Abstract: This dissertation focuses on efficiently forming reduced-order models for large, linear dynamic systems. Projections onto unions of Krylov subspaces lead to a class of reduced-order models known as rational interpolants. The cornerstone of this dissertation is a collection of theory relating Krylov projection to rational interpolation. Based on this theoretical framework, three algorithms for model reduction are proposed. The first algorithm, dual rational Arnoldi, is a numerically reliable approach involving orthogonal projection matrices. The second, rational Lanczos, is an efficient generalization of existing Lanczos-based methods. The third, rational power Krylov, avoids orthogonalization and is suited for parallel or approximate computations. The performance of the three algorithms is compared via a combination of theory and examples. Independent of the precise algorithm, a host of supporting tools are also develop ed to form a complete model-reduction package. Techniques for choosing the matching frequencies, estimating the modeling error, insuring the model's stability, treating multiple-input multiple-output systems, implementing parallelism, and avoiding a need for exact factors of large matrix pencils are all examined to various degrees

Abstract: A data-determined method for testing structural models of the errors in vector autoregressions is discussed. The method can easily be combined with prior economic knowledge and a subjective analysis of data characteristics to yield valuable information concerning model selection and specification. In one dimension, it turns out that standard t statistics can be used to test the various overidentifying restrictions that are implied by a model. In another dimension, the method compares a priori knowledge of a structural model for the errors with the properties exhibited by the data. Thus this method may help to ensure that orderings of the errors for impulse response and forecast error variance decomposition analyses are sensible, given the data. Two economic examples are used to illustrate the method.

Abstract: The paper describes an efficient frequency-domain modeling and simulation method of a coupled interconnect system using scattering parameters. First, low-order rational approximations of the multiport scattering parameters are derived over a wide frequency range using a robust interpolation technique. The method applies frequency normalization, shift, and Householder QR orthogonalization to improve the stability and the accuracy when solving the resulting systems of equations. For interconnects characterized with frequency-dependent parasitic parameters, the order of the rational of approximation is reduced by using appropriate reference system. Then, the generated multiport pole-residue models are incorporated into a circuit simulator using recursive convolution. Thus, the method avoids explicit convolution, numerical transform, and artificial filtering of a large number of points that are often necessary in conventional approaches. Examples with experimental and simulated results are given to illustrate the method.

Abstract: In [6] the Generalized Minimal Residual Method (GMRES) which constructs the Arnoldi basis and then solves the transformed least squares problem was studied. It was proved that GMRES with the Householder orthogonalization-based implementation of the Arnoldi process (HHA), see [9], is backward stable. In practical computations, however, the Householder orthogonalization is too expensive, and it is usually replaced by the modified Gram-Schmidt process (MGSA). Unlike the HHA case, in the MGSA implementation the orthogonality of the Arnoldi basis vectors is not preserved near the level of machine precision. Despite this, the MGSA-GMRES performs surprisingly well, and its convergence behaviour and the ultimately attainable accuracy do not differ significantly from those of the HHA-GMRES. As it was observed, but not explained, in [6], it is thelinear independence of the Arnoldi basis, not the orthogonality near machine precision, that is important. Until the linear independence of the basis vectors is nearly lost, the norms of the residuals in the MGSA implementation of GMRES match those of the HHA implementation despite the more significant loss of orthogonality.

Abstract: A new subgridding technique for finite difference (FD) methods is presented. The method is based on the integral form of Maxwell's equations combined with a simple yet efficient orthogonalization technique for the discretization geometry at subgrid interfaces. No additional correction factors or interpolations are required. This leads to spurious-mode free solutions when applied to FD approximations of eigenvalue problems and to stable difference formulations when applied to the finite difference time-domain (FD-TD) method. The high efficiency of the subgridding technique is demonstrated by the FD-TD analysis of an inter-digital filter with circular posts.

Abstract: Blind source separation and blind output decorrelation are two well-known problems in signal processing. For instantaneous mixtures, blind source separation is equivalent to a generalized eigen-decomposition, while blind output decorrelation can be considered as an iterative method of output orthogonalization. We propose a steepest descent procedure on a new cost function based on the Frobenius norm which measures the diagonalization of correlation matrices to perform blind source separation as well as blind decorrelation. The method is applicable to both stationary and nonstationary signals and instantaneous as well as convolutive mixture models. Simulation results by Monte Carlo trials are provided to show the consistent performance of the proposed algorithm.

Abstract: An efficient algorithm for deriving QSPR/QSAR models with nonorthogonal and ordered orthogonal descriptors, based on orthogonalization of topological indices, is presented. It is applied to structure-boiling point modeling of nonanes as the test case. The selection of the best descriptors from multivariate linear regression modeling is carried out using descriptors which are first orthogonalized. It is shown that such an algorithm is applicable for the selection of the best descriptors in a multivariate linear regression model even to very large sets of descriptors. A computationally-effective method for the (ordered) orthogonalization of topological indices is also introduced. By the use of an ordered orthogonalization procedure it is possible to select the best order of descriptors for orthogonalization. The orthogonalization in the selected order of descriptors produces models with a smaller number of significant descriptors. The comparison between QSPR/QSAR models with nonorthogonal and order...

Abstract: This paper presents a general rational block Lanczos algorithm for computing multipoint matrix Pade approximation of linear multiport networks, which model many important circuits in digital, analog, or mixed signal designs. This algorithm generalizes a novel block Lanczos algorithm with a reliable adaptive scheme for breakdown treatment to address two drawbacks of the single frequency Pade approximation: poor approximation of the transfer function in the frequency domain far away from the expansion point and the instability of the reduced model when the original system is stable. In addition, due to smaller Krylov subspace corresponding to each frequency point, the rational algorithm also alleviates the possible breakdowns when completing high order approximations. The cost of full backward orthogonalization with respect to all previous Lanczos vectors in a rational Lanczos algorithm, as compared to a partial backward orthogonalization in a single point Lanczos algorithm, is offset by more accurate and smaller order approximations.

