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.

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.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2009. / Title from PDF t.p. (viewed July 21, 2010). Additional advisors: Thomas Jannett, Tsun-Zee Mai, S. S. Ravindran, Günter Stolz, Gilbert Weinstein. Includes bibliographical references (p. 59-64).

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. 27 January 2016. / The direct and inverse spectral problems for an inhomogeneous string with an interval of zero density and a concentrated mass at the end point moving with damping are investigated. The partial differential equation is mapped into an ordinary differential equation using separation of variables which in turn is transformed into a Sturm-Liouville differential equation with boundary conditions depending on these parathion variable. The Marchenko approach is employed in the inverse problem to recover the potential, density and other parameters from the knowledge of the two spectra and length of the string.

Keung Yee Lo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 109-111). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Inverse problems and Parameter Identification --- p.1 / Chapter 1.2 --- Examples in inverse problems --- p.2 / Chapter 1.3 --- Applications in parameter identifications --- p.5 / Chapter 1.4 --- Difficulties arising in inverse problems --- p.7 / Chapter 2 --- Identifying Parameters in Parabolic Systems --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.2 --- An averaging-terminal status formulation and existence of its solutions --- p.12 / Chapter 2.3 --- Optimization approach and its convergence --- p.17 / Chapter 2.4 --- Unconstrained minimization problems --- p.26 / Chapter 2.5 --- Armijo algorithm --- p.28 / Chapter 2.6 --- Numerical experiments --- p.32 / Chapter 2.6.1 --- Convergence of the minimization problem --- p.40 / Chapter 2.7 --- Noisy data --- p.59 / Chapter 3 --- Identifying Parameters in Elliptic Systems --- p.68 / Chapter 3.1 --- Augmented Lagrangian Method --- p.68 / Chapter 3.2 --- The discrete saddle-point problem --- p.70 / Chapter 3.3 --- An Uzawa algorithm --- p.71 / Chapter 3.4 --- Formulation of the algorithm --- p.73 / Chapter 3.5 --- Numerical experiments --- p.76 / Chapter 3.6 --- Alternative formulation of the cost functional --- p.90 / Chapter 3.7 --- Iterative GMRES method --- p.102 / Bibliography --- p.109

