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Scaling limit of fluctuations for high contrast stochastic homogenization of the Helmholtz equation: second order moments

Olivier Pinaud

Abstract

This work is concerned with the high contrast stochastic homogenization of the Helmholtz equation. Our goal is to characterize the second order moments of the scaling limit of the fluctuations of the wavefield. We show that these moments are those of a random wavefield solution to a homogenized Helmholtz equation with a white noise source term and obtain expressions for its variance. Two factors contribute to the white noise: fluctuations in the inverse permittivity of the high contrast inhomogeneities, and fluctuations in their size. This problem is motivated by wave propagation in sea ice, which is a random compositive of ice and pockets of air and brine. The analysis hinges on three ingredients: a covariance formula due to Chatterjee for functions of independent random variables; small-volume expansions to quantify the fluctuations due to one inclusion; and the standard two-scale expansions for stochastic homogenization.

Scaling limit of fluctuations for high contrast stochastic homogenization of the Helmholtz equation: second order moments

Abstract

This work is concerned with the high contrast stochastic homogenization of the Helmholtz equation. Our goal is to characterize the second order moments of the scaling limit of the fluctuations of the wavefield. We show that these moments are those of a random wavefield solution to a homogenized Helmholtz equation with a white noise source term and obtain expressions for its variance. Two factors contribute to the white noise: fluctuations in the inverse permittivity of the high contrast inhomogeneities, and fluctuations in their size. This problem is motivated by wave propagation in sea ice, which is a random compositive of ice and pockets of air and brine. The analysis hinges on three ingredients: a covariance formula due to Chatterjee for functions of independent random variables; small-volume expansions to quantify the fluctuations due to one inclusion; and the standard two-scale expansions for stochastic homogenization.
Paper Structure (27 sections, 7 theorems, 171 equations, 1 figure)

This paper contains 27 sections, 7 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgment.
  3. Homogenization setting
  4. Probabilistic framework
  5. Two-scale expansion and correctors
  6. Main result
  7. Covariance formula
  8. Small-volume expansions
  9. Preliminaries
  10. Two-scale expansions for $G_\eta^{j,-}$.
  11. Expansions of $\Delta^\alpha_j u_{\eta}$
  12. The asymptotic second order moments
  13. Step 1: decomposing $J_{\eta,j}(z)$.
  14. Step 2: estimates on correctors.
  15. Step 3: averages.
  16. ...and 12 more sections

Key Result

Theorem 3.1

We have

Figures (1)

  • Figure 1: Random domain in 2D. The side of the squares is $\eta$, and only cells fully enclosed in $\mathcal{B}$ contain inclusions. The background has inverse permittivity of order $O(1)$, while that of the inclusions is of order $O(\eta^2)$.

Theorems & Definitions (12)

  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • Lemma 6.1
  • proof
  • Corollary 6.2
  • proof
  • ...and 2 more