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 ${ℝ}^d$. In practice, one can only consider a truncated version of these corrector problems, which is next discretized using e.g. a Finite Element method. The previous analysis has been extended to this practical situation by R. Costaouec [31] .

In the work mentioned above, the perturbation to the deterministic case is supposed to be small in the $L^{\infty }$ norm (that is, almost surely small). The team has also considered the case when the perturbation is small in a weaker norm, typically a $L^p$ norm with $p<\infty$ (the case when only the expectation of the perturbation is assumed to be small, rather than the perturbation itself, is covered by that framework). The approach proves to be very efficient from a computational viewpoint, in comparison to the standard approach of stochastic homogenization, as shown in [13] , [12] . In that setting, the computation of the homogenized matrix requires repeatedly solving a corrector-like equation for various configurations of the material. For this purpose, C. Le Bris and F. Thomines have shown how to adapt the Reduced Basis approach to the specific context, to even further reduce the computational cost [62] .

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 ${ℝ}^d$. This equation is therefore delicate and expensive to solve. In practice, the space ${ℝ}^d$ is truncated to some bounded domain, on which the corrector problem is numerically solved. In turn, this yields a converging approximation of the homogenized tensor, which happens to be a random matrix. For a given truncation of ${ℝ}^d$, R. Costaouec, C. Le Bris and F. Legoll, in collaboration with X. Blanc (CEA), have studied how to reduce the variance of this matrix, using the technique of antithetic variables, which is a method widely used in other fields of application. Its efficiency in the context of stochastic homogenization has been extensively studied, both numerically and theoretically [60] , [19] . R. Costaouec, C. Le Bris and F. Legoll are currently investigating the possibility to use other variance reduction approaches, such as control variate techniques.

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 $Lu=-\text{div}\left(A\nabla u\right)$, where $A$ is a periodic, coercive and bounded matrix. In collaboration with X. Blanc (CEA) and A. Anantharaman, F. Legoll has established some results of this question [11] . This contribution presents in a unified manner and complements several results already given in the literature.

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.