This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance.

Optical fibre communication using the Nonlinear Fourier transform (NFT) is one of the potential solutions to tackle the so-called capacity crunch problem in long-haul optical fibre networks. The NFT transforms the nonlinear propagation of temporal signal, governed by the nonlinear Schr ̈odinger equation (NLSE), into simple linear evolutions of continuous and discrete spectra in the so-called nonlinear spectral domain. These spectra and the corresponding nonlinear spectral domain, defined by the NFT, are the generalized counterparts of the linear spectrum and frequency domain defined by the ordinary Fourier transform. Using the NFT, the optical fibre channel is effectively linearised, and the basic idea is to utilize degrees of freedom in the nonlinear spectral domain for data transmission. However, many aspects of this concept require rigorous investigation due to complexity and infancy of the approach. In this thesis, the aim is to provide a comprehensive investigation of data transmission over mainly the continues spectrum (CS) and partly over of the discrete spectrum (DS) of nonlinear optical fibres. First, an optical fibre communication system is defined, in which solely the CS carries the information. A noise model in the nonlinear spectral domain is derived for such a system by asymptotic analysis as well as extensive simulations for different scenarios of practical interest. It is demonstrated that the noise added to the signal in CS is severely signal-dependent such that the effective signalling space is limited. The variance normalizing transform (VNT) is used to mathematically verify the limits of signalling spaces and also estimate the channel capacity. The numerical results predict a remarkable capacity for signalling only on the CS (e.g., 6 bits/symbol for a 2000-km link), yet it is demonstrated that the capacity saturates at high power. Next, the broadening effect of chromatic dispersion is analysed, and it is confirmed that some system parameters, such as symbol rate in the nonlinear spectral domain, can be optimized so that the required temporal guard interval between the subsequently transmitted data packets is minimized, and thus the effective data rate is significantly enhanced. Furthermore, three modified signalling techniques are proposed and analysed based on the particular statistics of the noise added to the CS. All proposed methods display improved performance in terms of error rate and reach distance. For instance, using one of the proposed techniques and optimized parameters, a 7100-km distance can be reached by signalling on the CS at a rate of 9.6 Gbps. Furthermore, the impact of polarization mode dispersion (PMD) is examined for the first time, as an inevitable impairment in long-haul optical fibre links. By semi-analytical and numerical investigation, it is demonstrated that the PMD affects the CS by causing signal-dependent phase shift and noise-like errors. It is also verified that the noise is still the dominant cause of performance degradation, yet the effect of PMD should not be neglected in the analysis of NFT-based systems. Finally, the capacity of soliton communication with amplitude modulation (part of the degrees of freedom of DS) is also estimated using VNT. For the first time, the practical constraints, such as the restricted signalling space due to limited bandwidth, are included in this capacity analysis. Furthermore, the achievable data rates are estimated by considering an appropriately defined guard time between soliton pulses. Moreover, the possibility of transmitting data on DS accompanied by an independent CS signalling is also validated, which confirms the potentials of the NFT approach for combating the capacity crunch.

This thesis presents a robust method for estimating the relaxations of a metallic object from its electromagnetic induction (EMI) response. The EMI response of a metallic object can be accurately modeled by a sum of real decaying exponentials. However, it is difficult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the relaxation constants are estimated by nonlinear iterative search that often leads to unsatisfactory results. An effective EMI modeling technique is developed by first linearizing the problem through enumeration and then solving the linearized model using a sparsity-regularized minimization. This approach overcomes several long-standing challenges in EMI signal modeling, including finding the unknown model order as well as handling the ill-posed nature of the problem. The resulting algorithm does not require a good initial guess to converge to a satisfactory solution. This new modeling technique is extended to incorporate multiple measurements in a single parameter estimation step. More accurate estimates are obtained by exploiting an invariance property of the EMI response, which states that the relaxation frequencies do not change for different locations and orientations of a metallic object. Using tests on synthetic data and laboratory measurement of known targets, the proposed multiple-measurement method is shown to provide accurate and stable estimates of the model parameters. The ability to estimate the relaxation constants of targets enables more robust subsurface target discrimination using the relaxations. A simple relaxation-based subsurface target detection algorithm is developed to demonstrate the potential of the estimated relaxations.

