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We present a new iterative approach to solving neutral-particle transport problems. The scheme divides the transport solution into its particular and homogeneous or “source-free” components. The particular problem is solved directly, while the homogeneous problem is found iteratively. To organize the iterative inversion of the homogeneous components, we exploit the structures of the so called Case-modes that compose it. The asymptotic Case-modes, those that vary slowly in space and angle, are assigned to a diffusion solver. The remaining transient Case-modes, those with large spatial gradients, are assigned to a transport solver. The scheme iterates on the contribution from each solver until the particular plus homogeneous solution converges. The iterative method is implemented successfully in slab geometry with isotropic scattering and one energy group. The convergence rate of the method is only weakly dependent on the scattering ratio of the problem. Instead, the rate of convergence depends strongly on the material thickness of the slab, with thick slabs converging in few iterations. The transient solution is obtained by applying a One Cell Inversion scheme instead of a Source Iteration based scheme. Thus, the transient unknowns are calculated with little coordination between them. This independence among unknowns makes our scheme ideally suited for transport calculations on parallel architectures. The slab geometry iterative scheme is adapted to XY geometry. Unfortunately, this attempt to extend the slab geometry iterative scheme to multiple dimensions has not been successful. The exact filtering scheme needed to discriminate asymptotic and transient modes has not been obtained and attempts to approximate this filtering process resulted in a divergent iterative scheme. However, the development of this iterative scheme yield valuable analysis tools to understand the Case-mode structure of any spatial discretization under arbitrary material properties.

This study proposed a numerical algorithm for estimating the angular misalignments between sensors of an ultra short baseline (USBL) positioning system. The algorithm is based on positioning a seabed transponder by moving a vessel along a predetermined straight-line path. Under the scheme of straight-line survey, mathematical representations of positioning error arising from heading, pitch, and roll misalignments were derived, respectively. The effect of each misalignment angle and how the differences can be used to calibrate each misalignment angle in turn are presented. A USBL calibration procedure that takes advantage of the geometry of position errors resulting from angular misalignments is then developed. During the USBL measurement, temporal and spatial variations of sound speed structure in water column are the major error sources. Therefore, this study used the sound speed profile together with a ray tracing method to correct observations of the USBL measurement. In addition, this study developed a method to compensate the effects of cross-track error on the estimation of alignment errors, and this makes the proposed algorithm applicable for using a vessel without dynamic positioning (DP) systems to collect USBL observations. The performance of the algorithm is evaluated through simulation and field experiment. The simulation and experimental results have demonstrated the effectiveness and robustness of the iterative scheme in finding alignment errors. The proposed algorithm yields a very rapid convergence of the solution series; usually the estimates obtained in the first iteration approximate to true values, and only a few iterations are necessary to achieve fairly accurate solutions.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In Chapter 1 a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. We also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme. In Chapter 2 we give a convergence analysis of the following iterative scheme: u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0, where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional approximations of T and K[superscript]* for solving stably Fredholm integral equations of the first kind with noisy data. In Chapter 3 a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this chapter.

<p dir="ltr">In this thesis, we consider the inverse problem of reconstructing the shape, position, and size of an unknown scattering object. We will talk about different methods used for nondestructive testing in scattering theory. We will consider qualitative reconstruction methods to understand and determine important information about the support of unknown scattering objects. We will also discuss the material properties of the system and connect them to certain crucial aspects of the region of interest, as well as develop useful techniques to determine physical information using inverse scattering theory. </p><p><br></p><p dir="ltr">In the first part of the analysis, we consider the transmission eigenvalue (TE) problem associated with the scattering of a plane wave for an isotropic scatterer. In particular, we examine the transmission eigenvalue problem with two conductivity boundary parameters. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter, whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues. In addition, we will study the dependence of the TE's on the physical parameters and connect the first transmission eigenvalue to the physical parameters of the problem by a monotone-type argument. Lastly, we will consider the limiting procedure as the second boundary parameter vanishes at the boundary of the scattering region and provide numerical examples to validate the theory presented in Chapter 2. </p><p><br></p><p dir="ltr">The connection between transmission eigenvalues and the system's physical parameters provides a way to do testing in a nondestructive way. However, to understand the region of interest in terms of its shape, size, and position, one needs to use different techniques. As a result, we consider reconstructing extended scatterers using an analogous method to the Direct Sampling Method (DSM), a new sampling method based on the Landweber iteration. We will need a factorization of the far-field operator to analyze the corresponding imaging function for the new Landweber direct sampling method. Then, we use the factorization and the Funk--Hecke integral identity to prove that the new imaging function will accurately recover the scatterer. The method studied here falls under the category of qualitative reconstruction methods, where an imaging function is used to retrieve the scatterer. We prove the stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.</p><p><br></p><p dir="ltr">Motivated by the work done with the transmission eigenvalue problem with two conductivity parameters, we also study the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. In such a problem, one analyzes the behavior of the scattered field as one of the conductivity parameters vanishes at the boundary. Consequently, we prove the convergence of the scattered field dealing with two conductivity parameters to the scattered field dealing with only one conductivity parameter. We consider the uniqueness of recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we consider the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. To this end, we study the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable concerning noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance using the direct sampling method. </p>