Studies On The Numerical Methods For Time Domain Scattering And Inverse Scattering Problems With Unbounded Scatterers

We are concerned with several time domain scattering and inverse scattering prob-lems with particular unbounded scatterers. Numerical methods are established to solve the forward and inverse problems and related analyses are given. The scattering problem is mainly about the scattering of waves by scatterers. The forward scattering problem is:given the incident field (acoustic wave or electromagnetic wave) and the informa-tion of the scatterer, find the scattered field or far field. The inverse scattering problem is to recover the scatterer with the given incident field and a partial knowledge of the scattered field or far field. Among all kinds of scattering problems, we are concerned with the scattering of acoustic waves by impenetrable scatterers. We develop our anal-ysis in time domain, which means the wave field is time-dependent non-harmonic and satisfies the wave equation. In comparison to frequency domain problems, time domain problems arise more naturally in diverse application areas such as geophysical explo-ration, medical imaging and nondestructive testing. Moreover, dynamic impulsive data is usually easier to obtain in reality and the information content of a temporal signal is usually much greater than that of the data available with only one or a few discrete frequencies.To solve the forward scattering problem, we use the boundary integral equation method based on the time domain layer potentials. The time domain potentials are de-veloped on the basis of the Green's function of the wave equation. With the analysis of the jump relations of the potentials on the boundary of the scatterer, we can get a representation formula of the solution to the time domain scattering problem and give the boundary integral equation. The time domain potentials are associated with a re-tarded time, thus the boundary integral equation we get is called the retarded potential boundary integral equation (RPBIE). In this thesis, a "first kind" RPBIE based on the single layer potential is employed and the RPBIEs may vary from problem to problem. In the numerical computation, the time discretization of the RPBIE is usually not given directly. We use the convolution quadrature (CQ) method for the time discretization due to the convolution structure of the RPBIE. Then the time domain problem is trans-formed to a system of Helmholtz problems and the spatial discretization can be applied to the Helmholtz problems for the numerical computation.To solve the inverse problems, we consider the linear sampling method (LSM) and a Newton type iterative approach based on the RPBIE for the forward problem. The basic idea of the LSM is reformulating the nonlinear ill-posed inverse scattering problem as a linear Fredholm integral equation of the first kind, which is known as the near field equation in time domain. The reconstruction of the scatterer with the LSM is based on the "blow-up" property of the near field equation, which is, the solution of the near field equation is bounded in the region of the scatterer and "blows up" to infinity when it gets to the outside region. The iterative method is a classic method to solve the inverse scattering problem. In theory, the reconstruction of the scatterer will be close enough to the exact scatterer after enough iteration steps. In this thesis, on the basis of the RPBIE for the forward problem and the Newton's method, we can get a iterative equation. Then, with a chosen initial data, we can get a better reconstruction of the scatterer in each iteration step.The main work of this thesis is as follows:1. We discuss the numerical methods for the time domain forward and inverse scat-tering problems with a class of unbounded scatterers:the locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalen-t symmetric problem defined in the whole plane. For the forward problem, we restrict the analysis of the symmetric scattering problem to the half space and redefine the time domain potentials with the half-space Green's function. Thus we get the RPBIE defined in the half space and give the unique solvability of the RPBIE. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain LSM is employed to deal with the inverse problem. To have our computation in the half space, we redefine the near field equation using the symmetric property of the scattering problem and prove the "blow-up" property of the new-defined near field equation. The computation schemes we proposed are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.2. We study the three dimensional extension of the locally perturbed half-plane, that is, the local perturbation in the three-dimensional space. For the forward problem, we attempt to tackle the scattering problem in the unbounded domain directly and anal-yse the integral equation defined on the unbounded boundary of the scatterer. We use the time domain single layer potential which is defined on the basis of the half-space Green's function to get the RPBIE. Then we prove that the RPBIE defined on the un-bounded boundary is equivalent to an integral equation defined on the bounded support of the integral kernel and give the unique solvability of the RPBIE on the unbounded boundary. To solve the inverse problem, we still use the time domain LSM and give the "blow-up" property of the near field equation in the three dimensional case.3. We consider the time domain scattering and inverse scattering problems of open cavities. The scatterer is a half plane with a locally low-lying part and the locally low-lying part is known as the open cavity. To solve the forward problem, we give a trans-parent boundary condition on the opening of the open cavity. Then we can get an initial boundary value problem in the region of the open cavity. The boundary conditions on the wall and the opening of the open cavity are different. Using the method of integral transform, we give the unique solvability of the forward problem in the sense of weak solution. On the basis of the boundary conditions, using the representation formula of the solution to the time domain scattering problem, we get the RPBIEs on the boundary of the cavity region and give the CQ method for the time discretization. For the inverse problem, based on the RPBIEs for the forward problem, we get a Newton type itera-tive method for the inverse problem after the analysis of the Frechet derivatives of the operators in the RPBIEs.These are our main researches in recent years and constitute the main part of this thesis. However, the research of the time domain scattering and inverse scattering prob-lems with unbounded scatterers is not over yet and we have much more work to do in the future. Moreover, we are also interested in other time domain scattering problems and briefly mention some in this thesis which would possibly lead to our future work.