Many interacting complex systems in biology, in physics, in technology and social systems, can be represented in a form of large networks. These large networks are mathematically represented by graphs. A graph is represented usually by the adjacency or the Laplacian matrix. Important features of the underlying structure and dynamics of them can be extracted from the analysis of the spectrum of the graphs. Spectral analysis of the so called normalized Laplacian of large networks became popular in the recent years. The Laplacian matrices of the empirical networks are in form of unstructured large sparse matrices. The aim of this thesis is the comparison of different eigenvalue solvers for large sparse symmetric matrices which arise from the graph theoretical epresentation of undirected networks. The spectrum of the normalized Laplacian is in the interval [0 2] and the multiplicity of the eigenvalue 1 plays a particularly important role for the network analysis. Moreover, the spectral analysis of protein-protein interaction networks has revealed that these networks have a different distribution type than other model networks such as scale free networks. In this respect, the eigenvalue solvers implementing the well-known implicitly restarted Arnoldi method, Lanczos method, Krylov-Schur and Jacobi Davidson methods are investigated. They exist as MATLAB routines and are included in some freely available packages. The performances of different eigenvalue solvers PEIG, AHBEIGS, IRBLEIGS, EIGIFP, LANEIG, JDQR, JDCG in MATLAB and the library SLEPc in C++ were tested for matrices of size between 100-13000 and are compared in terms of accuracy and computing time. The accuracy of the eigenvalue solvers are validated for the Paley graphs with known eigenvalues and are compared for large empirical networks using the residual plots and spectral density plots are computed.

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Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.

Because of the accuracy and ease of implementation, the Monte Carlo methodology is widely used in the analysis of nuclear systems. The estimated effective multiplication factor (keff) and flux distribution are statistical by their natures. In eigenvalue problems, however, neutron histories are not independent but are correlated through subsequent generations. Therefore, it is necessary to ensure that only the converged data are used for further analysis. Discarding a larger amount of initial histories would reduce the risk of contaminating the results by non-converged data, but increase the computational expense. This issue is amplified for large nuclear systems with slow convergence. One solution would be to use the convergence of keff or the flux distribution as the criterion for initiating accumulation of data. Although several approaches have been developed aimed at identifying convergence, these methods are not always reliable, especially for slow converging problems. This dissertation has attacked this difficulty by developing two independent but related methodologies. One aims to find a more reliable and robust way to assess convergence by statistically analyzing the local flux change. The other forms a basis to increase the convergence rate and thus reduce the computational expense. Eventually, these two topics will contribute to the ultimate goal of improving the reliability and efficiency of the Monte Carlo criticality calculations.

Consider (1) -yn1+ q1y1 = (λr11 + µr12)y1 on [0, 1] y’1(0) = cot α1 and = y’1(1) = a1λ + b1 y1(0) y1(1) c1λ+d1 (2) - yn2 + q2y2 = (λr21 + µr22)y2 on [0, 1] y’2(0) = cot α2 and = y’2(1) = a2µ + b2 y2(0) y2(1) c2µ + d2 subject to certain definiteness conditions; where qi and rij are continuous real valued functions on [0, 1], the angle αi is in [0, π) and ai, bi, ci, di are real numbers with δi = aidi − bici > 0 and ci = 0 for I, j = 1,2. Under the Uniform Left Definite condition we have proved an asymptotic theorem and an oscillation theorem. Analysis of (1) and (2) subject to the Uniform Ellipticity condition focus on the location of eigenvalues, perturbation theory and the local analysis of eigenvalues. We also gave a bound for the number of nonreal eigenvalues. We also have studied the system T1(x1) = (λA11 + µA12)(x1) and T2(x2) = (λA21 + µA22)(x2) where Aij (j =1, 2) and Ti are linear operators acting on finite dimensional Hilbert spaces Hi (i = 1, 2). For a pair of commutative operators Γ = (Γ0, Γ1) constructed from Aij and Ti on the Hilbert space tensor product H1 ⊗ H2, we can associate a natural Koszul complex namely Dºr-(λ,μ) D1 r-(λ,μ) 0 H H ø H H 0 We have constructed a basis for the Koszul quotient space N(D1Г−(λ,µ))/R(D0Г−( λ,µ)) in terms of the root basis of (Г0, Г1). (For equations pl refer the PDF file)

