Title: Stable determination of sound-soft polyhedral scatterers by a single measurement

Authors: Luca Rondi

Issue: Volume 57 (2008), Issue 3, 1377-1408

Abstract: We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in $\mathbb{R}^3$ by a single far-field measurement. The admissible multiple polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time obstacles, screens and even more complicated scatterers. We characterize any multiple polyhedral scatterer by a size parameter $h$ which is related to the minimal size of the cells of its boundary. In a first step we show that, provided the error $\varepsilon$ on the far-field measurementis small enough with respect to $h$, then the corresponding error, in the Hausdorff distance, on themultiple polyhedral scatterer can be controlled by an explicit function of $\varepsilon$ which approaches zero, as $\varepsilon \rightarrow 0^{+}$, in an essentially optimal, although logarithmic, way. Then, we show how to improve this stability estimate, provided we restrict our attention to multiple polyhedral obstacles and $\varepsilon$ is even smaller with respect to $h$. In this case we obtain an explicit estimate essentially of H\"{o}lder type.