Inversion Of The Qupot Potential Support Using A Sampling-like Algorithm

In this thesis,we consider the inverse scattering problem for stationary Schr(?)dinger equation.We will give two sampling methods for reconstructing the support of the potential from the fixed-energy scattered data.First,we consider the multilevel sampling method for reconstructing the support of the potential from the near field data.This method is a kind of iteration method.In each iteration,we refine the mesh inside the sampling domain obtained from the last iteration and renew the approximation of the support according to the criterion that if the approximation of the potential at the new grid point exceeds the new cut-off value,where the approximation of the potential is obtained from the state and field equations of the scattering problem and the backpropagation function.The numerical examples verify the viability of this method.Then,based on two factorizations of the far field operator,we study the generalized linear sampling method for reconstructing the support of the potential from the far field data.We add a penalty term to the cost functional in the linear sampling method and obtain the nearby solution of the far field equation using minimizing sequences of the new least squares cost functional.An indicator function related to the new penalty term is allowed to be constructed such that the value of the indicator function at the nearby solution is bounded is equivalent to that the sampling point is located at the support of the potential.We will give the mathematical theory of this method for the cases of noisy-free and noisy measurements,respectively.Also,we provide numerical examples to validate the effectiveness of this method.