Department of Mathematics
The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in RN, N≥2, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case.The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with N far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter h representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to h, does not depend on h. If we restrict to N=2,3 and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with N−1 measurements and then with a single measurement. In this case the dependence on h is not explicit anymore and the modulus of continuity depends on h as well.
Inverse scattering, Polyhedral scatterers, Sound-hard, Stability, Reflection principle
Source Publication Title
Journal of Differential Equations
© 2016 Elsevier Inc. All rights reserved.
The work of Hongyu Liu was supported by Hong Kong Baptist University (FRG grants), by Hong Kong RGC General Research Funds (grants No. 12302415 and 405513), and by NSFC (grant No. 11371115). Luca Rondi was partly supported by Università degli Studi di Trieste (FRA 2014 grants), and by GNAMPA, INdAM.
Link to Publisher's Edition
Liu, H., Petrini, M., Rondi, L., & Xiao, J. (2017). Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements. Journal of Differential Equations, 262 (3), 1631-1670. https://doi.org/10.1016/j.jde.2016.10.021