A new technique is presented to study radio propagation and rough surface scattering problems based on a reformulation of the Magnetic Field Integration Equation (MFIE) called the Single-Scatter Subtraction (S^3) method. This technique amounts to a physical preconditioning by separating the single- and multiple-scatter currents and removing the single-scattering contribution from the integral term that is present in the MFIE. This requires the calculation of a new quantity that is the kernel of the MFIE integral call the kernel integral or Gbar. In this work, 1-dimensional deterministically rough surfaces are simulated by surfaces consisting of single and multiple cosines. In order to truncate the problem domain, a beam illumination is used as the source term and it is shown that this also causes the kernel integral to have a finite support. Using the Single Scatter Subtraction method on these surfaces, closed-form expressions are found for the kernel integral and thus the single-scatter current for a well defined region of validity of surface parameters which may then be efficiently radiated into the far field numerically. Both the closed-form expressions, and the computed radiated fields are studied for their physical significance. This provides a clear physical intuition for the technique as an augmentation to existing ones as a bent-plane approximation as shown analytically and also validated by numeric results. Further analysis resolves a controversy on the nature of Bragg scatter which is found to be a multiple-scatter phenomenon. Error terms present in the kernel integral also raise new questions on the effect of truncation for any MFIE-based solution. Additionally, a dramatic enhancement of backscatter predicted by this new approach versus the Kirchhoff method is observed as the angle of incidence increases due to the error terms. / Doctor of Philosophy / A new technique is presented to study the interaction of electromagnetic waves with rough surfaces. Building on the technique called the Magnetic Field Integral Equation (MFIE) which allows the solution for the electromagnetic fields scattered from the surface by considering only the induced electric and magnetic currents on the surface, the Single-Scatter Substraction (S 3 ) method separates the surface currents into those that interact with the surface only once or single-scatter, and those that interact multiple times called multiple-scatter. Since this is the introduction of this technique, only the former is investigated. In this study, a new quantity which is an integral of one of the components of the standard MFIE is studied and closed-form approximations are presented along with bounds of validity. This provides closed form solutions for the single-scattering currents, from which the radiated fields may be efficiently found numerically. Since they are closed form, the expressions provide insight into the nature of the physical scattering process. Numerical results of these expressions are compared to the standard approximate technique as well as the ”exact” solution found by numerically solving the MFIE. Compared to the standard approximate technique which approximates the surface by a tangent plane at each point on the surface, the single-scatter currents approximate the surface with a bent-plane at each point. This shifts the scattered fields from certain directions to others, and highlights where single- and multiple-scattering have an effect.

A novel statistical technique introduced by Besieris is used to study solutions of the nonlinear stochastic complex parabolic equation in the presence of two profiles. Specifically, the randomly modulated linear potential and the randomly perturbed quadratic focusing medium. In the former, a class of solutions is shown to admit an exact statistical description in terms of the moments of the wave function. In the latter, all even-order moments are computed exactly, whereas the odd-order moments are solved asymptotically. Lastly, it is shown that this statistical technique is isomorphic to mappings of nonconstant coefficient partial differential equations to constant coefficient equations. A generalization of this mapping and its inherent restrictions are discussed. / Ph. D. / incomplete_metadata

We investigate the steady state solutions that can exist for a two dimensional swarm of biological organisms, which have pairwise social interaction forces. The three steady states we investigate using a continuum model are a ribbon migrating swarm, a circular migrating swarm, and a milling swarm. We solve these numerically by reformulating the integral equation that arises from the continuum model as an energy minimization problem. For the ribbon migrating solution, we are able to determine an analytic solution from Carleman's equation which arises after an asymptotic expansion of the social interaction potential. Using this technique we are able to show the existence of a square root singularity that emerges at the boundary of the compactly supported swarm. The analytic solution agrees with the numerical solution for certain parameter values in the social interaction potential. We then demonstrate the existence of solutions for a migrating and milling circular swarm which contain a square root singularity. The milling swarm looks similar to the infinite ribbon, so we are able to use an asymptotic expansion of the potential to obtain an analytic solution in this case as well. The singularities in the density of the swarm suggest that the Morse potential should not be used for modeling biological swarming.

