Today the multisensor array signal processing of noisy measurements has received much attention. The classical problem in array signal processing is determining the location of an energy-radiating source relative to the location of the array, in other words, direction-of-arrival (DOA) estimation. One is considering the signal estimation problem when together with the signal(s) of interest some noise and interfering signals are present. In this report a direction-of-arrival estimation system is described based on an antenna array for detecting arrival angles in azimuth plane of signals pitched by the antenna array. For this, the Multiple Signal Classication (MUSIC) algorithmis first of all considered. Studies show that in spite of its good reputation and popularity among researches, it has a certain limit of its performance. In this subspace-based method for DOA estimation of signal wavefronts, the term corresponding to additive noise is initially assumed spatially white. In our paper, we address the problem of DOA estimation of multiple target signals in a particular noise situation - in correlated spherically isotropic noise, which, in many practical cases, models a more real context than under the white noise assumption. The purpose of this work is to analyze the behaviour of the MUSIC algorithm and compare its performance with some other algorithms (such as the Capon and the Classical algorithms) and, uppermost, to explore the quality of the detected angles in terms of precision depending on different parameters, e.g. number of samples, noise variance, number of incoming signals. Some modifications of the algorithms are also done is order to increase their performance. Program MATLAB is used to conduct the studies. The simulation results on the considered antenna array system indicate that in complex conditions the algorithms in question (and first of all, the MUSIC algorithm) are unable to automatically detect and localize the DOA signals with high accuracy. Other algorithms andways for simplification the problem (for example, procedure of denoising) exist and may provide more precision but require more computation time.

Aeroelastic behavior prediction is often confined to analytical or highly computational methods, so I developed a low degree of freedom computational method using structural finite elements and unsteady loading to cover a gap in the literature. Finite elements are readily suitable for determination of the free vibration characteristics of eccentric, elastic structures, and the free vibration characteristics fundamentally determine the aeroelastic behavior. I used Theodorsen’s unsteady strip loading formulation to model the aerodynamic loading on linear elastic structures assuming harmonic motion. I applied Hassig’s ‘p-k’ method to predict the flutter boundary of nonsymmetric, aeroelastic systems. I investigated the application of a quintic interpolation assumed displacement shape to accurately predict higher order characteristic effects compared to linear analytical results. I show that quintic interpolation is especially accurate over cubic interpolation when multi-modal interactions are considered in low degree of freedom flutter behavior for high aspect ratio HALE aircraft wings.

A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations Scott W. Mosher 110 Pages Directed by Dr. Farzad Rahnema It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost. In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion. The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].

The variety of principal minors of nxn symmetric matrices, denoted Zn, can be described naturally as a projection from the Lagrangian Grassmannian. Moreover, Zn is invariant under the action of a group G C GL(2n) isomorphic to (SL(2)xn) x Sn. One may use this symmetry to study the defining ideal of Zn as a G-module via a coupling of classical representation theory and geometry. The need for the equations in the defining ideal comes from applications in matrix theory, probability theory, spectral graph theory and statistical physics. I describe an irreducible G-module of degree 4 polynomials called the hyperdeterminantal module (which is constructed as the span of the G-orbit of Cayley's hyperdeterminant of format 2 x 2 x 2) and show that it that cuts out Zn set theoretically. This result solves the set-theoretic version of a conjecture of Holtz and Sturmfels and gives a collection of necessary and sufficient conditions for when it is possible for a given vector of length 2n to be the principal minors of a symmetric n x n matrix. In addition to solving the Holtz and Sturmfels conjecture, I study Zn as a prototypical G-variety. As a result, I exhibit the use of and further develop techniques from classical representation theory and geometry for studying G-varieties.

Analysis of large networks in biology, science, technology and social systems have become very popular recently. These networks are mathematically represented as graphs. The task is then to extract relevant qualitative information about the empirical networks from the analysis of these graphs. It was found that a graph can be conveniently represented by the spectrum of a suitable difference operator, the normalized graph Laplacian, which underlies diffusions and random walks on graphs. When applied to large networks, this requires computation of the spectrum of large matrices. The normalized Laplacian matrices representing large networks are usually sparse and unstructured. The thesis consists in a systematic evaluation of the available eigenvalue solvers for nonsymmetric large normalized Laplacian matrices describing directed graphs of empirical networks. The methods include several Krylov subspace algorithms like implicitly restarted Arnoldi method, Krylov-Schur method and Jacobi-Davidson methods which are freely available as standard packages written in MATLAB or SLEPc, in the library written C++. The normalized graph Laplacian as employed here is normalized such that its spectrum is confined to the range [0, 2]. The eigenvalue distribution plays an important role in network analysis. The numerical task is then to determine the whole spectrum with appropriate eigenvalue solvers. A comparison of the existing eigenvalue solvers is done with Paley digraphs with known eigenvalues and for citation networks in sizes 400, 1100 and 4500 by computing the residuals.

The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov algorithm to yield a method that permits an adjustment of shift after every step of the computation, adding greatly to the flexibility of the algorithm. We call this new method rational SHIRA. A numerical example is presented to demonstrate its efficiency.

In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work

The present work introduces a methodology to obtain a discrete time state space representation of an electrical network using the nodal [G] matrix of the Electromagnetic Transients Program (EMTP) solution. This is the first time the connection between the EMTP nodal analysis solution and a corresponding state-space formulation is presented. Compared to conventional state space solutions, the nodal EMTP solution is computationally much more efficient. Compared to the phasor solutions used in transient stability analysis, the proposed approach captures a much wider range of eigenvalues and system operating states. A fundamental advantage of extracting the system eigenvalues directly from the EMTP solution is the ability of the EMTP to follow the characteristics of nonlinearities. The system's trajectory can be accurately traced and the calculated eigenvalues and eigenvectors correctly represent the system's instantaneous dynamics. In addition, the algorithm can be used as a tool to identify network partitioning subsystems suitable for real-time hybrid power system simulator environments, including the implementation of multi-time scale solutions. The proposed technique can be implemented as an extension to any EMTP-based simulator. Within our UBC research group, it is aimed at extending the capabilities of our real-time PC-cluster Object Virtual Network Integrator (OVNI) simulator.

Some of the theory of nonnegative matrices is first presented. The Perron-Frobenius theorem is highlighted. Some of the important linear algebraic methods of information retrieval are surveyed. Latent Semantic Indexing (LSI), which uses the singular value de-composition is discussed. The Hyper-Text Induced Topic Search (HITS) algorithm is next considered; here the power method for finding dominant eigenvectors is employed. Through the use of a theorem by Sinkohrn and Knopp, a modified HITS method is developed. Lastly, the PageRank algorithm is discussed. Numerical examples and MATLAB programs are also provided.