Browsing by Subject "inverse problems"
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Item Application of a Constrained Optimization Technique to the Imaging of Heterogeneous Objects Using Diffusion Theory(2011-02-22) Sternat, Matthew RyanThe problem of inferring or reconstructing the material properties (cross sections) of a domain through noninvasive techniques, methods using only input and output at the domain boundary, is attempted using the governing laws of neutron diffusion theory as an optimization constraint. A standard Lagrangian was formed consisting of the objective function and the constraints to satisfy, which was minimized through optimization using a line search method. The chosen line search method was Newton's method with the Armijo algorithm applied for step length control. A Gaussian elimination procedure was applied to form the Schur complement of the system, which resulted in greater computational efficiency. In the one energy group and multi-group models, the limits of parameter reconstruction with respect to maximum reconstruction depth, resolution, and number of experiments were established. The maximum reconstruction depth for one-group absorption cross section or multi-group removal cross section were only approximately 6-7 characteristic lengths deep. After this reconstruction depth limit, features in the center of a domain begin to diminish independent of the number of experiments. When a small domain was considered and size held constant, the maximum reconstruction resolution for one group absorption or multi-group removal cross section is approximately one fourth of a characteristic length. When finer resolution then this is considered, there is simply not enough information to recover that many region's cross sections independent of number of experiments or flux to cross-section mesh refinement. When reconstructing fission cross sections, the one group case is identical to absorption so only the multi-group is considered, then the problem at hand becomes more ill-posed. A corresponding change in fission cross section from a change in boundary flux is much greater then change in removal cross section pushing convergence criteria to its limits. Due to a more ill-posed problem, the maximum reconstruction depth for multi-group fission cross sections is 5 characteristic lengths, which is significantly shorter than the removal limit. To better simulate actual detector readings, random signal noise and biased noise were added to the synthetic measured solutions produced by the forward models. The magnitude of this noise and biased noise is modified and a dependency of the maximum magnitude of this noise versus the size of a domain was established. As expected, the results showed that as a domain becomes larger its reconstruction ability is lowered which worsens upon the addition of noise and biased noise.Item Inverse Problems for Fractional Diffusion Equations(2013-06-21) Zuo, LihuaIn recent decades, significant interest, based on physics and engineering applications, has developed on so-called anomalous diffusion processes that possess different spread functions with classical ones. The resulting differential equation whose fundamental solution matches this decay process is best modeled by an equation containing a fractional order derivative. This dissertation mainly focuses on some inverse problems for fractional diffusion equations. After some background introductions and preliminaries in Section 1 and 2, in the third section we consider our first inverse boundary problem. This is where an unknown boundary condition is to be determined from overposed data in a time- fractional diffusion equation. Based upon the fundamental solution in free space, we derive a representation for the unknown parameters as the solution of a nonlinear Volterra integral equation of second kind with a weakly singular kernel. We are able to make physically reasonable assumptions on our constraining functions (initial and given boundary values) to be able to prove a uniqueness and reconstruction result. This is achieved by an iterative process and is an immediate result of applying a certain fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In the fourth section a reaction-diffusion problem with an unknown nonlinear source function, which has to be determined from overposed data, is considered. A uniqueness result is proved and a numerical algorithm including convergence analysis under some physically reasonable assumptions is presented in the one-dimensional case. To show effectiveness of the proposed method, some results of numerical simulations are presented. In Section 5, we also attempted to reconstruct a nonlinear source in a heat equation from a number of known input sources. This represents a new research even for the case of classical diffusion and would be the first step in a solution method for the fractional diffusion case. While analytic work is still in progress on this problem, Newton and Quasi-Newton method are applied to show the feasibility of numerical reconstructions. In conclusion, the fractional diffusion equations have some different properties with the classical ones but there are some similarities between them. The classical tools like integral equations and fixed point theory still hold under slightly different assumptions. Inverse problems for fractional diffusion equations have applications in many engineering and physics areas such as material design, porous media. They are trickier than classical ones but there are also some advantages due to the mildly ill-conditioned singularity caused by the new kernel functions.Item Modeling Aspects and Computational Methods for Some Recent Problems of Tomographic Imaging(2012-02-14) Allmaras, MoritzIn this dissertation, two recent problems from tomographic imaging are studied, and results from numerical simulations with synthetic data are presented. The first part deals with ultrasound modulated optical tomography, a method for imaging interior optical properties of partially translucent media that combines optical contrast with ultrasound resolution. The primary application is the optical imaging of soft tissue, for which scattering and