Some Research On Numerical Methods Of Inverse Cavity Scattering Problems

This thesis is concerned with the mathematical and numerical analysis on inverse cavity scattering problems for boundary or point sources with near-field data,and present the uniqueness of inverse cavity scattering problems with phaseless near-field data.All scattering problems considered are modelled by the Helmholtz equation.The first chapter of this thesis is an introduction,which mainly expounded the background and current status at home and abroad of our research topic,and briefly introduces the structure of this thesis and some preliminary knowledge involved,which includes some related concepts of acoustic scattering,the regularization methods of illposed problems and the Nystr(?)m method.In the second chapter,we introduce the inverse cavity scattering problems with a Dirichlet or a Neumann boundary condition.A direct sampling method relying on the boundary conditions is proposed for reconstructing the shape of the cavity.The scattered fields are approximated by the Fourier-Bessel functions and the indicator functions are established by the superpositions of the total fields or their derivatives.We proved that the indicator functions vanish only on the boundary of the cavity.Finally,numerical results of two kinds of boundary conditions are presented to demonstrate the viability and effectiveness of the proposed method.In the third chapter,we introduce the co-inversion problem of inverse cavity scattering of determining the boundary of the cavity and the corresponding location of incident point sources with the measured total field data.Owing to the incident field and the scattered field superimposed in the measured total field information,the single-layer potential method is used to decouple the two field and the optimization method and direct sampling method is used to reconstruct the shape of cavity and determine the location of the point source respectively.Moreover,a new sampling scheme is proposed and we analyzed the mathematical theory of the optimization method and indicator functions.Several numerical examples are provided to illustrate the effectiveness and stability of the proposed algorithm.In the fourth chapter,with the superpositions of incident point sources,the aid of the reference ball and the introduction of admissible surface,we rigorously prove that the location and shape of the cavity as well as its boundary condition can be uniquely determined by phaseless near-field data at an admissible surface.The last chapter of this thesis is the conclusion.