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This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to the problem of finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is replaced by a piecewise polynomial approximation, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of a two dimensional scattering problem, integrals over four dimensional space-time manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two temporal integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions are generalised into a class of admissible kernel functions. A quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions is presented and proven to converge exponentially by using the theory of countably normed spaces. A priori error estimates for the Galerkin approximation scheme are recalled, enhanced and discussed. In particular, the scattered wave’s energy is studied as an alternative error measure. The numerical schemes are presented in such a way that allows the use of non-uniform meshes in space and time, in order to be used with adaptive methods that are based on a posteriori error indicators and which modify the computational domain according to the values of these error indicators. The theoretical analysis of these schemes demands the study of generalised mapping properties of time domain boundary layer potentials and integral operators, analogously to the well known results for elliptic problems. These mapping properties are shown for both two and three space dimensions. Using the generalised mapping properties, three types of a posteriori error estimators are adopted from the literature on elliptic problems and studied within the context of the two dimensional transient problem. Some comments on the three dimensional case are also given. Advantages and disadvantages of each of these a posteriori error estimates are discussed and compared to the a priori error estimates. The thesis concludes with the presentation of two adaptive schemes for the two dimensional scattering problem and some corresponding numerical experiments.

The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied. / by Drake Harmon. / Vita. / Thesis (Ph.D.)--Florida Atlantic University, 2013. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

L'équation intégrale du champ électrique (EFIE) et l'équation intégrale du champ combiné (CFIE) souffrent d'un mauvais conditionnement à haute discrétisation et à bassefréquence : si la taille moyenne des arrêtes du maillage est réduite ou si la fréquence est diminuée le conditionnement du système se dégrade rapidement. Cela provoque le ralentissement ou la non convergence des solveurs itératifs. Cette dissertation présente de nouveaux paradigmes permettant l'obtention de solveurs à convergence rapide pour équations intégrales; pour prévenir la dégradation du conditionnement nous avançons l'état de l'art des techniques de préconditionnement dites de Calderon et de celles reposant sur l'utilisation des bases hiérarchiques. Pour traiter l'EFIE, nous introduisons une base hiérarchique pour maillages structurés et non-structurés dérivant des pré-ondelettes primaires et duales de Haar. De plus, nous introduisons un nouveau cadre permettant de préconditionner efficacement l'EFIE dans le cas d'objets à connexion multiples. L'applicabilité à la CFIE des préconditionneurs à bases hiérarchiques fait l'objet d'une étude aboutissant à la formalisation d'une technique de préconditionnement. Nous présentons aussi un préconditionneur multiplicatif de type Calderon (RF-CMP) qui permet l'obtention d'une matrice système Hermitienne, définie positive (HDP) et bien conditionnée, sans avoir recours, contrairement aux préconditionneurs existants, au raffinement du maillage ni à l'utilisation de fonction duales. Puisque la matrice est HPD, la méthode du gradient conjugué peut servir de solveur itératif avec une convergence garantie. / The electric field integral equation (EFIE) and the combined field integral equation(CFIE) suffer from the dense-discretization and the low-frequency breakdown: if the average edgelength of the mesh is reduced, or if the frequency is decreased, then the condition number of the system matrix grows. This leads to slowly or non-converging iterative solvers. This dissertation presents new paradigms for rapidly converging integral equation solvers: to overcome the illconditioning, we advance and extend the state of the art both in hierarchical basis and in Calderón preconditioning techniques. For the EFIE, we introduce a hierarchical basis for structured and unstructured meshes based on generalized primal and dual Haar prewavelets. Furthermore, a framework is introduced which renders the hierarchical basis able to efficiently precondition the EFIE in the case that the scatterer is multiply connected. The applicability of hierarchical basis preconditioners to the CFIE is analyzed and an efficient preconditioning scheme is derived. In addition, we present a refinement-free Calderón multiplicative preconditioner (RF-CMP) that yields a system matrix which is Hermitian, positive definite (HPD), and well-conditioned. Different from existing Calderón preconditioners, no dual basis functions and thus no refinement of the mesh is required. Since the matrix is HPD—in contrast to standard discretizations of the EFIE—we can apply the conjugate gradient (CG) method as iterative solver, which guarantees convergence. Eventually, the RF-CMP is extended to the CFIE.