心臟表面電位能夠真實反映心肌的活動，因此以重建心臟表面電位為目標的心臟電生理反問題被廣泛研究。心臟電生理反問題是一個不適定問題，因此輸入數據中一個小的噪聲也有可能導致一個高度不穩定的解。因此，通常基於2 範數的規則化方法被用於解決這個病態問題。但是2 範數的懲罰函數會導致一定程度的模糊，使得分辨和定位心臟表面一些不正常或者病變部位不準確。而直接使用1 範數的懲罰函數，會由於其不可微分而增加計算復雜度。 / 我們首先提出一種基於 1 範數的方法來減少計算復雜度和能夠快速收斂。在這個方法中，使用變量分離技術使得1 範數的懲罰函數可微分。然後這個反問題被構造成一個有界約束二次優化問題，從而可以很容易地利用梯度映射法叠代求解。在試驗中，使用合成數據和真實數據來評估提出的方法。實驗表明，提出的方法可以很好地處理測量噪聲和幾何噪聲，而且能夠獲得比以前的1、2 範數方法更準確的實驗結果。 / 盡管提出的 1 範數方法能夠有效克服2 範數存在的問題，但是1 範數方法仍然只是0 範數的近似。因此我們采用了一種平滑0 範數的方法來求解心臟電生理反問題。平滑0 範數使用平滑函數，使得0 範數連續，從而能夠直接求解0 範數的反問題。實驗結果表明，使用平滑0範數方法可以獲得比1、2 範數更好、更準確的心臟表面電位。 / 在以往的心臟反問題研究中，使用的心臟幾何模型都是靜態的，與實際跳動的心臟不符，從而使得反問題方法難以進入臨床。因此我們提出了從動態心臟模型中重建心臟表面電位。動態心臟模型是從一系列核磁共振圖像中重建得到的。體表電位也同步獲得。仿真實驗獲得了很好的心臟表面電位結果。 / 在論文最後，我們提出一個基於心臟電生理反問題的系統，來輔助束支傳導阻滯的治療。在這個系統中，心臟模型和體表模型都從病人的數據中重建獲得，體表電位也得到收集。通過電生理反問題方法，在心臟表面重建電位及其分布。醫生通過觀察重建結果來輔助束支傳導阻滯的診斷和治療。 / The epicardial potentials (EPs) targeted inverse problem of electrocardiography (ECG) has been widely investigated as it is demonstrated that EPs reflect underlying myocardial activity. It is a wellknown ill-posed problem as small noises in input data may yield a highly unstable solution. Traditionally, L2-norm regularization methods have been proposed to solve this ill-posed problem. But L2-norm penalty function inherently leads to considerable smoothing of the solution, which reduces the accuracy of distinguishing abnormalities and locating diseased regions. In this thesis, we propose three new techniques in order to achieve more accurate reconstruction results of EPs and applied these techniques to a clinical application. We first propose a L1-norm regularization method in order to reduce the computational complexity and make rapid convergence possible. Variable splitting is employed to make the L1- norm penalty function differentiable based on the observation that both positive and negative potentials exist on the epicardial surface. Then, the inverse problem of ECG is further formulated as a boundconstrained quadratic problem, which can be efficiently solved by gradient projection in an iterative manner. Extensive experiments conducted on both synthetic data and real data demonstrate that the proposed method can handle both measurement noise and geometry noise and obtain more accurate results than previous L2- and L1- norm regularization methods, especially when the noises are large. / Although L1 norm regularization achieves better reconstructed results compared with L2 norm regularization, L1 norm is still an approximation of L0 norm which is more accurate than L1 norm. We further presented a smoothed L0 norm technique in order to directly solve the L0 norm constrained problem. Our method employs a smoothing function to make the L0 norm continuous. Extensive experiments showed that the proposed method reconstructs more accurate epicardial potentials compared with L1 norm and L2 norm. / In current research of ECG inverse problem, epicardial potentials are reconstructed from a static heart model which blocks the techniques to clinic applications. A novel strategy is presented to recovii er epicardial potentials using a dynamic heart model built from MRI image sequences and ECG data. We used MRI images to estimate the current density and visualize it on the surface of the heart model. The ECG data also be used to achieve the time synchronization when the propagation of the current density. Experiments are conducted on a set of real time MRI images, also with the real ECG data, and we get favorable results. / Finally, a non-invasive system is presented for enhancing the diagnosis of Bundle Branch Block (BBB). In this system, epicardial potential is estimated and visualized in the 3D heart model to improve the diagnosis of BBB. Using patient CT and BSPM data, the system is able to reconstruct details of the complete electrical activity of BBB on the 3D heart model. Through the analysis of the epicardial potential mapping in this system, patients with BBB are easily and accurately distinguished instead of from empirically checking ECG. Therefore the diagnosis of BBB is improved using this system. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Liansheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 103-124). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Inverse Problem of ECG --- p.6 / Chapter 2.1 --- Background --- p.6 / Chapter 2.2 --- Problem Formulations --- p.8 / Chapter 2.2.1 --- Potential Reconstruction Problem --- p.8 / Chapter 2.2.2 --- Coefficient Reconstruction Problem --- p.11 / Chapter 2.3 --- Solving Methods --- p.11 / Chapter 2.3.1 --- Regularization Methods --- p.11 / Chapter 2.3.2 --- Non-quadratic Regularization --- p.12 / Chapter 2.3.3 --- Activation Wavefronts Solution --- p.14 / Chapter 3 --- L1-Norm to EPs Reconstruction --- p.16 / Chapter 3.1 --- Related Work --- p.16 / Chapter 3.2 --- Method --- p.21 / Chapter 3.3 --- Experimental Results and Validation --- p.24 / Chapter 3.3.1 --- Error Evaluation --- p.26 / Chapter 3.3.2 --- Synthetic Data Cases --- p.26 / Chapter 3.3.3 --- Real Data Cases --- p.32 / Chapter 3.4 --- Discussion --- p.44 / Chapter 3.5 --- Summary --- p.48 / Chapter 4 --- L0-Norm to EPs Reconstruction --- p.49 / Chapter 4.1 --- Related Work --- p.49 / Chapter 4.2 --- Smoothed L0-norm Method --- p.54 / Chapter 4.3 --- Experimental Results and Protocols --- p.57 / Chapter 4.3.1 --- Data --- p.57 / Chapter 4.3.2 --- Evaluation Protocol --- p.60 / Chapter 4.3.3 --- Experiments and Results --- p.60 / Chapter 4.4 --- Discussion --- p.68 / Chapter 4.5 --- Summary --- p.69 / Chapter 5 --- EPs Reconstruction in A Dynamic Model --- p.71 / Chapter 5.1 --- Related Work --- p.71 / Chapter 5.2 --- Forward Model --- p.73 / Chapter 5.3 --- Parameters Estimation for Inverse Problem of ECG --- p.75 / Chapter 5.4 --- Experiments and Results --- p.77 / Chapter 5.5 --- Summary --- p.80 / Chapter 6 --- Diagnosis of BBB: an Application --- p.82 / Chapter 6.1 --- Related Work --- p.82 / Chapter 6.2 --- Method --- p.84 / Chapter 6.2.1 --- Data --- p.85 / Chapter 6.2.2 --- Signal Preprocessing of BSPM --- p.87 / Chapter 6.2.3 --- Epicardial Potential Estimation and Imaging --- p.88 / Chapter 6.3 --- Experiments and Results --- p.89 / Chapter 6.3.1 --- Population Under Study --- p.89 / Chapter 6.3.2 --- Results --- p.89 / Chapter 6.4 --- Summary --- p.92 / Chapter 7 --- Conclusion --- p.94 / Chapter 7.1 --- Summary of Contributions --- p.94 / Chapter 7.2 --- Future Works --- p.96 / Chapter A --- Barzilai and Borwein Approach --- p.97 / Chapter B --- List of Publications --- p.99 / Bibliography --- p.103

by Yu Zhenli. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (p. 193-202). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.