliptic operator
All the works previously mentioned are concerned with elliptic PDEs. F. Nier has studied various problems in the context of wave propagation in random heterogeneous media. In collaboration with S. Breteaux (PhD student in Rennes), he has derived a Boltzmann type equation from first principles of quantum mechanics, using the bosonic QFT presentation of Gaussian random fields. Various extensions of this result are currently under investigation.