Rich tomography is becoming increasingly popular since we have seen a substantial increase in computational power and storage. Instead of measuring one scalar for each ray, multiple measurements are needed per ray for various imaging modalities. This advancement has allowed the design of experiments and equipment which facilitate a broad spectrum of applications. We present new reconstruction results and methods for several imaging modalities including x-ray diffraction strain tomography, Photoelastic tomography and Polarimet- ric Neutron Magnetic Field Tomography (PNMFT). We begin with a survey of the Radon and x-ray transforms discussing several procedures for inversion. Furthermore we highlight the Singular Value Decomposition (SVD) of the Radon transform and consider some stability results for reconstruction in Sobolev spaces. We then move onto define the Non-Abelian Ray Transform (NART), Longitudinal Ray Transform (LRT), Transverse Ray Transform (TRT) and the Truncated Trans- verse Ray Transform (TTRT) where we highlight some results on the complete inver- sion procedure, SVD and mention stability results in Sobolev spaces. Thereafter we derive some relations between these transforms. Next we discuss the imaging modali- ties in mind and relate the transforms to their specific inverse problems, primarily being linear. Specifically, NART arises in the formulation of PNMFT where we want to im- age magnetic structures within magnetic materials with the use of polarized neutrons. After some initial numerical studies we extend the known Radon inversion presented by experimentalists, reconstructing fairly weak magnetic fields, to reconstruct PNMFT data up to phase wrapping. We can recover the strain field tomographically for a polycrystalline material using diffraction data and deduce that a certain moment of that data corresponds to the TRT. Quite naturally the whole strain tensor can be reconstructed from diffraction data measured using rotations about six axes. We develop an innovative explicit plane-by-plane filtered back-projection reconstruction algorithm for the TRT, using data from rotations about three orthogonal axes and state the reasoning why two- axis data is insufficient. For the first time we give the first published results of TRT reconstruction. To complete our discussion we present Photoelastic tomography which relates to the TTRT and implement the algorithm discussing the difficulties that arise in reconstructing data. Ultimately we return to PNMFT highlighting the nonlinear inverse problem due to phase wrapping. We propose an iterative reconstruction algorithm, namely the Modified Newton Kantarovich method (MNK) where we keep the Jacobian (Fréchet derivative) fixed at the first step. However, this is shown to fail for large angles suggesting to develop the Newton Kantarovich (NK) method where we update the Jacobian at each step of the iteration process.

In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example, image reconstruction and deblurring require the use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest, the number of unknown variables is huge, and some may depend nonlinearly on the data, such that we must solve nonlinear problems. It is quite different and significantly more challenging to solve nonlinear problems than linear inverse problems, and we need to use more sophisticated methods to solve these kinds of problems. / Master of Science / In various research areas, there are many required measurements which can't be observed due to physical and economical reasons. Instead, these unknown measurements can be recovered by known measurements. This phenomenon can be modeled and be solved by mathematics.

In this paper we discuss the identification of parameter functions in material models for elastic large deformations. A model of the the forward problem is given, where the displacement of a deformed material is found as the solution of a n onlinear PDE. Here, the crucial point is the definition of the 2nd Piola-Kirchhoff stress tensor by using several material laws including a number of material parameters. In the main part of the paper we consider the identification of such parameters from measured displacements, where the inverse problem is given as an optimal control problem. We introduce a solution of the identification problem with Lagrange and SQP methods. The presented algorithm is applied to linear elastic material with large deformations.

In this paper we discuss the identification of parameter functions in material models for elastic large deformations. A model of the the forward problem is given, where the displacement of a deformed material is found as the solution of a n onlinear PDE. Here, the crucial point is the definition of the 2nd Piola-Kirchhoff stress tensor by using several material laws including a number of material parameters. In the main part of the paper we consider the identification of such parameters from measured displacements, where the inverse problem is given as an optimal control problem. We introduce a solution of the identification problem with Lagrange and SQP methods. The presented algorithm is applied to linear elastic material with large deformations.