Problems of estimating nonnegative functions from nonnegative data induced by nonnegative mappings are ubiquitous in science and engineering. We address such problems by minimizing an information-theoretic discrepancy measure, namely Csiszar's I-divergence, between the collected data and hypothetical data induced by an estimate. Our applications can be summarized along the following three lines: 1) Deautocorrelation: Deautocorrelation involves recovering a function from its autocorrelation. Deautocorrelation can be interpreted as phase retrieval in that recovering a function from its autocorrelation is equivalent to retrieving Fourier phases from just the corresponding Fourier magnitudes. Schulz and Snyder invented an minimum I-divergence algorithm for phase retrieval. We perform a numerical study concerning the convergence of their algorithm to local minima. X-ray crystallography is a method for finding the interatomic structure of a crystallized molecule. X-ray crystallography problems can be viewed as deautocorrelation problems from aliased autocorrelations, due to the periodicity of the crystal structure. We derive a modified version of the Schulz-Snyder algorithm for application to crystallography. Furthermore, we prove that our tweaked version can theoretically preserve special symmorphic group symmetries that some crystals possess. We quantify noise impact via several error metrics as the signal-to-ratio changes. Furthermore, we propose penalty methods using Good's roughness and total variation for alleviating roughness in estimates caused by noise. 2) Deautoconvolution: Deautoconvolution involves finding a function from its autoconvolution. We derive an iterative algorithm that attempts to recover a function from its autoconvolution via minimizing I-divergence. Various theoretical properties of our deautoconvolution algorithm are derived. 3) Linear inverse problems: Various linear inverse problems can be described by the Fredholm integral equation of the first kind. We address two such problems via minimum I-divergence methods, namely the inverse blackbody radiation problem, and the problem of estimating an input distribution to a communication channel (particularly Rician channels) that would create a desired output. Penalty methods are proposed for dealing with the ill-posedness of the inverse blackbody problem.

The spherical Radon transform (SRT) integrates a function over the set of all spheres with a given set of centers. Such transforms play an important role in some newly developing types of tomography as well as in several areas of mathematics including approximation theory, integral geometry, inverse problems for PDEs, etc. In Chapter I we give a brief description of thermoacoustic tomography (TAT or TCT) and introduce the SRT. In Chapter II we consider the injectivity problem for SRT. A major breakthrough in the 2D case was made several years ago by M. Agranovsky and E. T. Quinto. Their techniques involved microlocal analysis and known geometric properties of zeros of harmonic polynomials in the plane. Since then there has been an active search for alternative methods, which would be less restrictive in more general situations. We provide some new results obtained by PDE techniques that essentially involve only the finite speed of propagation and domain dependence for the wave equation. In Chapter III we consider the transform that integrates a function supported in the unit disk on the plane over circles centered at the boundary of this disk. As is common for transforms of the Radon type, its range has an in finite co-dimension in standard function spaces. Range descriptions for such transforms are known to be very important for computed tomography, for instance when dealing with incomplete data, error correction, and other issues. A complete range description for the circular Radon transform is obtained. In Chapter IV we investigate implementation of the recently discovered exact backprojection type inversion formulas for the case of spherical acquisition in 3D and approximate inversion formulas in 2D. A numerical simulation of the data acquisition with subsequent reconstructions is made for the Defrise phantom as well as for some other phantoms. Both full and partial scan situations are considered.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--School of Industrial and Systems Engineering, Georgia Institute of Technology, 2004. Directed by Joel S. Sokol. / Vita. Includes bibliographical references (leaves 114-116).

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In 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.

Elastography is an emerging functional imaging technique of current clinical research interest due to a direct relation between mechanical material parameters, especially the tissue stiffness, and tissue pathologies such as cancer. Digital Image Elasto-Tomography (DIET) is a new method that aims to develop elastographic techniques and create a simplified, improved breast cancer screening process. The elastic material information of breast tissue is reconstructed in the DIET concept from mechanically excited steady-state harmonic motion observed on the surface of the breast. While this inversion process has been traditionally approached using finite element methods, this surface-orientated problem is naturally suited to the use of Boundary Element Methods (BEMs) requiring the discretization only on the surface of the domain and on the interface of a potential inclusion. As only approximate information is available about breast tissue material parameters, this thesis presents the development of BEM based inverse problem algorithms suitable for the reconstruction of all material parameters in a proportionally damped isotropic linear elastic solid, where only the material density is known. The highly nonlinear identification process of a potential inclusion is treated through the combination of a systematic Grid-Search with gradient descent techniques. This algorithm is extended to a three-step algorithm that performs a background material parameter estimation before the subsequent identification of an inclusion and thus provides a confident indication for the differentiation between cancerous and healthy breast tissue. The development of these algorithms is illustrated by several simulation studies highlighting important reconstruction behaviors relevant to the elastographic inverse problem. A first experimental test on a silicon based breast phantom is presented.

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this thesis, the effects of adding lumped masses to an Euler-Bernoulli beam on its frequencies and their corresponding mode shapes are investigated for simply-supported as well as fixed-free boundary conditions. This investigation paves the way for proposing a method to impose two frequencies on a system consisting of a beam and a lumped mass by determining the magnitude of the mass as well as its position along the beam.

In this thesis, algorithms for blind and non-blind motion deblurring of digital images are proposed. The non-blind algorithm is based on a convex program consisting of a data fitting term and a sparsity-promoting regularization term. The data fitting term is the squared l_2 norm of the residual between the blurred image and the latent image convolved with a known blur kernel. The regularization term is the l_1 norm of the latent image under a wavelet frame (framelet) decomposition. This convex program is solved with the first-order primal-dual algorithm proposed by Chambolle and Pock. The proposed blind deblurring algorithm is based on the work of Cai, Ji, Liu, and Shen. It works by embedding the proposed non-blind algorithm in an alternating minimization scheme and imposing additional constraints in order to deal with the challenging non-convex nature of the blind deblurring problem. Numerical experiments are performed on artificially and naturally blurred images, and both proposed algorithms are found to be competitive with recent deblurring methods. / Graduate / 0544 / tdanniels@gmail.com

A number of researchers have proposed several Bayesian methods for portfolio selection, which combine statistical information from financial time series with the prior beliefs of the portfolio manager, in an attempt to reduce the impact of estimation errors in distribution parameters on the portfolio selection process and the effect of these errors on the performance of 'optimal' portfolios in out-of-sample-data. This thesis seeks to reverse the direction of this process, inferring portfolio managers’ probabilistic beliefs about future distributions based on the portfolios that they hold. We refer to the process of portfolio selection as the forward problem and the process of retrieving the implied probabilities, given an optimal portfolio, as the inverse problem. We attempt to solve the inverse problem in a general setting by using a finite set of scenarios. Using a discrete time framework, we can retrieve probabilities associated with each of the scenarios, which tells us the views of the portfolio manager implicit in the choice of a portfolio considered optimal. We conduct the implied views analysis for portfolios selected using expected utility maximization, where the investor's utility function is a globally non-optimal concave function, and in the mean-variance setting with the covariance matrix assumed to be given. We then use the models developed for inverse problem on empirical data to retrieve the implied views implicit in a given portfolio, and attempt to determine whether incorporating these views in portfolio selection improves portfolio performance out of sample.