Effective models and numerical homogenization for wave propagation in heterogeneous media on arbitrary timescales

Abdulle, Assyr; Pouchon, Timothée Noé

doi:10.48550/arxiv.1905.09062

Cited by 3 publications

(11 citation statements)

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“…Similar formulas have been obtained for the wave equation in heterogeneous media, see e.g. Theorem 3.5 in [2], and also [1,48].…”

Section: The Traditional Ansatz: Definition Of the Cell Tensors X Ksupporting

confidence: 64%

“…is in general not well-posed [9,1]. Several techniques have been proposed in the literature to obtain well-posed homogenized models of finite order in the context of the conductivity or of the wave equation [13,10,1,2,12]. The derivation of the well-posed homogenized equation (1.6) relies on a minimization principle inspired from Smyshlyaev and Cherednichenko [52] and is marked by two surprising facts.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, N 0 is of average Y N 0 (y)dy = 1 and N k is of average Y N k (y)dy = 0 for k ≥ 1 (proposition 8): u * ε is consistently the average of u ε with respect to the fast variable x/ε. While the derivation of (1.7) is very standard in periodic homogenization [50,41,18,14], the existence of such a relation (1.11) (when compared to (1.7)) between the oscillating solution is less obvious; it has been noticed for the first time by Bakhvalov and Panasenko [15] for the conductivity equation, and then in further homogenization contexts in [51,52,23,12,2,1]. Following the denomination of [12], we call the ansatz (1.11) "criminal" because the function u * ε has the structure of a formal power series in ε.…”

Section: Introductionmentioning

confidence: 99%

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10 High-order homogenization of the Poisson equation in a perforated periodic domain

Feppon¹

2022

Optimization and Control for Partial Differential Equations

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We derive high order homogenized models for the Poisson problem in a cubic domain periodically perforated with holes where Dirichlet boundary conditions are applied. These models have the potential to unify the three possible kinds of limit problems derived by the literature for various asymptotic regimes (namely the "unchanged" Poisson equation, the Poisson problem with a strange reaction term, and the zeroth order limit problem) of the ratio η ≡ aε/ε between the size aε of the holes and the size ε of the periodic cell. The derivation relies on algebraic manipulations on formal two-scale power series in terms of ε and more particularly on the existence of a "criminal" ansatz, which allows to reconstruct the oscillating solution uε as a linear combination of the derivatives of its formal average u * ε weighted by suitable corrector tensors. The formal average is itself the solution of a formal, infinite order homogenized equation. Classically, truncating the infinite order homogenized equation yields in general an ill-posed model. Inspired by a variational method introduced in [52, 23], we derive, for any K ∈ N, well-posed corrected homogenized equations of order 2K + 2 which yields approximations of the original solutions with an error of order O(ε 2K+4 ) in the L 2 norm. Finally, we find asymptotics of all homogenized tensors in the low volume fraction regime η → 0 and in dimension d ≥ 3. This allows us to show that our higher order effective equations converge coefficient-wise to either of the classical homogenized regimes of the literature which arise when η is respectively equivalent, or greater than the critical scaling η crit ∼ ε 2/(d−2) .

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“…Similar formulas have been obtained for the wave equation in heterogeneous media, see e.g. Theorem 3.5 in [2], and also [1,48].…”

Section: The Traditional Ansatz: Definition Of the Cell Tensors X Ksupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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10 High-order homogenization of the Poisson equation in a perforated periodic domain

Feppon¹

2022

Optimization and Control for Partial Differential Equations

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“…i) As shown in [3] the Boussinesq trick can be generalized to arbitrary order M ∈ N. This fact can be used to derive a higher-order effective optimal control problem that is well-posed independently of ε.…”

Section: A Well-posed Effective Problemmentioning

confidence: 97%

“…which is well-posed independently of the choice of ε. In the recent publication [3] the Boussinesq trick has been generalized to arbitrary order M > 2 using an induction argument.…”

Section: A Well-posed Effective Problemmentioning

confidence: 99%

High-order homogenization in optimal control by the Bloch wave method

Lamacz-Keymling¹,

Yousept²

2020

Preprint

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This article examines a linear-quadratic elliptic optimal control problem in which the cost functional and the state equation involve a highly oscillatory periodic coefficient A ε . The small parameter ε > 0 denotes the periodicity length. We propose a high-order effective control problem with constant coefficients that provides an approximation of the original one with error O(ε M ), where M ∈ N is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal control problems via the corresponding variational inequalities.

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Higher-order pathwise theory of fluctuations in stochastic homogenization

Duerinckx

Otto

2019

Stoch PDE: Anal Comp

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We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order d 2 , where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the "homogenization commutator", thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón-Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

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Effective models and numerical homogenization for wave propagation in heterogeneous media on arbitrary timescales

Cited by 3 publications

References 0 publications

10 High-order homogenization of the Poisson equation in a perforated periodic domain

10 High-order homogenization of the Poisson equation in a perforated periodic domain

High-order homogenization in optimal control by the Bloch wave method

Higher-order pathwise theory of fluctuations in stochastic homogenization

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