Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 101-103).

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 418-432). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 27-29). Also available on the Internet.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (p. [77]-79).

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 27-29). Also available on the Internet.

Thesis (Ph. D.)--Florida State University, 2006. / Advisors: Kevin Speer, Martin Wegkamp, Florida State University, College of Arts and Sciences, Dept. of Statistics. Title and description from dissertation home page (viewed Sept. 20, 2006). Document formatted into pages; contains x, 48 pages. Includes bibliographical references.

Andrew E. Yagle. / Bibliography: leaf 16. / "December, 1983." / Air Force Office of Scientific Research Grant AFOSR-82-0135A

In this thesis, we focus on the numerical study on some inverse problems and optimal control problems. In the first part, we consider some linear inverse problems with discontinuous or piecewise constant solutions. We use the total variation to regularize these inverse problems and then the finite element technique to discretize the regularized problems. These discretized problems are treated from the saddle-point perspective; and some primal-dual numerical schemes are proposed. We intensively investigate the convergence of these primal-dual type schemes, establishing the global convergence and estimating their worst-case convergence rates measured by the iteration complexity. We test these schemes by some experiments and verify their efficiency numerically. In the second part, we consider the finite difference and finite element discretization for an optimal control problem which is governed by time fractional diffusion equation. The prior error estimate of the discretized model is analyzed, and a projection gradient method is applied for iteratively solving the fully discretized surrogate. Some numerical experiments are conducted to verify the efficiency of the proposed method. Overall speaking, the thesis has been mainly inspired by some most recent advances developed in optimization community, especially in the area of operator splitting methods for convex programming; and it can be regarded as a combination of some contemporary optimization techniques with some relatively mature inverse and control problems. Keywords: Total variation minimization, linear inverse problem, saddle-point problem, finite element method, primal-dual method, convergence rate, optimal control problem, time fractional diffusion equation, projection gradient method.

Inverse problems are problems where causes for desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development, including radar/sonar, medical imaging, geophysical exploration, invisibility cloaking and remote sensing, to name just a few. In this thesis, we focus on the theoretical study and applications of some intriguing inverse problems. Precisely speaking, we are concerned with two typical kinds of problems in the field of wave scattering and nonlocal inverse problem, respectively. The first topic is on the geometric structures of eigenfunctions and their applications in wave scattering theory, in which the conductive transmission eigenfunctions and Laplacian eigenfunctions are considered. For the study on the intrinsic geometric structures of the conductive transmission eigenfunctions, we first present the vanishing properties of the eigenfunctions at corners both in R2 and R3, based on microlocal analysis with the help of a particular type of planar complex geometrical optics (CGO) solution. This significantly extends the previous study on the interior transmission eigenfunctions. Then, as a practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter. For the study on the geometric structures of Laplacian eigenfunctions, we separately discuss the two-dimensional case and the three-dimensional case. In R2, we introduce a new notion of generalized singular lines of Laplacian eigenfunctions, and carefully investigate these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We provide an accurate and comprehensive quantitative characterization of the relationship. In R3, we study the analytic behaviors of Laplacian eigenfunctions at places where nodal or generalized singular planes intersect, which is much more complicated. These theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal (polyhedral) setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Our second topic is concerning the fractional partial differential operators and some related nonlocal inverse problems. We present some prelimilary knowledge on fractional Sobolev Spaces and fractional partial differential operators first. Then we focus on the simultaneous recovery results of two interesting nonlocal inverse problems. One is simultaneously recovering potentials and the embedded obstacles for anisotropic fractional Schrödinger operators based on the strong uniqueness property and Runge approximation property. The other one is the nonlocal inverse problem associated with a fractional Helmholtz equation that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. The main method utilized here is the low-frequency asymptotics combining with the variational argument. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.

Iterative Learning Control (ILC) considers systems which perform the given desired trajectory repetitively. The command for the upcoming iteration is updated after every iteration based on the previous recorded error, aiming to converge to zero error in the real-world. Iterative Learning Control can be considered as an inverse problem, solving for the needed input that produces the desired output. However, digital control systems need to convert differential equations to digital form. For a majority of real world systems this introduces one or more zeros of the system z-transfer function outside the unit circle making the inverse system unstable. The resulting control input that produces zero error at the sample times following the desired trajectory is unstable, growing exponentially in magnitude each time step. The tracking error between time steps is also growing exponentially defeating the intended objective of zero tracking error. One way to address the instability in the inverse of non-minimum phase systems is to use basis functions. Besides addressing the unstable inverse issue, using basis functions also has several other advantages. First, it significantly reduces the computation burden in solving for the input command, as the number of basis functions chosen is usually much smaller than the number of time steps in one iteration. Second, it allows the designer to choose the frequency to cut off the learning process, which provides stability robustness to unmodelled high frequency dynamics eliminating the need to otherwise include a low-pass filter. In addition, choosing basis functions intelligently can lead to fast convergence of the learning process. All these benefits come at the expense of no longer asking for zero tracking error, but only aiming to correct the tracking error in the span of the chosen basis functions. Two kinds of matched basis functions are presented in this dissertation, frequency-response based basis functions and singular vector basis functions, respectively. In addition, basis functions are developed to directly capture the system transients that result from initial conditions and hence are not associated with forcing functions. The newly developed transient basis functions are particularly helpful in reducing the level of tracking error and constraining the magnitude of input control when the desired trajectory does not have a smooth start-up, corresponding to a smooth transition from the system state before the initial time, and the system state immediately after time zero on the desired trajectory. Another topic that has been investigated is the error accumulation in the unaddressed part of the output space, the part not covered by the span of the output basis functions, under different model conditions. It has been both proved mathematically and validated by numerical experiments that the error in the unaddressed space will remain constant when using an error-free model, and the unaddressed error will demonstrate a process of accumulation and finally converge to a constant level in the presence of model error. The same phenomenon is shown to apply when using unmatched basis functions. There will be unaddressed error accumulation even in the absence of model error, suggesting that matched basis functions should be used whenever possible. Another way to address the often unstable nature of the inverse of non-minimum phase systems is to use the in-house developed stable inverse theory Longman JiLLL, which can also be incorporated into other control algorithms including One-Step Ahead Control and Indirect Adaptive Control in addition to Iterative Learning Control. Using this stable inverse theory, One-Step Ahead Control has been generalized to apply to systems whose discrete-time inverses are unstable. The generalized one-step ahead control can be viewed as a Model Predictive Control that achieves zero tracking error with a control input bounded by the actuator constraints. In situations where one feels not confident about the system model, adaptive control can be applied to update the model parameters while achieving zero tracking error.