This thesis presents a robust method for estimating the relaxations of a metallic object from its electromagnetic induction (EMI) response. The EMI response of a metallic object can be accurately modeled by a sum of real decaying exponentials. However, it is diffcult to obtain the model parameters from measurements when the number of exponentials in the sum is unknown or the terms are strongly correlated. Traditionally, the time constants and residues are estimated by nonlinear iterative search that often leads to unsatisfactory results. In this thesis, a constrained linear method of estimating the parameters is formulated by enumerating the relaxation parameter space and imposing a nonnegative constraint on the parameters. The resulting algorithm does not depend on a good initial guess to converge to a solution. Using tests on synthetic data and laboratory measurement of known targets the proposed method is shown to provide accurate and stable estimates of the model parameters.

Le but de cette thèse est d’obtenir des informations sur le spectre discret d’opérateurs non auto-adjoints définis par des perturbations relativement compactes d’opérateurs auto-adjoints. Ces opérateurs auto-adjoints sont choisis parmi les opérateurs classiques de mécanique quantique. Il s’agit des opérateurs de Dirac, de Klein-Gordon et le laplacien fractionnaire qui généralise l’opérateur de Schrödinger habituellement considéré pour de tels problèmes. La principale méthode utilisée ici relève d’un théorème d’analyse complexe donnant une condition de type Blaschke sur les zéros d’une fonction holomorphe du disque unité. Cette condition traduit lecomportement des valeurs propres de l’opérateur perturbé sous forme d’inégalités de type Lieb-Thirring. Une autre méthode venant d’analyse fonctionnelle a été employée pour obtenir de telles inégalités et les deux méthodes sont comparées entre elles. / The topic of this thesis concerns the discrete spectrum of non-selfadjoint operators defined by relatively compact perturbation of selfadjoint operators. These selfadjoint operators are choosen among classical operators of quantum mechanics. These areDirac operator, Klein-Gordon operator, and the fractional Laplacian who generalize the Schrödinger operator. The main method is based on a theorem of complex analysis which gives Blaschke-type condition on the zeros of a holomorphic function on the unit disc. This Blaschke condition gives the information on the behaviour of eigenvalues of the perturbed operator by mean of Lieb-Thirring-type inequalities. Another method using functional analysis is also used to obtain these kind of inequalities and both methods are compared to each other.

This thesis is a blend of neutron transport theory and numerical analysis. We start with the study of the problem of the Mika/Case eigenexpansion used in the solution process of the homogeneous one-speed Boltzmann neutron transport equation with anisotropic scattering for plane symmetry. The anisotropic scattering is expressed as a finite Legendre series in which the coefficients are the ``scattering coefficients'. This eigenexpansion consists of a discrete spectrum of eigenvalues with its corresponding eigenfunctions and the continuous spectrum [-1,+1] with its corresponding eigendistributions. In the general case where the anisotropic scattering can be of any (finite) order, multiple discrete eigenvalues exist and these have to be located to have the complete spectrum. We have devised a stable and robust method that locates all these discrete eigenvalues. The method is a two-step process: first the number of discrete eigenvalues is calculated and this is followed by the calculation of the discrete eigenvalues themselves, now being able to count them down and make sure none are forgotten. <p><p>During our numerical experiments, we came across what we called near-singular eigenvalues: discrete eigenvalues that are located extremely close to the continuum and hence lead to near-singular behaviour in the eigenfunction. Our solution method has been adapted and allows for the automatic detection of such a near-singular eigenvalue. <p><p>For the elements of the continuous spectrum [-1,+1], there is no non-zero function satisfying the associated eigenequation but there is a non-zero distribution that does satisfy it. It is not feasible to compute a distribution as such but one can evaluate integrals in which this distribution appears. The continuum part of the eigenexpansion can hence only be characterised by its (angular) moments. Accurate and fast numerical quadrature is needed to evaluate these integrals. Several quadrature methods have been evaluated on a representative test function. <p><p><p>The eigenexpansion was proved to be orthogonal and complete and hence can be used to represent the infinite medium Green's function. The latter is the building block of the Boundary Sources Method, an integral solution method for the neutron transport equation. Using angular and angular/spatial moments of the Green's function, it is possible to solve with high accuracy slab problems. We have written a one-dimensional slab code implementing this Boundary Sources Method allowing for media with arbitrary order anisotropic scattering. Our results are very good and the code can be considered as a benchmark code for others. <p><p><p>As a final application, we have used our code to study the discrete spectrum of a well-known scattering kernel in radiative transfer, the Henyey-Greenstein kernel. This kernel has one free parameter which is used to fit the kernel to experimental data. Since the kernel is a continuous function, a finite Legendre approximation needs to be adopted. Depending on the free parameter, the approximation order and the number of secondaries per collision, the number of discrete eigenvalues ranges from two to thirty and even more. Bounds for the minimum approximation order are derived for different requirements on the approximation: non-negativity, an absolute and relative error tolerance. <p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished