Numerical Solutions Of Inverse Scattering Problems Based On Statistical Computing Methods

With the improvement of the computer performance and the practical needs of engineering applications,the study on the inverse scattering problems is no longer limited to the theoretical analysis.More often,one expects to get effective numerical methods to simulate the solution of the inverse scattering problems.However,compared with the forward scattering problem,the inverse scattering problem is often ill posed,i.e.,the existence,uniqueness and stability of the solution are not always available.From the numerical perspective,the methods for inverse scattering problem are often quite different from those for the direct scattering problem.In this dissertation,we will study several inverse scattering problems in acoustics and elastics using statistical methods.The main contents and contributions are stated as follows:1.A new technique combining the extended sampling method(ESM)and the Bayesian approach is proposed to reconstruct the location and shape of a sound-soft obstacle in acoustics with limited aperture data.The inverse problem is reformulated as a statistical model by Bayes' formula.The well-posedness is proved and a Markov chain Monte Carlo(MCMC)algorithm is proposed to explore the posterior probability distribution.It is critical to know the location of the target for the convergence of the MCMC algorithm.To this end,we resort to the extended sampling method.The ESM-Bayesian method has two steps: i)ESM to obtain the location of the target,and ii)Bayesian method to reconstruct the boundary.Both steps are based on the same physical model and use the same set of measured data.2.A deterministic-statistic approach by combining the ESM and the ensemble Kalman filter(En KF)is proposed to solve an inverse elastic obstacle scattering problem.As the first step,we propose the ESM to reconstruct the approximate location of the obstacle.In order to make the algorithm more suitable for the limited aperture data,we improved the ESM on the basis of some existing literatures.In the second step,the En KF is used to optimize the location obtained by the ESM,and the shape of the unknown obstacles is reconstructed using the same set of measurement data.The approximate location obtained by the ESM is used to construct the initial particles of the En KF.Numerical experiments show that this method inherits the advantages of the two methods,and it can effectively recover the obstacle using limited aperture data.3.Inspired by the ideas of the previous two chapters,in this chapter,we propose a high-quality method for an inverse acoustic source problem by combining the direct sampling method(DSM)and the Bayesian approach.In the first step,the DSM is used to retrieve the number and locations of the unknown sources.In the second step,the Bayesian approach is used to recover the detailed information of the sources.In the Bayesian framework,all parameters in the model are regarded as random variables.The number and locations of the sources obtained in the first step are encoded in the prior information of the Bayesian approach.Then,the posterior distribution of the unknown parameters is obtained by using Bayes' formula.The well-posedness of the algorithm is proved,and the preconditioned Crank Nicolson Metropolis Hastings(p CN-MH)algorithm is used to solve the posterior density function.Numerical experiments show that the proposed method is robust and can recover the number,locations and intensities of the unknown sources effectively under limited observation data.4.An inverse medium scattering problem of identifying the geometry of the interface of penetrable layered scatterers in the homogeneous medium background is considered.The ensemble Kalman method is used as an inversion solver.In order to deal with the topological changes of the scatterer interface flexibly,we introduce the level set technique into the En KF algorithm.In the whole inversion process,the Fréchet derivative of the forward operator and its adjoint does not need to compute.This is obviously an advantage compared with the traditional gradient type algorithm.Numerical experiments show the effectiveness and flexibility of the algorithm.The algorithm can deal with not only one scatterer,but also multiple scatterers and ring-type scatterers.5.An inverse scattering problem of recovering the shape of a cavity in an elastic medium background is considered in this chapter.Due to the high computational cost of the direct scattering problem,if the MCMC algorithm is considered as an inversion solver,it will inevitably lead to a large amount of calculations of the likelihood potential function in the Bayes' formula,which will make the whole inversion process very time-consuming.In order to improve the inversion efficiency,we use the Kriging surrogate model to replace the likelihood potential function in the Bayes' formula.Numerical experiments show that the Kriging surrogate model can not only provide reasonable calculation accuracy,but also greatly shorten the inversion time of the MCMC algorithm.