Thesis (M.S.)--Ohio University, June, 2000. / Title from PDF t.p.

Thesis (Ph. D.)--University of Washington, 2008. / Vita. Includes bibliographical references (p. 99-102).

Thesis (Ph. D.)--West Virginia University, 2009. / Title from document title page. Document formatted into pages; contains v, 58 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 56-58).

Thesis (Ph. D.)--University of Missouri-Columbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 105-115). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2004. / Typescript. Vita. Includes bibliographical references (leaves 105-115). Also available on the Internet.

Thesis (Ph.D.)--University of Delaware, 2007. / Principal faculty advisor: Robert Gilbert, Dept. of Mathematical Sciences. Includes bibliographical references.

Thesis (M.Sc.Eng.) PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This thesis considers the efficient Hessian computation in inverse problems with specific application to the elastography inverse problem. Inverse problems use measurements of observable parameters to infer information about model parameters, and tend to be ill-posed. They are typically formulated and solved as regularized constrained optimization problems, whose solutions best fit the measured data. Approaching the same inverse problem from a probabilistic Bayesian perspective produces the same optimal point called the maximum a posterior (MAP) estimate of the parameter distribution, but also produces a posterior probability distribution of the parameter estimate, from which a measure of the solution's uncertainty may be obtained. This probability distribution is a very high dimensional function with which it can be difficult to work. For example, in a modest application with N = 104 optimization variables, representing this function with just three values in each direction requires 3^10000 U+2248 10^5000 variables, which far exceeds the number of atoms in the universe. The uncertainty of the MAP estimate describes the shape of the probability distribution and to leading order may be parameterized by the covariance. Directly calculating the Hessian and hence the covariance, requires O(N) solutions of the constraint equations. Given the size of the problems of interest (N = O(10^4 - 10^6)), this is impractical. Instead, an accurate approximation of the Hessian can be assembled using a Krylov basis. The ill-posed nature of inverse problems suggests that its Hessian has low rank and therefore can be approximated with relatively few Krylov vectors. This thesis proposes a method to calculate this Krylov basis in the process of determining the MAP estimate of the parameter distribution. Using the Krylov space based conjugate gradient (CG) method, the MAP estimate is computed. Minor modifications to the algorithm permit storage of the Krylov approximation of the Hessian. As the accuracy of the Hessian approximation is directly related to the Krylov basis, long term orthogonality amongst the basis vectors is maintained via full reorthogonalization. Upon reaching the MAP estimate, the method produces a low rank approximation of the Hessian that can be used to compute the covariance. / 2031-01-01

The non-invasive differential diagnosis of breast masses through ultrasound imaging motivates the following class of elastic inverse problems: Given one or more measurements of the displacement field within an elastic material, determine the material property distribution within the material. This thesis is focused on uncertainty quantification in inverse problem solutions, with application to inverse problems in linear and nonlinear elasticity. We consider the inverse nonlinear elasticity problem in the context of Bayesian statistics. We show the well-known result that computing the Maximum A Posteriori (MAP) estimate is consistent with previous optimization formulations of the inverse elasticity problem. We show further that certainty in this estimate may be quantified using concepts from information theory, specifically, information gain as measured by the Kullback-Leibler (K-L) divergence and mutual information. A particular challenge in this context is the computational expense associated with computing these quantities. A key contribution of this work is a novel approach that exploits the mathematical structure of the inverse problem and properties of conjugate gradient method to make these calculations feasible. A focus of this work is estimating the spatial distribution of the elastic nonlinearity of a material. Measurement sensitivity to the nonlinearity is much higher for large (finite) strains than for smaller strains, and so large strains tend to be used for such measurements. Measurements of larger deformations, however, tend to show greater levels of noise. A key finding of this work is that, when identifying nonlinear elastic properties, information gain can be used to characterize a trade-off between larger strains with higher noise levels and smaller strains with lower noise levels. These results can be used to inform experimental design. An approach often used to estimate both linear and nonlinear elastic property distributions is to do so sequentially: Use a small strain deformation to estimate the linear properties, and a large strain deformation to estimate the nonlinearity. A key finding of this work is that accurate characterization of the joint posterior probability distribution over both linear and nonlinear elastic parameters requires that the estimates be performed jointly rather than sequentially. All the methods described above are demonstrated in applications to problems in elasticity for both simulated data as well as clinically measured data (obtained in vivo). In the context of the clinical data, we evaluate repeatability of measurements and parameter reconstructions in a clinical setting.

<p dir="ltr">Inverse problems in infinite dimensions are ubiquitously encountered across the scien- tific disciplines. These problems are defined by the need to reconstruct continuous fields from incomplete, noisy measurements, which oftentimes leads to ill-posed problems. Almost universally, the solutions to these problems are constructed in a Bayesian framework. How- ever, in the infinite-dimensional setting, the theory is largely restricted to the Gaussian case, and the treatment of prior physical knowledge is lacking. We develop a new framework for Bayesian reconstruction of infinite-dimensional fields which encodes our physical knowledge directly into the prior, while remaining in the continuous setting. We then prove various characteristics of the method, including situations in which the problems we study have unique solutions under our framework. Finally, we develop numerical sampling schemes to characterize the various objects involved.</p>

Variational inverse problems integrate computational simulations of physical phenomena with physical measurements in an informational feedback control system. Control parameters of the computational model are optimized such that the simulation results fit the physical measurements.The solution procedure is computationally expensive since it involves running the simulation computer model (the emph{forward model}) and the associated emph {adjoint model} multiple times. In practice, our knowledge of the underlying physics is incomplete and hence the associated computer model is laden with emph {model errors}. Similarly, it is not possible to measure the physical quantities exactly and hence the measurements are associated with emph {data errors}. The errors in data and model adversely affect the inference solutions. This work develops methods to address the challenges posed by the computational costs and by the impact of data and model errors in solving variational inverse problems. Variational inverse problems of interest here are formulated as optimization problems constrained by partial differential equations (PDEs). The solution process requires multiple evaluations of the constraints, therefore multiple solutions of the associated PDE. To alleviate the computational costs we develop a parallel in time discretization algorithm based on a nonlinear optimization approach. Like in the emph{parareal} approach, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. Solution continuity equations across interval boundaries are added as constraints. All the computational steps - forward solutions, gradients, and Hessian-vector products - involve only ideally parallel computations and therefore are highly scalable. This work develops a systematic mathematical framework to compute the impact of data and model errors on the solution to the variational inverse problems. The computational algorithm makes use of first and second order adjoints and provides an a-posteriori error estimate for a quantity of interest defined on the inverse solution (i.e., an aspect of the inverse solution). We illustrate the estimation algorithm on a shallow water model and on the Weather Research and Forecast model. Presence of outliers in measurement data is common, and this negatively impacts the solution to variational inverse problems. The traditional approach, where the inverse problem is formulated as a minimization problem in $L_2$ norm, is especially sensitive to large data errors. To alleviate the impact of data outliers we propose to use robust norms such as the $L_1$ and Huber norm in data assimilation. This work develops a systematic mathematical framework to perform three and four dimensional variational data assimilation using $L_1$ and Huber norms. The power of this approach is demonstrated by solving data assimilation problems where measurements contain outliers. / Ph. D.