Inverse obstacle scattering: uniqueness and reconstruction algorithms
Authors: Jun Zou
Venue:  (2007)

In this thesis, we will address two most important topics in inverse acoustic and electromagnetic obstacle scattering problems: uniqueness and reconstruction algorithms. The first part is devoted to the uniqueness issues. A detailed exposition of the background of this problem and a comprehensive discussion of the existing results are presented. The focus of this part is on our contribution to this field, especially on the unique determination of polygonal or polyhedral scatterers with a single or finitely many far-field measurements. In summary, we have shown the following results when the polyhedral type scatterers are concerned in inverse acoustic obstacle scattering: if the scatterer consists of finitely many solid polyhedral obstacles, which may be either sound-soft, sound-hard or two types mixed together, and it may also contain some crack-type obstacles but only sound-soft ones, then one can uniquely determine the scatterer by a single incident plane wave at some fixed k0 > 0 and d0 &isin; SN-1 . This statement is affirmatively verified in any dimensions whenever there is no any sound-hard obstacle present; when there is any sound-hard obstacle, the uniqueness is validated in the R2 case, but still incomplete in the RN case with N &ge; 3, which is proved to be true only by N different incident plane waves. Whenever the scatterer contains some sound-hard crack-type obstacles, we have constructed some examples to show that one cannot uniquely determine the scatterer by any less than N incident waves. So in the case with the additional presence of sound-hard crack-type obstacles, another result we have established that one can uniquely determine such a scatterer by N incident waves at any fixed wave number and arbitrary N linearly independent incident directions is optimal. We also consider more general polyhedral type scatterers with partially coated components, and some uniqueness results are established to determine the underlying physical properties. Besides, we have also collected a global uniqueness result for balls or discs. It is shown that in the resonance region, the shape of a sound-soft/sound-hard ball in R3 or a soundhard/sound-soft disc in R2 is uniquely determined by a single far-field datum measured at some fixed spot corresponding to a single incident plane wave. This seems to be an important result in the uniqueness study field as it is the first to establish the unique determination by a single far-field datum measured at one fixed spot. While all the other existing uniqueness results require far-field data observed at least in one open subset on the unit sphere with non-zero measure. To pave the way for the uniqueness study with such simple balls or discs, we also present a systematic and rather complete study of the interlacing character of the zeros for Bessel and spherical Bessel functions and their respective derivatives. Finally, all the uniqueness results for inverse acoustic obstacle scattering associated with general polyhedral scatterers have been extended to the inverse electromagnetic scattering. The second part of this thesis is concerned with the reconstruction algorithms. We will present a novel multilevel linear sampling method (MLSM) which is developed in our recent work. The new method resembles the popular multi-level techniques in scientific computing and is shown to possess the asymptotically optimal computational complexity. For an n &times; n sampling mesh-grid in R2 or an n &times; n &times; n sampling mesh-grid in R3 , the proposed algorithm only requires to solve O (nN-1)( N = 2,3) far-field equations for a RN problem, and this is in sharp contrast to the original version of the linear sampling method which needs to solve n N far-field equations instead. Numerical experiments have illustrated the promising feature of the new algorithm in significantly reducing the computational costs.