Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1909. / "Estratto dal tomo XXVIII (2 sem. 1909)del Rendiconti del Circolo matematico di Palermo"--P. 1. Vita.

Thèse--La Faculté des sciences de Paris, 1901.

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.

In Chapter 2 we study the uniqueness problem of sign-changing solutions for a nonlinear scalar equation. It is well-known that positive solution is radially symmetric and unique up to a translation. Recently, there are many works on the existence and multiplicity of sign-changing solutions. However much less is known for uniqueness, even in the radially symmetric class. In Chapter 2, we solve this problem for nearly critical nonlinearity by Lyaponov-Schmidt reduction. Moreover, we can also prove the non-degeneracy. / In Chapter 3 we are concerned with the uniqueness problem for coupled nonlinear Schrodinger equations. The problem is to classify all positive solutions. In Chapter 3, some sufficient conditions are given. In particular, we have a sufficient and necessary condition in one dimension. The proof is elementary because only the implicit function theorem, integration by parts, and the uniqueness for scalar equation are needed. / In Chapter 4 we go back to the nonlinear scalar equation and consider the traveling wave solutions. Using an infinite dimensional Lyaponov-Schmidt reduction, new examples of traveling wave solutions are constructed. Our approach explains the difference between two dimension and higher dimensions, and also explores a connection between moving fronts and the mean curvature flow. This is the first such traveling waves connecting the same states. / This thesis is devoted to the study of nonlinear elliptic equations and systems. It is divided into two parts. In the first part, we study the uniqueness problem, and in the second part, we are concerns with traveling wave solutions. / Yao, Wei. / Adviser: Jun Cheng Wei. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 132-142). / 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, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In this master thesis, it is proposed to solve in the large thedifferential equationZ2(d2y/dz2) + z (dy/dz) (b0+b1zm) + (c0+c1zm)y = 0Here, m is an arbitrary positive integer, the variable z is complex as are the constants bi, ci (i = 0, 1). It is also assumed that the roots of the indicial equation about the regular singular point z=0 are such that their difference is incongruent to zero modulo m.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1927. / "Sonderabdruck aus der "Mathematischen Zeitschrift", Band 27, Heft 1"--T.p. verso. Vita. Includes bibliographical references.

Thesis (Ph. D.)--Kent State University, 2008. / Title from PDF t.p. (viewed Nov. 10, 2009). Advisor: Hassan Allouba. Keywords: stochastic partial differential equations; compact support property. Includes bibliographical references (p. 30).

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