Browsing by Subject "Matrix algebra"
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Item Open Access Accuracy and efficiency considerations in the solution of extremely large electromagnetics problems(IEEE, 2011) Gürel, Levent; Ergül, ÖzgürThis study considers fast and accurate solutions of extremely large electromagnetics problems. Surface formulations of large-scale objects lead to dense matrix equations involving millions of unknowns. Thanks to recent developments in parallel algorithms and high-performance computers, these problems can easily be solved with unprecedented levels of accuracy and detail. For example, using a parallel implementation of the multilevel fast multipole algorithm (MLFMA), we are able to solve electromagnetics problems discretized with hundreds of millions of unknowns. Unfortunately, as the problem size grows, it becomes difficult to assess the accuracy and efficiency of the solutions, especially when comparing different implementations. This paper presents our efforts to solve extremely large electromagnetics problems with an emphasis on accuracy and efficiency. We present a list of benchmark problems, which can be used to compare different implementations for large-scale problems. © 2011 IEEE.Item Open Access Analysis of an arbitrary profile reflector antenna system having resistive type surface–E-polarization case(IEEE, 2006-06) Oǧuzer, T.; Altıntaş, Ayhan; Nosich, A. I.The regularized solution is performed for arbitrary shape conic section profile geometry. In this case the reflector surface is taken as made up of resistive type material. The problem is formulated depending on the circular In periodicity and then Fourier series coefficients of the surface current density are obtained. The resultant matrix equation is in the regularized form. Then the various numerical results are obtained for different eccentricity factor of the conic section and the resisitvity of the reflector surface. © 2006 IEEE.Item Open Access Analysis of cross-correlations between financial markets after the 2008 crisis(Elsevier BV, 2013) Sensoy, A.; Yuksel, S.; Erturk, M.We analyze the cross-correlation matrix C of the index returns of the main financial markets after the 2008 crisis using methods of random matrix theory. We test the eigenvalues of C for universal properties of random matrices and find that the majority of the cross-correlation coefficients arise from randomness. We show that the eigenvector of the largest deviating eigenvalue of C represents a global market itself. We reveal that high volatility of financial markets is observed at the same times with high correlations between them which lowers the risk diversification potential even if one constructs a widely internationally diversified portfolio of stocks. We identify and compare the connection and cluster structure of markets before and after the crisis using minimal spanning and ultrametric hierarchical trees. We find that after the crisis, the co-movement degree of the markets increases. We also highlight the key financial markets of pre and post crisis using main centrality measures and analyze the changes. We repeat the study using rank correlation and compare the differences. Further implications are discussed.Item Open Access Analysis of finite arrays of axially directed printed dipoles on electrically large circular cylinders(IEEE, 2004) Ertürk, V. B.; Rojas, R. G.; Lee, K. W.Various arrays consisting of finite number of printed dipoles on electrically large dielectric coated circular cylinders are investigated using a hybrid method of moments/Green's function technique in the spatial domain. This is basically an "element by element" approach in which the mutual coupling between dipoles through space as well as surface waves is incorporated. The efficiency of the method comes from the computation of the Green's function, where three types of spatial domain Green's function representations are used interchangeably, based on their computational efficiency and regions where they remain accurate. Numerical results are presented in the form of array current distributions, active reflection coefficient and far-field pattern to indicate the efficiency and accuracy of the method. Furthermore, these results are compared with similar results obtained from finite arrays of printed dipoles on grounded planar dielectric slabs. It is shown that planar approximations, except for small separations, can not be used due to the mutual coupling between the array elements. Consequently, basic performance metrics of printed dipole arrays on coated cylinders show significant discrepancies when compared to their planar counterparts. © 2004 IEEE.Item Open Access Analysis of finite arrays of circumferentially oriented printed dipoles on electrically large cylinders(Wiley, 2004) Ertürk, V. B.; Güner, B.An efficient and accurate hybrid method of moments (MoM)/Green's function technique in the spatial domain is developed for the rigorous analysis of large, finite phased arrays of circumferentially oriented printed dipoles on electrically large, dielectric-coated, circular cylinders. Basic performance metrics (in the form of array current distribution, active reflection coefficient, far-field patterns, and so forth) of several arrays have been obtained and compared with