Another extension of Nonlinear Integral, Upper and Lower Nonlinear Integrals, which is a pair of extreme nonlinear integrals to contain all types of Nonlinear Integrals in the same scheme, is also proposed. It can give a set of upper and lower bounds which include all types of Nonlinear Integrals. We tried to find a solution with the smallest distance between the upper and lower bounds and the smallest error which is a NP hard problem. So we use the multi-objective optimization method to find a set of results for the regression model based on the Upper and Lower Nonlinear Integrals. We can just select one or more optimal solution(s) for a specific problem from the set of results. A weather predictor based on this model has been constructed to predict the next days temperature changing trend and range. / Finally, a NI based data mining framework has been established for identifying the chance of developing liver cancer based on the Hepatitis B Virus DNA sequence data. We have shown that the framework obtains the best diagnosing performance amongst many existing classifiers. / Nonlinear Integral (NI) is a useful integration tool. It has been applied to many areas including classification and regression. The classical method relies on a large number of training data, which lead to large time and space complexity. Moreover, the classical Nonlinear Integral has many limitations. For dealing with different situation, we propose Double Nonlinear Integrals and Nonlinear Integrals with Polynomial Kernel to deal with the problems transversely and longitudinally. / The classical Nonlinear Integrals implement projection along a line with respect to the features. But in many cases the linear projection cannot achieve good performance for classification or regression due to the limitation of the integrand. The linear function used for the integrand is just a special type of polynomial functions with respect to the features. We propose Nonlinear Integral with Polynomial Kernel (NIPK) in which a polynomial function is used as the integrand of Nonlinear Integral. It enables the projection to be along different types of curves on the virtual space, so that the virtual values gotten by the Nonlinear Integrals with Polynomial Kernel can be better regularized and easier to deal with. Experiments show that there is evident improvement of performance for NIPK compared to classical NI. / When the data to be classified have special distribution in the data space, the projection may overlap and the classification accuracy will be lowered. For example, when one group of the data is surrounded by the data of another group, or the number of classes for the data is large. To handle this kind of problems; we propose a new classification model based on the Double Nonlinear Integrals. Double Nonlinear Integral means projecting to a 2-Dimensional space by using the Nonlinear Integral twice in succession and classifying the virtual values in the 2-D space corresponding to the original data. Double Nonlinear Integrals can lessen loss of information due to the intersection of different classes on real axis. Accuracy will also be increased accordingly. / Wang, Jinfeng. / Advisers: Kwong Sak Leung; Kin Hong Lee. / Source: Dissertation Abstracts International, Volume: 72-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 139-151). / 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 Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

The radiative transfer equation (RTE) models the transport of radiation through a participating medium. In particular, it captures how radiation is scattered, emitted, and absorbed as it interacts with the medium. This process arises in numerous application areas, including: neutron transport in nuclear reactors, radiation therapy in cancer treatment planning, and the investigation of forming galaxies in astrophysics. As a result, there is great interest in the solution of the RTE in many different fields. We consider the energy dependent form of the RTE and allow media containing regions of negligible absorption. This particular case is not often considered due to the additional dimension and stability issues which arise by allowing vanishing absorption. In this thesis, we establish the existence and uniqueness of the underlying boundary value problem. We then proceed to develop a stable numerical algorithm for solving the RTE. Alongside the construction of the method, we derive corresponding error estimates. To show the validity of the algorithm in practice, we apply the algorithm to four different example problems. We also use these examples to validate our theoretical results.

Spectral methods for partial differential equations with boundary conditions in complex domains are developed with the help of a fictitious domain approach. For rectangular embedding, spectral-Galerkin formulations with special trial and test functions are presented and discussed, as well as the well-posedness and the error analysis. For circular and annular embedding, dimension reduction is applied and a sequence of 1-D problems with artificial boundary values are solved. Applications of our methods include the fractional Laplace problem and the Helmholtz equations. In numerical examples, our methods show good performance on the boundary value problems in both smooth and polygonal complex domains, and the L2 errors decay exponentially for smooth solutions. For singular problems, high-order convergence rates can also be obtained.