Inverse problems aim at estimating the unknown excitations or properties of a physical system based on available measurements of the system response. For example, wave tomography is used in geophysics for seismic waveform inversion; in biomedical engineering, optical tomography is used to detect breast cancer tissue; in structural engineering, inversion techniques are used for health monitoring and damage detection in structural safety evaluation. Inverse solvers depend on the type of measurement data the unknown parameters to be estimated. The work in this thesis focuses on structural parameter identification based on static and dynamic measurements. As an integral part of the formulated inverse solver, the associated forward problem is studded and deeply investigated. In reality, the data are associated with uncertainties caused by measurement devices or unfriendly environmental conditions during data acquisition. Traditional approaches use probability theory and model uncertainties as random variables. This approach has its own limitation due to a prior assumption on the probability structure of uncertainty. This is usually too optimistic or not realistic. However, in practice, it is usually difficult to reliably assess the statistical nature of uncertainties. Instead, only bounds on the uncertain variables and some partial information about their probabilities are known. The main source of uncertainty is due to the accuracy of measuring devices; these are designed to operate within specific allowable tolerances, as defined by National Institute of Standards and Technology (NIST). Tolerances are performance requirements that fix the limit of allowable error or departure from true performance or value. Thus closed intervals are the most realistic way to model uncertainty in measurements. In this work, uncertainties in measurement data are modeled as interval variables bounded by their endpoints. It is proven that interval analysis provides guaranteed enclosure of the exact solution set regardless of the underlying nature of the associated uncertainties. This work presents a solution of inverse problems under measurements uncertainty within the framework of Interval Finite Element Methods (IFEM) and adjoint-based optimization techniques. The solution consists of a two-step algorithm: first, an estimate of the parameters is obtained by means of a deterministic iterative solver. Then, the algorithm switches to a full interval solution, using the previous deterministic estimate as an initial guess. In general, the solution of an inverse problem requires iterative solutions of the forward problem. Efficient and accurate interval forward solutions in static and dynamic domains have been developed. In particular, overestimation due to interval dependency has been drastically reduced using a new decomposition of the load, stiffness, and mass matrices. Further improvements in the available interval iterative solvers have been achieved. Conjugate gradient and Newton-Raphson methods to gether with an inexact line search are used in the newly formulated optimization procedure. Moreover Tikhonov regularization is used to improve the conditioning of the ill-posed inverse problem. The developed interval solution for the inverse problem under uncertainty has been tested in a wide range of applications in static and dynamic domains. By comparing current solutions with other available methods in the literature, it is proven that the developed method provides guaranteed sharp bounds on the exact solution sets at a low computational cost. In addition, it contains those solutions provided by probabilistic approaches regardless of the used probability distributions. In conclusion, the developed method provides a powerful tool for the analysis of structural inverse problem under uncertainty.

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Constitutive model parameters are identified during model calibration through trial-and-error process driven to fit test data. In this research, the calibration of constitutive models is formally handled as an inverse problem. The first phase of this research explores error propagation. Data errors, experimental biases (e.g. improper boundary conditions), and model errors affect the inversion of model parameters and ensuing numerical predictions. Drained and undrained tests are simulated to study the effect of these three classes of errors. Emphasis is placed on the analysis of error surfaces computed by successive forward simulations. The second phase of this research centers on test procedures. Conventional soil tests were developed to create uniform stress and strain fields; consequently, they provide limited amount of information, the inversion is ill-posed, and results enhance uncertainty and error propagation. This research examines soil testing using new, non-conventional loading and boundary conditions to create rich, diverse, non-uniform strain and stress fields. In particular, the flexural excitation of cylindrical soil specimens is shown to provide rich data leading to a more informative test than conventional geotechnical tests. The new test is numerically optimized. Then a set of unique experimental studies is conducted.

In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue problem on (0, 1), −(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) £fw(x)y(p−1) (0.1) Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case p = 2. The spectrum {£fk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding to£fn has exactly n−1 zeros in (0, 1). Using a Pr¡Lufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) £f 1. Corresponding Ambarzumyan problems are also solved. We also study an associated boundary value problem with a nonlinear nonhomogeneous term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz continuous on R+, and odd on R such that f0 := lim y!0+ f(y) yp−1 , f1 := lim y!1 f(y) yp−1 . are not equal. We extend Kong¡¦s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped with any linear separated boundary conditions. Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr¡Lufer substitution techniques.

