Inverse problems with partial data have many applications in science and engineering. They are more challenging than the complete data cases since the lack of data increases ill-posedness and nonlinearity. The use of only deterministic or statistical methods might not provide satisfactory results. We propose to combine the deterministic and statistical methods to treat such inverse problems. The thesis is organized as follows.
In Chapter 1, we briefly introduce the inverse problems and their applications. The classical deterministic methods and Bayesian inversion are discussed. The chapter is concluded with a summary of contributions.
Chapter 2 considers the reconstruction of the unknown acoustic sources using partial data. First, we extend the direct sampling method to approximate the source locations. Second, the inverse problem is formulated as a statistical inference problem using the Bayes\u27 formula. The source locations obtained in the first step are coded in the priors. Then a Metropolis-Hastings algorithm is used to explore the posterior density.
In Chapter 3, a two-step deterministic-statistical approach is proposed to recover the trajectories of moving sources using partial measured data. In the first step, an approximate direct sampling method is developed to obtain the locations of the sources at different times. Such information is coded in the priors, which is critical for the success of the Bayesian method in the second step. The combined approach inherits the merits of the deterministic method and Bayesian inversion as demonstrated by the numerical examples.
Chapter 4 studies the reconstruction of Stekloff eigenvalues and the index of refraction of an inhomogeneous medium from Cauchy data. The inverse spectrum problem of Stekloff eigenvalues is investigated by the reciprocity gap method. Then a Bayesian approach is proposed to estimate the index of refraction using a few reconstructed eigenvalues.
In Chapter 5, we consider the inverse spectral problem to determine the material properties given a few transmission eigenvalues. The lack of theoretical results motivates us to propose a Bayesian approach to formulate a statistical inference problem. The MCMC algorithm is used to explore the posterior density. Due to the non-uniqueness nature of the problem, we adopt the local conditional means (LCM) to characterize the posterior density function
