Direct Sampling Method For Inversion Of Qupot Potential Support Based On Fixed Energy Scattering Data

Inverse scattering problem is a class of important inverse problems in the fields of acoustic,electromagnetic and quantum scattering,and studies to recover the location,shape and material properties of the scatter by using the measured near-field data or far-field data of the scattered field.In this thesis we consider the inverse scattering problem of Schr?dinger equation,where we assume that the quantum potential has compact support.We focus on studying the direct sampling method to recover the support of the potential by employing the scattered data with fixed energy.To begin with,we supplement the numerical examples for the existed study of the direct sampling method,where the numerical investigation has not been provided yet,to recover the support of the potential from the near-field data.The numerical results verify the validity of this method.In what follows,the direct sampling method to recover the support of the quantum potential from the far-field data is considered.By constructing an indicator function taking values between 0 and 1,we illustrate qualitatively that the chosen sampling point might locate inside the support of the potential if the value of the indicator function at this point is relatively large,otherwise,the sampling point is more likely to locate outside the support of the potential.The effectiveness of this method is verified by numerical examples as well.The direct sampling method involves only the calculations of the inner products,hence,is easy to be implemented numerically.Moreover,the far-field data corresponding to only one or a few incident directions are usually sufficient to obtain the satisfactory reconstructions.Furthermore,this method is not sensitive to the noise and thus is a stable inversion method.