Thesis (Ph. D.)--University of Washington, 1996. / Vita. Includes bibliographical references (p. [138]-148).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 152-155).

Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 06/21/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

Thesis (M.S.)--University of North Carolina at Wilmington, 2003. / Includes bibliographical references (leaf : 44).

by Choi Chun-Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 56-58). / Abstract also in Chinese. / Chapter 1 --- Radial symmetry for solutions of a semilinear el- liptic equation on a bounded domain --- p.6 / Chapter 2 --- Asymptotic symmetry of singular solutions to a semilinear elliptic equation --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Some preliminary analysis --- p.14 / Chapter 2.3 --- Proof of Theorem 2.1 --- p.20 / Chapter 3 --- Classification of non-negative solutions to Yamabe type equations --- p.32 / Chapter 3.1 --- Introduction --- p.32 / Chapter 3.2 --- The Proof of Theorem 3.1 for k > 0 --- p.38 / Chapter 3.3 --- Case k <0 --- p.48 / Bibliography

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 68-70. / Ph.D.

by Li Jingzhi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Parameter identification problems --- p.1 / Chapter 1.2 --- Overview of existing numerical methods --- p.2 / Chapter 1.3 --- Outline of the thesis --- p.4 / Chapter 2 --- General Framework --- p.6 / Chapter 2.1 --- Abstract inverse problem --- p.6 / Chapter 2.2 --- Abstract multilevel models --- p.7 / Chapter 2.3 --- Abstract MMC algorithm --- p.9 / Chapter 3 --- Dual Viewpoint and Convergence Condition --- p.15 / Chapter 3.1 --- Dual viewpoint of nonlinear multigrid method --- p.15 / Chapter 3.2 --- Convergence condition of MMC algorithm --- p.16 / Chapter 4 --- Applications of MMC Algorithm for Parameter Identification in Elliptic and Parabolic Systems --- p.20 / Chapter 4.1 --- Notations --- p.20 / Chapter 4.2 --- Parameter identification in elliptic systems I --- p.21 / Chapter 4.3 --- Parameter identification in elliptic systems II --- p.23 / Chapter 4.4 --- Parameter identification in parabolic systems I --- p.24 / Chapter 4.5 --- Parameter identification in parabolic systems II --- p.25 / Chapter 5 --- Numerical Experiments --- p.27 / Chapter 5.1 --- Test problems --- p.27 / Chapter 5.2 --- Smoothing property of gradient methods --- p.28 / Chapter 5.3 --- Numerical examples --- p.29 / Chapter 6 --- Conclusion Remarks --- p.55 / Bibliography --- p.56

Lee, Ho Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 75-79). / Abstracts in English and Chinese. / Chapter 1 --- Upscaling methods for high contrast problems --- p.6 / Chapter 1.1 --- Review on upscaling methods --- p.7 / Chapter 1.2 --- Upscaling method with high contrast of the conductivity --- p.11 / Chapter 2 --- Multiscale finite element methods for high contrast problems --- p.19 / Chapter 2.1 --- Review on Multiscale finite element methods --- p.20 / Chapter 2.2 --- Local spectral basis functions --- p.23 / Chapter 2.3 --- Discussion for MsFEM with spectral multiscale basis functions . --- p.25 / Chapter 3 --- Elliptic equations in high-contrast heterogeneous media --- p.28 / Chapter 3.1 --- Preliminaries --- p.29 / Chapter 3.2 --- Integral representation --- p.32 / Chapter 3.3 --- The well-posedness of the integral equation --- p.37 / Chapter 4 --- A numerical approach for the Elliptic equations in high-contrast heterogeneous media --- p.45 / Chapter 4.1 --- Introduction --- p.46 / Chapter 4.2 --- A new approach --- p.47 / Chapter 4.3 --- Discussion of the results --- p.50 / Chapter 4.4 --- Numerical experiment --- p.51 / Chapter 5 --- A preconditioner for high contrast problems using reduced-contrast approximations --- p.62 / Chapter 5.1 --- Reduced-contrast approximations for the solution of elliptic equations --- p.63 / Chapter 5.2 --- Review on multigrid methods --- p.66 / Chapter 5.3 --- Preconditioning and numerical experiments --- p.70 / Bibliography --- p.75

Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ao, Weiwei. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 178-186). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Interior Spike Solutions for Lin-Ni-Takagi Problem --- p.7 / Chapter 1.1.1 --- Background and Main Results --- p.7 / Chapter 1.1.2 --- Sketch of the Proof of Theorem 1.1.1 --- p.12 / Chapter 1.2 --- The A2 and B2 Chern-Simons System --- p.14 / Chapter 1.2.1 --- Background --- p.14 / Chapter 1.2.2 --- Previous Results --- p.19 / Chapter 1.2.3 --- Main Results --- p.20 / Chapter 1.2.4 --- Sketch of the Proof for A₂ Case --- p.21 / Chapter 1.2.5 --- Sketch of the Proof for B₂ Case --- p.26 / Chapter 1.3 --- Organization of the Thesis --- p.27 / Chapter 2 --- The Lin-Ni-Takagi Problem --- p.29 / Chapter 2.1 --- Notation and Some Preliminary Analysis --- p.29 / Chapter 2.2 --- Linear Theory --- p.35 / Chapter 2.3 --- The Non Linear Projected Problem --- p.40 / Chapter 2.4 --- An Improved Estimate --- p.43 / Chapter 2.5 --- The Reduced Problem: A Maximization Procedure --- p.50 / Chapter 2.6 --- Proof of Theorem 1.1.1 --- p.58 / Chapter 2.7 --- More Applications and Some Open Problems --- p.60 / Chapter 3 --- The Chern-Simons System --- p.66 / Chapter 3.1 --- Proof of Theorem 1.2.1 in the A₂ Case --- p.66 / Chapter 3.1.1 --- Functional Formulation of the Problem --- p.66 / Chapter 3.1.2 --- First Approximate Solution --- p.68 / Chapter 3.1.3 --- Invertibility of Linearized Operator --- p.72 / Chapter 3.1.4 --- Improvements of the Approximate Solution: O(ε) Term --- p.76 / Chapter 3.1.5 --- Next Improvement of the Approximate Solution: O(ε²) Term --- p.78 / Chapter 3.1.6 --- A Nonlinear Projected Problem --- p.82 / Chapter 3.1.7 --- Proof of Theorem 1.2.1 for A₂ under Assumption (i) --- p.85 / Chapter 3.1.8 --- Proof of Theorem 1.2.1 for A₂ under Assumption (ii) --- p.94 / Chapter 3.1.9 --- Proof of Theorem 1.2.1 for A₂ under Assumption (iii) --- p.99 / Chapter 3.2 --- Proof of Theorem 1.2.1 in the B₂ Case --- p.100 / Chapter 3.2.1 --- Functional Formulation of the Problem for B₂ Case --- p.100 / Chapter 3.2.2 --- Classi cation and Non-degeneracy for B₂ Toda system --- p.101 / Chapter 3.2.3 --- Invertibility of Linearized Operator --- p.105 / Chapter 3.2.4 --- Improvements of the Approximate Solution --- p.106 / Chapter 3.2.5 --- Proof of Theorem 1.2.1 for B₂ under Assumption (i) --- p.112 / Chapter 3.2.6 --- Proof of Theorem 1.2.1 for B₂ under Assumption (ii) --- p.122 / Chapter 3.2.7 --- Proof of Theorem 1.2.1 for B₂ under Assumption (iii) --- p.127 / Chapter 3.3 --- Open Problems --- p.128 / Chapter 4 --- Appendix --- p.129 / Chapter 4.1 --- B₂ and G₂ Toda System with Singular Source --- p.129 / Chapter 4.1.1 --- Case 1: B₂ Toda system with singular source --- p.130 / Chapter 4.1.2 --- Case 2: G₂ Toda system with singular source --- p.136 / Chapter 4.2 --- The Calculations of the Matrix Q₁ --- p.148 / Chapter 4.3 --- The Calculations of the Matrix Q₁ --- p.169 / Bibliography --- p.178

Inverse problems are mathematically and numerically very challenging due to their inherent ill-posedness in the sense that a small perturbation of the data may cause an enormous deviation of the solution. Regularization methods have been established as the standard approach for their stable numerical solution thanks to the ground-breaking work of late Russian mathematician A.N. Tikhonov. However, existing studies mainly focus on general-purpose regularization procedures rather than exploiting mathematical structures of specific problems for designing efficient numerical procedures. Moreover, the stochastic nature of data noise and model uncertainties is largely ignored, and its effect on the inverse solution is not assessed. This thesis attempts to design some problem-specific efficient numerical methods for the Robin inverse problem and to quantify the associated uncertainties. It consists of two parts: Part I discusses deterministic methods for the Robin inverse problem, while Part II studies stochastic numerics for uncertainty quantification of inverse problems and its implication on the choice of the regularization parameter in Tikhonov regularization. / Key Words: Robin inverse problem, variational approach, preconditioning, Modica-Motorla functional, spectral stochastic approach, Bayesian inference approach, augmented Tikhonov regularization method, regularization parameter, uncertainty quantification, reduced-order modeling / Part I considers the variational approach for reconstructing smooth and nonsmooth coefficients by minimizing a certain functional and its discretization by the finite element method. We propose the L2-norm regularization and the Modica-Mortola functional from phase transition for smooth and nonsmooth coefficients, respectively. The mathematical properties of the formulations and their discrete analogues, e.g. existence of a minimizer, stability (compactness), convexity and differentiability, are studied in detail. The convergence of the finite element approximation is also established. The nonlinear conjugate gradient method and the concave-convex procedure are suggested for solving discrete optimization problems. An efficient preconditioner based on the Sobolev inner product is proposed for justifying the gradient descent and for accelerating its convergence. / Part II studies two promising methodologies, i.e. the spectral stochastic approach (SSA) and the Bayesian inference approach, for uncertainty quantification of inverse problems. The SSA extends the variational approach to the stochastic context by generalized polynomial chaos expansion, and addresses inverse problems under uncertainties, e.g. random data noise and stochastic material properties. The well-posedness of the stochastic variational formulation is studied, and the convergence of its stochastic finite element approximation is established. Bayesian inference provides a natural framework for uncertainty quantification of a specific solution by considering an ensemble of inverse solutions consistent with the given data. To reduce its computational cost for nonlinear inverse problems incurred by repeated evaluation of the forward model, we propose two accelerating techniques by constructing accurate and inexpensive surrogate models, i.e. the proper orthogonal decomposition from reduced-order modeling and the stochastic collocation method from uncertainty propagation. By observing its connection with Tikhonov regularization, we propose two functionals of Tikhonov type that could automatically determine the regularization parameter and accurately detect the noise level. We establish the existence of a minimizer, and the convergence of an alternating iterative algorithm. This opens an avenue for designing fully data-driven inverse techniques. / This thesis considers deterministic and stochastic numerics for inverse problems associated with elliptic partial differential equations. The specific inverse problem under consideration is the Robin inverse problem: estimating the Robin coefficient of a Robin boundary condition from boundary measurements. It arises in diverse industrial applications, e.g. thermal engineering and nondestructive evaluation, where the coefficient profiles material properties on the boundary. / Jin, Bangti. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3541. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 174-187). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.