We consider the two scale asymptotic expansion for a transmission problem modeling scattering by a bounded inhomogeneity with a periodic coefficient in the lower order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size . Since the convergence of the boundary correctors to their limits is in general slow, we explore in detail their use in a second order approximation and show a new convergence estimate for the second order boundary corrector on a square. We show numerical examples of the higher order forward approximation in one and two dimensions. We then use the first order boundary correction as an asymptotic model for inversion and show numerical examples of inversion in the two dimensional case.

中文翻译：

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周期性微观结构和逆均质化系列的限制边界校正器。

我们考虑了传输问题的两个尺度渐近展开式，该问题通过亥姆霍兹方程的低阶项中具有周期性系数的有界不均匀性来建模散射。折射率的平方被假定为快速变量的周期函数，在具有特征尺寸 的晶胞上指定。由于边界校正器收敛到它们的极限通常很慢，我们详细探讨了它们在二阶近似中的使用，并展示了正方形上二阶边界校正器的新收敛估计。我们展示了一维和二维高阶正向逼近的数值例子。然后我们使用一阶边界校正作为反演的渐近模型，并展示二维情况下反演的数值例子。