Special quasirandom structures: a selection approach for stochastic homogenization (Q254494)

The authors propose and investigate a variance reduction approach for the homogenization of elliptic equations in divergence form NEWLINE\[NEWLINE-\operatorname{div}\left(A\left({\cdot\over\varepsilon},\omega\right)\nabla u^{\varepsilon}(\cdot,\omega)\right)=f \text{ in } D\subset\mathbb R^d, \quad u^{\varepsilon}(\cdot,\omega)=0\text{ on } \partial D,NEWLINE\]NEWLINE where the domain \(D\) is bounded and regular, \(f\in H^{-1}(D)\) is deterministic function, the field \(A\) is a fixed matrix-valued uniformly elliptic, uniformly bounded and stationary in a discrete sense random field and \(\varepsilon>0\) is a small parameter. Let us consider the corrector problem \(-\operatorname{div}[A(p+\nabla w_{p})]=0\) in \(\mathbb R^d\) almost surely with \(\int_{Q}\mathrm{E}(\nabla w_{p})=0\), \(Q=(0,1)^d\) and \(\nabla w_{p}\) stationary. The solution to the corrector problem gives the deterministic and constant coefficient \(A^*\) of the homogenized equation that in turn serves as the approximation of the given problem for the elliptic equation. The deterministic homogenized matrix \(A^*\) is approximated by the random variable \(A^*_{N}(\omega)\) defined by NEWLINE\[NEWLINEA^*_{N}(\omega)p={1\over| Q_{N}|}\int_{Q_{N}}A(\cdot,\omega)(p+\nabla w_{p}^{N}(\cdot,\omega)) \quad \text{for all } p\in\mathbb R^d,\quad Q_{N}=(0,N)^d.NEWLINE\]NEWLINE The main theoretical result of this paper is the following. Let \((X_{k})_{k\in\mathbb Z^d}\) be a sequence of i.i.d. scalar random variables following a common law \(\mu\). Let us assume that \(\mu\) is absolutely continuous with respect to the Lebesgue measure on \(\mathbb R\), and that, for any \(k\in\mathbb Z^d\), \(X_{k}(\omega)\in [-1,1]\) almost surely. Let us consider the stationary random field NEWLINE\[NEWLINEA(y,\omega)=C_0+\sum\limits_{k\in\mathbb Z^d}X_{k}(\omega){\mathbf 1}_{Q+k}(y)C_1(y),NEWLINE\]NEWLINE where \(C_0\) is constant and \(C_1\) is \(\mathbb Z^d\)-periodic and bounded. If \(C_0+C_1(y)\) and \(C_0-C_1(y)\) are uniformly coercive, \(C_0\) and \(C_1\) are symmetric, and let \(f:\mathbb R\to\mathbb R, f\neq \text{ constant}\), be a measurable function with compact level sets. Then, NEWLINE\[NEWLINE\mathrm{E}\left[A^*_{N}\left|{1\over| Q_{N}|}\sum\limits_{k\in Q_{N}\cap\mathbb Z^d}f(X_{k})=\mathrm{E}[f(X_0)]\right.\right]\to A^*,NEWLINE\]NEWLINE as \(N\to\infty\), where NEWLINE\[NEWLINE A^* p=\mathrm{E}\left[\int_{Q}A(x,\cdot)(p+\nabla w_{p}(x,\cdot))dx\right], \text{ for all } p\in\mathbb R^d.NEWLINE\]NEWLINE The conclusions from the presented numerical tests are the following. The systematic error is kept approximately constant by the approach, while the variance is reduced by several orders of magnitude.