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This thesis reports on the research carried out in the area of Microwave Tomography (MWT) where the study aims to develop inversion algorithms to obtain cheap and stable solutions of MWT inverse scattering problems which are mathematically formulated as nonlinear ill posed problems. The study develops two algorithms namely Inexact Newton Backtracking Method (INBM) and Newton Iterative-Conjugate Gradient on Normal Equation (NI-CGNE) which are based on Newton method. These algorithms apply implicit solutions of the Newton equations with unspecific manner functioning as the regularized step size of the Newton iterative. The two developed methods were tested by the use of numerical examples and experimental data gained by the MWT system of the University of Manchester. The numerical experiments were done on samples with dielectric contrast objects containing different kinds of materials and lossy materials. Meanwhile, the quality of the methods is evaluated by comparingthem with the Levenberg Marquardt method (LM). Under the natural assumption that the INBM is a regularized method and the CGNE is a semi regularized method, the results of experiments show that INBM and NI-CGNE improve the speed, the spatial resolutions and the quality of direct regularization method by means of the LM method. The experiments also show that the developed algorithms are more flexible to theeffect of noise and lossy materials compared with the LM algorithm.

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².

This dissertation describes research conducted on the development and improvement of microwave tomography algorithms for imaging the bulk-electrical parameters of unknown objects. The full derivation of a new inversion algorithm based on the state-of-the-art contrast source inversion (CSI) algorithm coupled to a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The algorithm is applied to two-dimensional (2D) scalar and vectorial configurations, as well as three-dimensional (3D) full-vectorial problems. The unknown electrical properties of the object are distributed on the elements of arbitrary meshes with varying densities. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The CSI algorithm is used in conjunction with multiplicative regularization (MR), as it is typical in most implementations of CSI. Due to the use of arbitrary meshes in the present implementation, new techniques are introduced to perform the necessary spatial gradient and divergence operators of MR. The approach is different from other MR-CSI implementations where the unknown variables are located on a uniform grid of rectangular cells and represented using pulse basis functions; with rectangular cells finite-difference operators can be used, but this becomes unwieldy in FEM-CSI. Furthermore, an improvement for MR is proposed that accounts for the imbalance between the real and imaginary parts of the electrical properties of the unknown objects. The proposed method is not restricted to any particular formulation of the contrast source inversion. The functionality of the new inversion algorithm with the different enhancements is tested using a wide range of synthetic datasets, as well as experimental data collected by the University of Manitoba electromagnetic imaging group and research centers in Spain and France.

This dissertation describes research conducted on the development and improvement of microwave tomography algorithms for imaging the bulk-electrical parameters of unknown objects. The full derivation of a new inversion algorithm based on the state-of-the-art contrast source inversion (CSI) algorithm coupled to a finite-element method (FEM) discretization of the Helmholtz differential operator formulation for the scattered electromagnetic field is presented. The algorithm is applied to two-dimensional (2D) scalar and vectorial configurations, as well as three-dimensional (3D) full-vectorial problems. The unknown electrical properties of the object are distributed on the elements of arbitrary meshes with varying densities. The use of FEM to represent the Helmholtz operator allows for the flexibility of having an inhomogeneous background medium, as well as the ability to accurately model any boundary shape or type: both conducting and absorbing. The CSI algorithm is used in conjunction with multiplicative regularization (MR), as it is typical in most implementations of CSI. Due to the use of arbitrary meshes in the present implementation, new techniques are introduced to perform the necessary spatial gradient and divergence operators of MR. The approach is different from other MR-CSI implementations where the unknown variables are located on a uniform grid of rectangular cells and represented using pulse basis functions; with rectangular cells finite-difference operators can be used, but this becomes unwieldy in FEM-CSI. Furthermore, an improvement for MR is proposed that accounts for the imbalance between the real and imaginary parts of the electrical properties of the unknown objects. The proposed method is not restricted to any particular formulation of the contrast source inversion. The functionality of the new inversion algorithm with the different enhancements is tested using a wide range of synthetic datasets, as well as experimental data collected by the University of Manitoba electromagnetic imaging group and research centers in Spain and France.

Spectral geometry is a mathematical discipline that studies the relationship between the geometry of Riemannian manifolds and the spectra of natural differential operators defined on them. The spectra of Laplacians are the ones most studied in this context. A sub-field of this discipline, called inverse spectral geometry, studies how much geometric information one can recover from such spectra. The motivation behind our study of inverse spectral geometry is a physical one. It has recently been proposed that inverse spectral geometry could be the missing mathematical link between quantum field theory and general relativity needed to unify those theories into a single theory of quantum gravity. Unfortunately, this proposed link is not well understood. Most of the efforts in inverse spectral geometry were historically concentrated on the generation of counterexamples to the most general formulation of inverse spectral geometry and the few positive results that exist are quite limited. In order to remedy to that, it has been proposed to linearize the problem, and study an infinitesimal version of inverse spectral geometry. In this thesis, I begin by reviewing the theory of pseudodifferential operators and using it to prove the spectral theorem for elliptic operators. I then define the commonly used Laplacians and survey positive and negative results in inverse spectral geometry. Afterwards, I briefly discuss a coordinate free reformulation of Riemannian geometry via the notion of spectral triple. Finally, I introduce a formulation of inverse spectral geometry adapted for numerical implementations and apply it to the inverse spectral geometry of a particular class of star-shaped domains in ℝ².