Bayesian Method For The Inverse Scattering Problem By The Obstacle Inside The Medium

Bayesian method has been born for more than 300 years,and it is still a dynamic research field.Bayesian method which has been widely applied in real life combines the prior information with likelihood function,and compared to classical statistical methods Bayesian method is more flexible.Especially in the inverse scattering problem of optics and electronics data analysis,Bayesian method has attracted more and more attentions in recent years.Inverse scattering problem is a very important mathematical and physical problem,due to the need of engineering applications,the solutions of various complex inverse scattering problems are becoming more and more important.However,the inverse scattering model that is more complex often brings stronger nonlinearity and ill-posedness,which makes that the study of this kind of problem is mathematically and practically significant.Therefore,this paper is based on statistical Bayesian theory and applies Bayesian theory to inverse scattering problems with medium.1)In this paper,Bayesian method is proposed to reconstruct single obstacle and multiple obstacles.The acoustic inverse scattering problem is transformed into solving the posterior probability distribution of the unknown variables by Bayesian formula,and the well-posedness of the posterior probability measure in Bayesian method is proved in the case of multiple obstacles.For the case of single obstacle,numerical experiments show that knowing the approximate location of the obstacle in advance is advantageous to improve the reconstruction effect and convergence speed of Bayesian method,which plays an important role in the reconstructions of multiple obstacles.When the structure of the model consists of multiple obstacles,the complexity of the inverse scattering problem will be greatly improved,which will greatly affect the convergence speed of Markov Chain Monte Carlo method.Therefore,we first employ the linear sampling method to obtain the number,approximate positions and shapes of multiple obstacles,and then use the Bayesian method to reconstruct the quantitative and more refined results based on the recoveries of the linear sampling method.Numerical experiments show that this idea not only fully inherits the advantages of these two methods,but also speeds up the convergence speed of Bayesian method.2)An inverse obstacle scattering problem in a layered medium is considered.Because too many layers inside the medium will bring a lot of research difficulties and time-consuming calculation,this paper is mainly concerned with the inverse scattering problem of single obstacle and multiple obstacles in the two-layered medium.When there is only one obstacle in the two-layered medium,we use Bayesian method to simultaneously reconstruct the external interface and the boundary of the embedded obstacle.When there are many obstacles in the two-layered homogeneous medium,this paper first uses the modified linear sampling method to obtain the number,approximate locations and shapes of multiple obstacles inside the two-layered medium,then employs Bayesian method to obtain more detailed reconstruction results.Numerical experiments show that the proposed method is effective.3)The Bayesian method is studied to solve the inverse scattering problem of a two-layered cavity.The main purpose is to reconstruct the internal interface of the cavity from the scattered field data.Bayesian method considers the acoustic inverse scattering problem as a statistical problem,that is solving the posterior distribution of unknown variables.Considering that the prior of unknown variables is Gaussian distribution,the well-posedness of a posteriori probability measure is proved in this paper.Numerical experiments show that Bayesian method is feasible.