similar printed arrays on grounded planar substrates. Certain discrepancies have been observed and discussed. © 2004 Wiley Periodicals, Inc.Item Open Access Analysis of the elliptic-profile cylindrical reflector with a non-uniform resistivity using the complex source and dual-series approach: H-polarization case(Springer, 2013) Oğuzer, T.; Altintaş, A.; Nosich, A. I.An elliptic-profile reflector with varying resistivity is analyzed under the illumination by an H-polarized beam generated by a complex-source-point (CSP) feed. The emphasis is done on the focusing ability that is potentially important in the applications in the optical range related to the partially transparent mirrors. We formulate the corresponding electromagnetic boundary-value problem and derive a singular integral equation from the resistive-surface boundary conditions. This equation is treated with the aid of the regularization technique called Riemann Hilbert Problem approach, which inverts the stronger singular part analytically, and converted to an infinite-matrix equation of the Fredholm 2nd kind. The resulting numerical algorithm has guaranteed convergence. This type of solution provides more accurate and faster results compared to the known method of moments. In the computations, a CSP feed is placed into a more distant geometrical focus of the elliptic reflector, and the near-field values at the closer focus are plotted and discussed. Various far-field radiation patterns including those for the non-uniform resistive variation on the reflector are also presented.Item Open Access Analyzing large sparse Markov chains of Kronecker products(IEEE, 2009) Dayar, TuğrulKronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. In the Kronecker based approach, the generator matrix underlying the MC is represented using Kronecker products [6] of smaller matrices and is never explicitly generated. The implementation of transient and steady-state solvers rests on this compact Kronecker representation, thanks to the existence of an efficient vector-Kronecker product multiplication algorithm known as the shuffle algorithm [6]. The transient distribution can be computed through uniformization using vector-Kronecker product multiplications. The steady-state distribution also needs to be computed using vector-Kronecker product multiplications, since direct methods based on complete factorizations, such as Gaussian elimination, normally introduce new nonzeros which cannot be accommodated. The two papers [2], [10] provide good overviews of iterative solution techniques for the analysis of MCs based on Kronecker products. Issues related to reachability analysis, vector-Kronecker product multiplication, hierarchical state space generation in Kronecker based matrix representations for large Markov models are surveyed in [5]. Throughout our discussion, we make the assumption that the MC at hand does not have unreachable states, meaning it is irreducible. And we take an algebraic view [7] to discuss recent results related to the analysis of MCs based on Kronecker products independently from modeling formalisms. We provide background material on the Kronecker representation of the generator matrix underlying a CTMC, show that it has a rich structure which is nested and recursive, and introduce a small CTMC whose generator matrix is expressed as a sum of Kronecker products; this CTMC is used as a running example throughout the discussion. We also consider preprocessing of the Kronecker representation so as to expedite numerical analysis. We discuss permuting the nonzero structure of the underlying CTMC symmetrically by reordering, changing the orders of the nested blocks by grouping, and reducing the size of the state space by lumping. The steady-state analysis of CTMCs based on Kronecker products is discussed for block iterative methods, multilevel methods, and preconditioned projection methods, respectively. The results can be extended to DTMCs based on Kronecker products with minor modifications. Areas that need further research are mentioned as they are discussed. Our contribution to this area over the years corresponds to work along iterative methods based on splittings and their block versions [11], associated preconditioners to be used with projection methods [4], near complete decomposability [8], a method based on iterative disaggregation for a class of lumpable MCs [9], a class of multilevel methods [3], and a recent method based on decomposition for weakly interacting subsystems [1]. © 2009 IEEE.Item Open Access Block SOR for Kronecker structured representations(Elsevier, 2004) Buchholz, P.; Dayar, TuğrulThe Kronecker structure of a hierarchical Markovian model (HMM) induces nested block partitionings in the transition matrix of its underlying Markov chain. This paper shows how sparse real Schur factors of certain diagonal blocks of a given partitioning induced by the Kronecker structure can be constructed from smaller component matrices and their real Schur factors. Furthermore, it shows how the column approximate minimum degree (COLAMD) ordering algorithm can be used to reduce fill-in of the remaining diagonal blocks that are sparse LU factorized. Combining these ideas, the paper proposes three-level block successive over-relaxation (BSOR) as a competitive steady state