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

We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in ℝ3 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 ε on the far-field measurement is small enough with respect to h, then the corresponding error, in the Hausdorff distance, on the multiple polyhedral scatterer can be controlled by an explicit function of ε which approaches zero, as ε → 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 ε is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Hölder type.