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.

Sunnyson Y.F. Seid. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 65-66). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Applications in parameter identifications --- p.1 / Chapter 1.2 --- Inverse problems --- p.6 / Chapter 1.3 --- Difficulties arising in inverse problems --- p.7 / Chapter 2 --- Methods in Parameter Identifications --- p.9 / Chapter 2.1 --- Output Least Squares Method --- p.9 / Chapter 2.2 --- Equation Error Method --- p.11 / Chapter 2.3 --- Augmented Lagrangian Techniques --- p.12 / Chapter 2.4 --- Variational Techniques --- p.14 / Chapter 2.5 --- Adaptive Control Methods --- p.15 / Chapter 2.6 --- Method of Characteristics --- p.16 / Chapter 2.7 --- Our Proposed Method --- p.17 / Chapter 3 --- Parameter Identifications in Elliptic Systems --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Finite element approach and its convergence --- p.21 / Chapter 3.3 --- Unconstrained minimization problems --- p.28 / Chapter 3.4 --- Armijo algorithm --- p.31 / Chapter 3.5 --- Numerical experiments --- p.34 / Chapter 3.6 --- Multi-level coarse grid techniques --- p.55 / Bibliography --- p.65

Geng Di. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 116-119). / Acknowledgement --- p.i / Abstract --- p.ii / Introduction --- p.iii / Part I --- p.1 / Chapter 1 --- Critical Dimension of a Semilinear Degenerate Elliptic Equation Involving Critical Sobolev-Hardy Exponent --- p.2 / Chapter 1.1 --- Introduction --- p.2 / Chapter 1.2 --- Non-existence (I) --- p.5 / Chapter 1.3 --- Non-existence (II) --- p.11 / Chapter 1.4 --- Existence --- p.13 / Chapter 1.5 --- Radial Symmetry of Solutions --- p.16 / Appendix A --- p.20 / Appendix B --- p.23 / Chapter 2 --- Critical Dimension of a Hessian Equation Involving Critical Ex- ponent --- p.27 / Chapter 2.1 --- Introduction --- p.27 / Chapter 2.2 --- Preliminary Results --- p.29 / Chapter 2.3 --- Existence Results --- p.32 / Chapter 2.4 --- Non-existence Results --- p.43 / Chapter 3 --- Absence of Critical Dimension for the Subelliptic Laplacian on the Heisenberg Group --- p.48 / Chapter 3.1 --- Introduction and Main Result --- p.48 / Chapter 3.2 --- Proof of the Theorem --- p.49 / Part2 --- p.55 / Chapter 4 --- Asymptotic Behavior for Weighted p-Laplace Equations Involv- ing Critical Growth on the Ball --- p.56 / Chapter 4.1 --- Introduction --- p.56 / Chapter 4.2 --- A Crucial Lemma --- p.59 / Chapter 4.3 --- Proof of the Main Theorems --- p.61 / Chapter 5 --- Asymptotics for a Semilinear Weighted Elliptic Equation In- volving Critical Sobolev-Hardy Exponent --- p.71 / Chapter 5.1 --- Introduction --- p.71 / Chapter 5.2 --- Some Preliminary Results --- p.73 / Chapter 5.3 --- A Crucial Estimate --- p.80 / Chapter 5.4 --- Proof of the Main Theorem --- p.85 / Appendix --- p.88 / Chapter 6 --- Asymptotics for Positive Solutions for a Biharmonic Equation Involving Critical Exponent --- p.93 / Chapter 6.1 --- Introduction --- p.93 / Chapter 6.2 --- Preliminary Results --- p.94 / Chapter 6.3 --- Pohozaev's identity and Green's Function --- p.98 / Chapter 6.4 --- A Crucial Lemma --- p.103 / Chapter 6.5 --- Proof of Main Theorem --- p.112 / Bibliography --- p.115

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 56-58). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 212-219). Available also in a digital version from Dissertation Abstracts.

Not available / text

Simulation of flow through a heterogeneous porous medium with fine-scale features can be computationally expensive if the flow is fully resolved. Coarsening the problem gives a faster approximation of the flow but loses some detail. We propose an algorithm that obtains the fully resolved approximation but only iterates on a sequence of coarsened problems. The sequence is chosen by optimizing the shapes of the coarse finite element basis functions. As a stand-alone method, the algorithm converges globally and monotonically with a quadratic asymptotic rate. Computational experience indicates the number of iterations needed is independent of the resolution and heterogeneity of the medium. However, an externally provided error estimate is required; the algorithm could be combined as an accelerator with another iterative algorithm. A single "inner" iteration of the other algorithm would yield an error estimate; following it with an "outer" iteration of our algorithm would give a viable method. / text

Thesis (DTech. degree in the Dept. of Mathematics and Statistics.)-Tshwane University of Technology, 2008.

No description available.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.