Bayesian Approach For Inverse Scattering Problems In Acoustics And Elasticity

Inverse scattering problems have attracted much attention due to their wide applica-tions in areas of science and engineering.These problems are usually ill-posed admitting great theoretical and computational challenges.In many practical applications,because of the non-negligible errors,uncertainties and limited information of the observation data,the deterministic method usually cannot deal with the uncertainty of the solution to the inverse problem,while the Bayesian method reformulates the inverse problem into a statistical in-ference problem and provides a systematic framework to quantify the uncertainty of the solutions.Based on the Bayesian method,this thesis focuses on several kinds of acous-tic and elastic inverse scattering problems,including inverse interior scattering problems,inverse transmission scattering problems,inverse medium scattering problems in acous-tics,and inverse medium scattering problems in elasticity.The research content of this dissertation is divided into the following four parts:In chapter 2,we focus on the Bayesian method for solving an inverse interior scat-tering problem.The main purpose is to reconstruct the shape of the scatterer from the limited-aperture observational data.The Bayesian approach reformulates the problem into a statistical problem,and the solution of the inverse problem is a posterior distribution by using Bayes' formula.Considering the Gaussian prior,we could prove the well-posedness of the posterior distribution.The Markov chain Monte Carlo sampling method is usually used to extract posterior distribution information,and numerical results demonstrate the effectiveness of the proposed method.In chapter 3,we investigate the Bayesian method for solving a two-dimensional time-harmonic acoustic inverse scattering problem from a penetrable obstacle,namely,the in-verse transmission scattering problem.For the inverse transmission scattering problem,its main purpose is to reconstruct the shape of the scatterer from the full-aperture data and the limited-aperture data.Based on Chapter 2,the well-posedness of the posterior distribution could be proved under the Gaussian prior.Finally,the Markov chain Monte Carlo method will be applied to extract samples of the posterior distribution,and numerical results show the effectiveness of the proposed method.In chapter 4,we mainly study the Bayesian level set method for solving an inverse medium scattering problem in acoustics.Suppose that the scatterer is a piecewise constant function with known values,and its support is represented by level set functions.Then the inverse problem is reformulated into a statistical problem of shape reconstruction by combining the Bayesian method with level set method from multi-frequency data.Con-sidering the Whittle-Matérn Gaussian random field as the prior distribution,in this case the discontinuous set of the level set has zero Lebesgue measure,almost surely.Thus,the Bayes' theorem would be used to prove the well-posedness of the posterior distribution.Finally,the samples of the posterior distribution are drawn by the Markov chain Monte Carlo method.The numerical results demonstrate that the proposed method is effective.In chapter 5,we pay attention to the Bayesian level set method for an inverse medium scattering problem in elasticity.Assuming that the Lamé parameters of the elastic medium are known constants,which are the same as those of the background medium,and the den-sity of the elastic medium is a piecewise constant function with a given value.Therefore,the inverse medium scattering problem can be expressed as a problem where the support of the scatterer needs to be recovered from the measurement data.Based on the Chapter 4,the shape of the scatterer is characterized by level set functions,and the prior of the level set function is achieved by Whittle-Matérn Gaussian random field.Furthermore,we can get that the posterior distribution is locally Lipschitz continuous.Numerical experiments are implemented by using the Markov chain Monte Carlo method,and numerical results present the feasibility and the effectiveness of proposed method.