<p>Directional wave-field decomposition for heterogeneousanisotropic media with in-stantaneous response is establishedfor both the acoustic and the electromagnetic equations.</p><p>We derive a sufficient condition for ellipticity of thesystem's matrix in the Laplace domain and show that theconstruction of the splitting matrix via a Dunford-Taylorintegral over the resolvent of the non-compact, non-normalsystem's matrix is well de ned. The splitting matrix also hasproperties that make it possible to construct the decompositionwith a generalized eigenvector procedure. The classical way ofobtaining the decomposition is equivalent to solving analgebraic Riccati operator equation. Hence the proceduredescribed above also provides a solution to the algebraicRiccati operator equation.</p><p>The solution to the wave-field decomposition for theisotropic wave equation is expressed in terms of theDirichlet-to-Neumann map for a plane. The equivalence of thisDirichlet-to-Neumann map is the acoustic admittance, i.e. themapping between the pressure and the particle velocity. Theacoustic admittance, as well as the related impedance aresolutions to algebraic Riccati operator equations and are keyelements in the decomposition. In the electromagnetic case thecorresponding impedance and admittance mappings solve therespective algebraic Riccati operator equations and henceprovide solutions to the decomposition problem.</p><p>The present research shows that it is advantageous toutilize the freedom implied by the generalized eigenvectorprocedure to obtain the solution to the decomposition problemin more general terms than the admittance/impedancemappings.</p><p>The time-reversal approach to steer an acoustic wave eld inthe cavity and half space geometries are analyzed from aboundary control perspective. For the cavity it is shown thatwe can steer the field to a desired final configuration, withthe assumption of local energy decay. It is also shown that thetime-reversal algorithm minimizes a least square error forfinite times when the data are obtained by measurements. Forthe half space geometry, the boundary condition is expressedwith help of the wave-field decomposition. In the homogeneousmaterial case, the response of the time-reversal algorithm iscalculated analytically. This procedure uses the one-wayequations together with the decomposition operator.</p>

This thesis describes improvements of the periodic QZ algorithm and several variants of the Schur algorithm for block Toeplitz matrices. Documentation of the available software is included.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

The presence of structure in a computational problem can often be exploited and can lead to a more efficient numerical algorithm. In this dissertation, we look at structured numerical problems that arise from applications in wireless communications and machine learning that also impact other areas of scientific computing. In wireless communication system designs, certain structured matrices (frames) need to be generated. The design of such matrices is equivalent to a symmetric inverse eigenvalue problem where the values of the diagonal elements are prescribed. We present algorithms that are capable of generating a larger set of these constructions than previous algorithms. We also discuss the existence of equiangular tight frames---frames that satisfy additional structural properties. Kernel learning is an important class of problems in machine learning. It often relies on efficient numerical algorithms that solve underlying convex optimization problems. In our work, the objective functions to be minimized are the von Neumann and the LogDet Bregman matrix divergences. The algorithm that solves this optimization problem performs matrix updates based on repeated eigendecompositions of diagonal plus rank-one matrices in the case of von Neumann matrix divergence, and Cholesky updates in case of the LogDet Bregman matrix divergence. Our contribution exploits the low-rank representations and the structure of the constraint matrices, resulting in more efficient algorithms than previously known. We also present two specialized zero-finding algorithms where we exploit the structure through the shape and exact formulation of the objective function. The first zero-finding task arises during the matrix update step which is part of the above-mentioned kernel learning application. The second zero-finding problem is for the secular equation; it is equivalent to the computation of the eigenvalues of a diagonal plus rank-one matrix. The secular equation arises in various applications, the most well-known is the divide-and-conquer eigensolver. In our solutions, we build upon a somewhat forgotten zero-finding method by P. Jarratt, first described in 1966. The method employs first derivatives only and needs the same amount of evaluations as Newton's method, but converges faster. Our contributions are the more efficient specialized zero-finding algorithms. / text

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p &gt; n, assuming that the ratio <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%5Cfrac%7Bp%7D%7Bn%7D" /> converges when the number of parameters and the sample size increase. We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cfrac%7B1%7D%7Bp%7DE%5BTr%5C%7B%5Ccdot%5C%7D%5D" />. Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set. Furthermore, we investigate the normalized <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" /> and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers. In this thesis we also prove that the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D" />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20W%5Csim%5Cmathcal%7BW%7D_p(I_p,n)" />, is a consistent estimator of the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" />. We consider <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20Y_t=%5Csqrt%7Bnp%7D%5Cbig(%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D-m%5E%7B(t)%7D_1%20(n,p)%5Cbig)," />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20m%5E%7B(t)%7D_1%20(n,p)=E%5Cbig%5B%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D%5Cbig%5D" />, which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p &gt; n) and the multivariate data (p ≤ n).