solver for HMMs. Finally, on a set of numerical experiments it demonstrates how these ideas reduce storage required by the factors of the diagonal blocks and improve solution time compared to an all LU factorization implementation of the BSOR solver. © 2004 Elsevier Inc. All rights reserved.Item Open Access Compact representation of solution vectors in Kronecker-based Markovian analysis(Springer, 2016-08) Buchholz, P.; Dayar, Tuğrul; Kriege, J.; Orhan, M. CanIt is well known that the infinitesimal generator underlying a multi-dimensional Markov chain with a relatively large reachable state space can be represented compactly on a computer in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. Nevertheless, solution vectors used in the analysis of such Kronecker-based Markovian representations still require memory proportional to the size of the reachable state space, and this becomes a bigger problem as the number of dimensions increases. The current paper shows that it is possible to use the hierarchical Tucker decomposition (HTD) to store the solution vectors during Kroneckerbased Markovian analysis relatively compactly and still carry out the basic operation of vector-matrix multiplication in Kronecker form relatively efficiently. Numerical experiments on two different problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases. © Springer International Publishing Switzerland 2016.Item Open Access Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering(Elsevier, 2005) Pekergin, N.; Dayar T.; Alparslan, D. N.This paper presents an improved version of a componentwise bounding algorithm for the state probability vector of nearly completely decomposable Markov chains, and on an application it provides the first numerical results with the type of algorithm discussed. The given two-level algorithm uses aggregation and stochastic comparison with the strong stochastic (st) order. In order to improve accuracy, it employs reordering of states and a better componentwise probability bounding algorithm given st upper- and lower-bounding probability vectors. Results in sparse storage show that there are cases in which the given algorithm proves to be useful. © 2004 Elsevier B.V. All rights reserved.Item Open Access Compressive sampling and adaptive multipath estimation(IEEE, 2010) Pilancı, Mert; Arıkan, OrhanIn many signal processing problems such as channel estimation and equalization, the problem reduces to a linear system of equations. In this proceeding we formulate and investigate linear equations systems with sparse perturbations on the coefficient matrix. In a large class of matrices, it is possible to recover the unknowns exactly even if all the data, including the coefficient matrix and observation vector is corrupted. For this aim, we propose an optimization problem and derive its convex relaxation. The numerical results agree with the previous theoretical findings of the authors. The technique is applied to adaptive multipath estimation in cognitive radios and a significant performance improvement is obtained. The fact that rapidly varying channels are sparse in delay and doppler domain enables our technique to maintain reliable communication even far from the channel training intervals. ©2010 IEEE.Item Open Access Computationally efficient wavelet affine invariant functions for shape recognition(IEEE, 2004) Bala, E.; Çetin, A. EnisAn affine invariant function for object recognition is constructed from wavelet coefficients of the object boundary. In previous works, undecimated dyadic wavelet transform was used to construct affine invariant functions. In this paper, an algorithm based on decimated wavelet transform is developed to compute an affine invariant function. As a result computational complexity is reduced without decreasing recognition performance. Experimental results are presented. © 2004 IEEE.Item Open Access Continuous and discrete fractional fourier domain decomposition(IEEE, 2000) Yetik, İ. Şamil; Kutay, M. A.; Özaktaş, H.; Özaktaş, Haldun M.We introduce the fractional Fourier domain decomposition for continuous and discrete signals and systems. A procedure called pruning, analogous to truncation of the singular-value decomposition, underlies a number of potential applications, among which we discuss fast implementation of space-variant linear systems.Item Open Access Correlations between the ranks of submatrices and weights of random codes(2009) Klyachko, A. A.; Özen, I.The results of our study are twofold. From the random matrix theory point of view we obtain results on the rank distribution of column submatrices. We give the moments and the covariances between the ranks (q- rank) of such submatrices. We conjecture the counterparts of these results for arbitrary submatrices. The case of higher correlations gets drastically complicated even in the case of three submatrices. We give a formula for the correlation of ranks of three submatrices and a conjecture for its closed form. From the code theoretical point of view our study yields the covariances of the coefficients of the weight enumerator of a random code. Particularly interesting is that the coefficients of the weight enumerator of a code with random parity check matrix are uncorrelated. We give a conjecture for the triple correlations between the