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Microwave imaging techniques have shown excellent capabilities in various fields such as civil engineering, nondestructive testing, industrial applications, and have in recent decades experienced strong growth as a research topic in biomedical diagnostics. Many research groups throughout the world work on prototype systems for producing images of human tissues in different biomedical applications, particularly breast tumor detection. However, the research community faces many challenges and in order to be competitive to other imaging modalities one of the means is to put emphasis on experimental work. Consequently, the use of flexible and accurate measurement systems, together with the design and fabrication of suitable antennas, are essential to the development of efficient microwave imaging systems. The first part of this thesis focuses on measurement systems for microwave imaging in terms of antenna design and development, robot controlled synthetic array geometries, permittivity measurements, and calibration. The aim was to investigate the feasibility of a flexible system for measuring the fields around an inhomogeneous object and to create quantitative images. Hence, such an aim requires solving of a nonlinear inverse scattering problem, which in turn requires accurate measurements for producing good quality experimental data. The presented solution by design of a flexible measurement system is validated by examination of microwave imaging from experimental data with a breast phantom. The second part of the thesis deals with the research challenges of designing high performance antennas to be placed in direct contact with or in close proximity to the imaged object. The need for novel antenna applicators is envisaged in the framework of the Mamacell measurement system, where the antenna applicators have to be designed and constructed to effectively couple the energy into the imaging object. For this purpose the main constraints and design requirements are a narrow lobe of the antenna, very small near-field effects, and small size. Numerical simulations and modeling shows that the proposed ridged waveguide antenna is capable of fulfilling the design requirements and the performance goals, demonstrating the potential for the future microwave imaging system called Mamacell.

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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.

<p>This study focuses on developing a new online recursive numerical algorithm for a coupled nonlinear inverse heat conduction-microstructure problem. This algorithm is essential in identifying, designing and controlling many industrial applications such as the quenching process for heat treating of materials, chemical vapor deposition and industrial baking. In order to develop the above algorithm, a systematic four stage research plan has been conducted. </P> <p> The first and second stages were devoted to thoroughly reviewing the existing inverse heat conduction techniques. Unlike most inverse heat conduction solution methods that are batch form techniques, the online input estimation algorithm can be used for controlling the process in real time. Therefore, in the first stage, the effect of different parameters of the online input estimation algorithm on the estimate bias has been investigated. These parameters are the stabilizing parameter, the measurement errors standard deviation, the temporal step size, the spatial step size, the location of the thermocouple as well as the initial assumption of the state error covariance and error covariance of the input estimate. Furthermore, three different discretization schemes; namely: explicit, implicit and Crank-Nicholson have been employed in the input estimation algorithm to evaluate their effect on the algorithm performance. </p> <p> The effect of changing the stabilizing parameter has been investigated using three different forms of boundary conditions covering most practical boundary heat flux conditions. These cases are: square, triangular and mixed function heat fluxes. The most important finding of this investigation is that a robust range of the stabilizing parameter has been found which achieves the desired trade-off between the filter tracking ability and its sensitivity to measurement errors. For the three considered cases, it has been found that there is a common optimal value of the stabilizing parameter at which the estimate bias is minimal. This finding is important for practical applications since this parameter is usually unknown. Therefore, this study provides a needed guidance for assuming this parameter. </p> <p> In stage three of this study, a new, more efficient direct numerical algorithm has been developed to predict the thermal and microstructure fields during quenching of steel rods. The present algorithm solves the full nonlinear heat conduction equation using a central finite-difference scheme coupled with a fourth-order Runge-Kutta nonlinear solver. Numerical results obtained using the present algorithm have been validated using experimental data and numerical results available in the literature. In addition to its accurate predictions, the present algorithm does not require iterations; hence, it is computationally more efficient than previous numerical algorithms. </p> <p> The work performed in stage four of this research focused on developing and applying an inverse algorithm to estimate the surface temperatures and surface heat flux of a steel cylinder during the quenching process. The conventional online input estimation algorithm has been modified and used for the first time to handle this coupled nonlinear problem. The nonlinearity of the problem has been treated explicitly which resulted in a non-iterative algorithm suitable for real-time control of the quenching process. The obtained results have been validated using experimental data and numerical results obtained by solving the direct problem using the direct solver developed in stage three of this work. These results showed that the algorithm is efficiently reconstructing the shape of the convective surface heat flux. </P> / Thesis / Doctor of Philosophy (PhD)

In this paper we deal with aspects of characterizing the ill-posedn ess of nonlinear inverse problems based on the discussion of specific examples. In particular, a parameter identification problem to a second order differential equation and its ill-posed linear components are under consideration. A new approach to the classification ofill-posedness degrees for multiplication operators completes the paper.

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 study, the implementation of modern control techniques, that can be used both for the stable recovery of the aircraft from the undesired high angle of attack flight state (stall) and the agile maneuvering of the aircraft in various air combat or defense missions, are performed. In order to accomplish this task, the thrust vectoring control (TVC) actuation is blended with the conventional aerodynamic controls. The controller design is based on the nonlinear dynamic inversion (NDI) control methodologies and the stability and robustness analyses are done by using robust performance (RP) analysis techniques. The control architecture is designed to serve both for the recovery from the undesired stall condition (the stabilization controller) and to perform desired agile maneuvering (the attitude controller). The detailed modeling of the aircraft dynamics, aerodynamics, engines and thrust vectoring paddles, as well as the flight environment of the aircraft and the on-board sensors is performed. Within the control loop the human pilot model is included and the design of a fly-by-wire controller is also investigated. The performance of the designed stabilization and attitude controllers are simulated using the custom built 6 DoF aircraft flight simulation tool. As for the stabilization controller, a forced deep-stall flight condition is generated and the aircraft is recovered to stable and pilot controllable flight regimes from that undesired flight state. The performance of the attitude controller is investigated under various high angle of attack agile maneuvering conditions. Finally, the performances of the proposed controller schemes are discussed and the conclusions are made.

In this paper we deal with aspects of characterizing the ill-posedn ess of nonlinear inverse problems based on the discussion of specific examples. In particular, a parameter identification problem to a second order differential equation and its ill-posed linear components are under consideration. A new approach to the classification ofill-posedness degrees for multiplication operators completes the paper.