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.)--University of Washington, 1993. / Vita. Includes bibliographical references (leaves [54]-55).

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

When an emerging outbreak occurs, stable parameter estimation and reliable projections of future incidence cases using limited (early) data can play an important role in optimal allocation of resources and in the development of effective public health intervention programs. However, the inverse parameter identification problem is ill-posed and cannot be solved with classical tools of computational mathematics. In this dissertation, various regularization methods are employed to incorporate stability in parameter estimation algorithms. The recovered parameters are then used to generate future incident curves as well as the carrying capacity of the epidemic and the turning point of the outbreak. For the nonlinear generalized Richards model of disease progression, we develop a novel iteratively regularized Gauss-Newton-type algorithm to reconstruct major characteristics of an emerging infection. This problem-oriented numerical scheme takes full advantage of a priori information available for our specific application in order to stabilize the iterative process. Another important aspect of our research is a reliable estimation of time-dependent transmission rate in a compartmental SEIR disease model. To that end, the ODE-constrained minimization problem is reduced to a linear Volterra integral equation of the first kind, and a combination of regularizing filters is employed to approximate the unknown transmission parameter in a stable manner. To justify our theoretical findings, extensive numerical experiments have been conducted with both synthetic and real data for various infectious diseases.

Inverse problems arise in many fields such as radar imaging, medical imaging and geophysics. It draws much attention in both mathematical communities and industrial members. Mathematically speaking, many inverse problems can be formulated by one or several physical equations and mathematical models. For example, the signal used in radar imaging is governed by Maxwell's equation, and most of geophysical studies can be formulated using elastic equation. Therefore, rigorous mathematical theories can be applied to study the inverse problems coming from this complex world. Random inverse problem is a fascinating area studying how to extract useful statistical information from unknown object coming from real world. In this thesis, we focus on the study of inverse problem related to random Schrödinger operators. We are particularly interested in the case where both the source and the potential of the Schrödinger system are random. In our first topic, we are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrödinger equation with unknown random source and unknown potential. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. Our second topic also focuses on the case where only the source is random. But in the second topic, the random model is different from our first topic. The second random model has received intensive study in recent years due to the reason that this random model has more flexibility fitting with different regularities. The recovering framework is similar to our first topic, but we shall develop different asymptotic estimates of the higher order terms, which is more difficult than the first one. Lastly, based on the previous two results, we study the case where both the source and the potential are random and unknown. The ergodicity is used to establish the single realization recovery. The asymptotic estimates of higher order terms are based on pseudodifferential operators and microlocal analysis. Three major novelties of our works in this thesis are that, first, we studied the case where both the source and the potential are unknown; second, both passive and active scattering measurements are used for the recovery in different scenarios; finally, only a single realization of the random sample is required to establish the recovery of useful information.