coefficients of the weight enumerator of a random code. © 2009 Elsevier Inc. All rights reserved.Item Open Access Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations(IEEE, 1998-05) Kutay, Mehmet Alper; Erden, M. F.; Özaktaş, Haldun M.; Arıkan, Orhan; Candan, Ç.; Güleryüz, Ö.It is possible to obtain either exact realizations or useful approximations of linear systems or matrix-vector products arising in many different applications, by synthesizing them in the form of repeated or multi-channel filtering operations in fractional Fourier domains, resulting in much more efficient implementations with acceptable decreases in accuracy. By varying the number and configuration of filter blocks, which may take the form of arbitrary flow graphs, it is possible to trade off between accuracy and efficiency in the desired manner. The proposed scheme constitutes a systematic way of exploiting the information inherent in the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.Item Open Access Coupled deconvolution for frequency extrapolation of electromagnetic solutions with matrix pencil method(IEEE, 2005) Gürel, Levent; Yıldırım, FerhatMatrix pencil method (MPM) has been widely used to estimate the parameters of complex-exponential based models. An important application is the extrapolation of the frequency-domain solutions of electromagnetic problems. In this paper, we present a mathematical tool, namely, coupled deconvolution, which improves the performance of the MPM-based extrapolation of electromagnetic solutions.Item Open Access Coupled matrix pencil method for frequency extrapolation of electromagnetic solutions(IEEE, 2005) Yıldırım, Ferhat; Gürel, LeventMatrix pencil method (MPM) is used to extrapolate the available electromagnetic solutions in frequency domain to estimate the high-frequency solutions. A new approach, namely, coupled MPM, is introduced to obtain the electromagnetic solutions at intermediate frequencies using the available low-frequency and high-frequency data.Item Open Access Decentralized blocking zeros and the decentralized strong stabilization problem(IEEE, 1995) Ünyelioğlu, K. A.; Özgüler, A. B.; Özgüner, Ü.This paper is concerned with a new system theoretic concept, decentralized blocking zeros, and its applications in the design of decentralized controllers for linear time-invariant finite-dimensional systems. The concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control. Decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant. As an application of this concept, we consider the decentralized strong stabilization problem (DSSP) where the objective is to stabilize a plant using a stable decentralized controller. It is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the DSSP to be solvable. That parity interlacing property is also sufficient for the solution of the DSSP for a large class of plants satisfying a certain connectivity condition. The DSSP is exploited in the solution of a special decentralized simultaneous stabilization problem, called the decentralized concurrent stabilization problem (DCSP). Various applications of the DCSP in the design of controllers for large-scale systems are also discussed.Item Open Access Decomposing linear programs for parallel solution(Springer, 1995-08) Pınar, Ali; Çatalyürek Ümit V.; Aykanat, Cevdet; Pınar, MustafaCoarse grain parallelism inherent in the solution of Linear Programming (LP) problems with block angular constraint matrices has been exploited in recent research works. However, these approaches suffer from unscalability and load imbalance since they exploit only the existing block angular structure of the LP constraint matrix. In this paper, we consider decomposing LP constraint matrices to obtain block angular structures with specified number of blocks for scalable parallelization. We propose hypergraph models to represent LP constraint matrices for decomposition. In these models, the decomposition problem reduces to the well-known hypergraph partitioning problem. A Kernighan-Lin based multiway hypergraph partitioning heuristic is implemented for experimenting with the performance of the proposed hypergraph models on the decomposition of the LP problems selected from NETLIB suite. Initial results are promising and justify further research on other hypergraph partitioning heuristics for decomposing large LP problems. © Springer-Verlag Berlin Heidelberg 1996.Item Open Access Disorder and localization in the lowest Landau level in the presence of dilute point scatterers(Pergamon Press, 1999) Gedik, Z.; Bayındır, MehmetWe study the localization properties of a two-dimensional noninteracting electron gas in the presence of randomly distributed short-range scatterers in very high magnetic fields. We evaluate the participation number of the eigenstates obtained by exact diagonalization technique. At low impurity concentrations we obtain self-averaged values showing that all states, except those exactly at the Landau level, are localized with finite localization length. We conclude that in this dilute regime the localization length does not diverge. We also find that the maximum localization length increases exponentially with impurity concentration. Our calculations suggest that scaling behavior may be absent even for higher concentrations of scatterers.