absorption rates contain important functional and structural information about the physiological state of tissue cells. We developed a mathematical model based on the diffusion approximation for photon propagation in highly scattering media. Simple reconstruction schemes for recovering optical absorption rates from boundary measurements with focused ultrasound are presented. We show numerical reconstructions from synthetic data generated for mathematical absorption phantoms. The results indicate that high resolution imaging with quantitatively correct values of absorption is possible. Synthetic focusing techniques are suggested that allow reconstruction from measurements with certain types of non-focused ultrasound signals. A preliminary stability analysis for a linearized model is given that provides an initial explanation for the observed stability of reconstruction. In the second part, backprojection schemes are proposed for the detection of small amounts of highly enriched nuclear material inside 3D volumes. These schemes rely on the geometrically singular structure that small radioactive sources represent, compared to natural background radiation. The details of the detection problem are explained, and two types of measurements, collimated and Compton-type measurements, are discussed. Computationally, we implemented backprojection by counting the number of particle trajectories intersecting each voxel of a regular rectangular grid covering the domain of detection. For collimated measurements, we derived confidence estimates indicating when voxel trajectory counts are deviating significantly from what is expected from background radiation. Monte Carlo simulations of random background radiation confirm the estimated confidence values. Numerical results for backprojection applied to synthetic measurements are shown that indicate that small sources can be detected for signal-to-noise ratios as low as 0.1%.Item Optimal shape design for a layered periodic structure(Texas A&M University, 2004-09-30) Flanagan, Michael BradyA multi-layered periodic structure is investigated for optimal shape design in diffraction gratings. A periodic dielectric material is used as the scattering profile for a planar incident wave. Designing optimal profiles for scattering is a type of inverse problem. The ability to fabricate such materials on the order of the wavelength of the incoming light is key for design strategies. We compute a finite element approximation on a variational setup of the forward problem. On the inverse and optimal design problem, we discuss the stability of the designs and develop computational strategies based on a level-set evolutionary approach.Item The Phase Retrieval Problem and Its Applications in Optics(2014-10-15) Trahan, Russell EIn this dissertation the various forms and applications of the phase retrieval problem in imaging are discussed. The phase retrieval problem in general refers to the estimation of the phase of a complex-valued function based on knowledge of its magnitude. Here the phase retrieval problem is applied to the estimation of the phase of an electromagnetic wave field based on knowledge of its magnitude. The magnitude (not the phase) can be measured using several devices as discussed. There are many applications of phase retrieval which have been explored where the mapping between the detected wave field magnitude and the light source is the Fourier transform. Within these applications phase retrieval solutions are used to estimate the phase of the Fourier transform so as to obtain the image or the shape of the light emitter. These solutions necessitate a model of the propagation of the wave field, a method of detecting the field?s magnitude, and a method of estimating the phase of the observed field. The first two considerations are discussed here historically and reference many significant scientific discoveries, namely the Huygens-Fresnel principle, the Van Cittert-Zernike theorem, and the Michelson interferometer. Within the field of interferometry where the Fourier transform is the mapping between the light source and observed wave field, most solutions utilize a discrete Fourier transform. The estimate of the light source takes the form of a two-dimensional pixelated image. These solutions have been explored for many years and have many variations for particular applications. Not many of these solution methods, however, have confronted the problem of measurement noise. Measurement noise here refers to noise in the quantification of the magnitude of the wave field at the observed locations. Within this dissertation the negative effects of noise are analyzed and a method of filtering the noise from the data is derived, tested, and shown to be effective. In a separate analysis the use of the discrete Fourier transform as opposed to the continuous Fourier transform is questioned. A phase solution is proposed which is capable of estimating the source of the observed wave field and takes discrete magnitude data and outputs a continuous image function formed from Gaussian bases. This method is beneficial from an analytical point-of-view since it is not an iterative solution. It also has an error metric which definitively determines whether the true solution has been found?unlike the traditional solution methods. The phase retrieval problem is also explored in the case where the Fourier transform is not the mapping between the image and the observed wave field. Particularly, the case of a small asteroid occulting a star is analyzed with the goal of characterizing the shape of the asteroid?s silhouette. A solution is formulated capable of resolving the asteroid silhouette based on time histories of the intensity of the wave field measured at multiple spatial locations. The solution is based on an analysis of the shadow that the occulter casts. The phase retrieval problem is present in many current fields of imaging and remains a prominent source of inquiry. Although many solution methods exist, there are still many improvements that can be made. This dissertation addresses some potential improvements to existing solutions and proposes new applications and formulations of the phase retrieval problem.