Robust and Efficient Reconstruction Methods and Their Mathematical Theories for Inverse Scattering Problems

In this dissertation we shall develop a series of new efficient reconstruction algorithms for inverse medium and scattering problems, and provide mathematical justifications for these algorithms and their related physical observations.;We start with the acoustic and transverse electric (TE) and transverse magnetic (TM) inverse scattering problems. A new notion of scattering coefficients for heterogeneous media is introduced and analysed mathematically. Explicit reconstruction formulae are provided for the linearized case. Based on this novel concept of scattering coefficients, sensitivity and resolution analysis are performed to mathematically assess the reconstruction quality and justify the super-resolution phenomenon in imaging high contrast targets.;We then turn to the electric impedance tomography (EIT) problem. We first develop a new direct sampling methods (DSM) for EIT to provide a concise estimate of the location of inhomogeneities inside a homogeneous background with a single boundary data. Then a shape design problem related to EIT and plasmon resonance is investigated. A novel effective numerical method is proposed to recover the Fredholm eigenvalues from polarization tensors by making use of holomorphic functional calculus, then a regularized Gauss-Newton optimization method is established for shape reconstruction.;Following the electric impedance tomography, we shall study another severely ill-posed inverse problem, the diffusive optical tomography (DOT). A novel DSM is derived and proposed to estimate small inclusions with one boundary data in both full and limited aperture cases.;Our final interest is on the application of non-negative matrix factorizations (NMF) to some highly nonlinear and ill-posed imaging and inverse problems. A sparse approximation of big data is proposed in terms of tensor products of positive vectors, and its effectiveness is rogorously analysed. A new concept of multi-level analysis framework based on NMF is suggested to extract major components representing structures of different resolutions. Then a semi-smooth Newton method based on primal-dual active sets is derived for effective numerical implementations of NMF and its effectiveness and robustness are tested through a series of challenging imaging and inverse problems.