Effective approximations of solutions to highly oscillatory diffusion equations from coarse measurements

Abstract

We approximate a diffusion equation with highly oscillatory coefficients with a diffusion equation with constant coefficients. The approach is put in action in contexts where only partial information (namely the global energy stored in the physical system) is available. While the reconstruction of the microstructure is known to be an ill-posed problem, we show that the reconstruction of effective coefficients is possible and this even with only some coarse information. The strategy we present takes the form of a non-convex optimization problem. Homogenization theory provides elements for a rigorous foundation of the approach. Some algorithmic aspects are discussed in details. We provide a comprehensive set of numerical illustrations that demonstrate the practical interest of our strategy. The present work improves on the earlier works [C. Le Bris, F. Legoll and S. Lemaire, COCV 2018; C. Le Bris, F. Legoll and K. Li, CRAS 2013].

Figures (12)

• Figure 1: In the one dimensional setting, we plot $\overline{A} \in (0, +\infty) \rightarrow \Psi_\varepsilon(\overline{A})$ with $A_\varepsilon(x) = 2 + \cos(2\pi x/\varepsilon)$ and $\varepsilon = 10^{-3}$.
• Figure 2: Convergence history of the algorithm for $\varepsilon=0.05$ and $P=3$.
• Figure 3: Component $11$ of $A_{\text{per}}$ defined in \ref{['eq:A11A22A12per']} and displayed on the periodic cell $(0,1)^2$.
• Figure 4: Two realizations of $a_{\text{rand}}(\cdot, \omega)$ restricted to the square $(-20, 20)^2$.
• Figure 5: Error \ref{['eq:errorAB']} between the homogenized coefficient $A_\star$ and the effective coefficients $\overline{A}_\varepsilon^{\text{ME}}$, $\overline{A}_\varepsilon^{\text{MS}}$ and $\overline{A}_\varepsilon^{\text{MV}}$, as a function of $\varepsilon$.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Proposition 3: Convergence of $I_\varepsilon$ to $0$
• proof : Proof of Proposition \ref{['prop:Iepsto0']}
• Proposition 4: Existence of minimizers
• proof : Proof of Proposition \ref{['prop:exist_min']}
• Proposition 5: Convergence of minimizers
• Remark 6
• Lemma 7
• proof : Proof of Lemma \ref{['lemma:lemma1']}.