Abstract: Starting from a specific implementation of the Lanczos biorthogonalization algorithm, an iterative process for the solution of systems of linear equations with general non-Hermitian coefficient matrix is derived. Due to the orthogonalization of the underlying Lanczos process the resulting iterative scheme involves inner products leading to global communication and synchronization on parallel processors. For massively parallel computers, these effects cause considerable delays often preventing the scalability of the implementation. In the process proposed, all inner product-like operations of an iteration step are independent such that the implementation consists of only a single global synchronization point per iteration. In exact arithmetic, the process is shown to be mathematically equivalent to the biconjugate gradient method. The efficiency of this new variant is demonstrated by numerical experiments on a PARAGON system using up to 121 processors.

Abstract: An algorithm was developed to obtain reconstructions from truncated projections by utilizing cross-correlation of a "knowledge set" of a priori nontruncated cross-sections with a similar structure. A cross-correlation matrix was constructed for the known set of cross-sectional images. The eigenvectors of this matrix form a set of orthogonal basis vectors for the reconstructed image. The basis set is optimal in the sense that the average of the differences between members of a given set of a priori images, and their truncated linear expansion for any basis set, is minimal for this particular basis set. A procedure for finding optimal basis vectors is fundamental for deriving the Karhunen-Loeve (K-L) transform. Therefore, one can represent an image not in the "knowledge set", but of similar structure by a linear combination of basis vectors corresponding to the larger eigenvalues; thus, the number of basis vectors is reduced to a number less than the total number of pixels. The projection of an image represented by this linear combination of basis vectors is a linear combination of projected basis vectors which are not necessarily orthogonal. A constrained least-squares method was used to evaluate the coefficients of this expansion by minimizing the sum of squares difference between the expansion and the projection measurements taking into account the distribution of coefficients over basis vectors. The constrained least-squares estimates of the coefficients were used in an expansion of the orthogonal basis to obtain the reconstructed image. The constrained solution has a reduced noise level in this inverse problem. It is shown that the reconstruction of truncated projections can be significantly improved over that of commonly used iterative reconstruction algorithms.

Abstract: A method for a multidimensional search for a close neighbor of an input vector, amongst a first set of reference vectors, comprises the prior determination of a first set of hyperplanes in the space containing the reference vectors, then selection of a first hyperplane from the first set, formation of a second set of reference vectors, by eliminating reference vectors which are on the other side of the first hyperplane selected, compared with the input vector, formation of a second set of hyperplanes, by eliminating the said first hyperplane, reiteration, a predetermined number of times, of the selection and formation operations, taking, as the first set of reference vectors and as the first set of hyperplanes, respectively, the second sets formed previously, and searching for the closest neighbor of the input vector in the second set of reference vectors.

Abstract: A novel algorithm is presented for the determination of a wavelet neural network architecture with the addition of autoregressive external input. The network is constructed on the basis of a multiresolution grid and refined using a modified Gram-Schmidt orthogonalization. The network was tested against a one dimensional test function of varying frequency, an eight dimensional sunspot series, and a transfer function model of a U-tube heat exchanger.

Abstract: A new intracell orthogonalization technique for a code division multiple access (CDMA) wireless local communication system is posed. On the forward link of the system, it is necessary to achieve high bit rate transmission and to reduce the complexity of the receiving terminal. To meet these two requirements, cyclically shifted spreading sequences are utilized. However, it is necessary to orthogonalize the multiple signals since these sequences are cross-correlated. One way of achieving orthogonalization is pre-decorrelation. However, the orthogonality does not clearly appear in a multipath fading channel because of the interchip interference caused by multipath. Therefore in this paper, pre-decorrelation is combined with multicarrier modulation. The results obtained through computer simulation show that the proposed technique can achieve orthogonality in a multipath fading channel. Also the proposed technique makes efficient forward link power control possible, and reduces intercell interference.

Abstract: Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution tool, GMRES(k), is adapted to deal with the sequence of such systems. The proposed adaptive strategy of GMRES(k) allows tuning of the restart parameter k based on the GMRES convergence rate for the given problem. Adaptive GMRES(k) is shown to be superior to several other iterative techniques on analog circuit simulation problems and on postbuckling structural analysis problems. Developing parallel techniques for robust but expensive sequential computations, such as globally convergent homotopy methods, is important. The design of these techniques encompasses the functionality of the iterative method (adaptive GMRES(k)) implemented sequentially and is based on the results of a parallel performance analysis of several implementations. An implementation of adaptive GMRES(k) with Householder reflections in its orthogonalization phase is developed. It is shown that the efficiency of linear system solution by the adaptive GMRES(k) algorithm depends on the change in problem difficulty when the problem is scaled. In contrast, a standard GMRES(k) implementation using Householder reflections maintains a constant efficiency with increase in problem size and number of processors, as concluded analytically and experimentally. The supporting numerical results are obtained on three distributed memory homogeneous parallel architectures: CRAY T3E, Intel Paragon, and IBM SP2.

Abstract: In this paper we are concerned with parallel implementation of row-oriented Gram-Schmidt orthogonalization. For the data partitioning four types of columnwise partitioning schemes were considered: column (1-col), block, cyclic and block-cyclic (b-c) partitioning. Analytical models for parallel execution time required by these implementations are derived and compared with numerical results. The best partitioning scheme is shown theoretically and by numerical results.

Abstract: In this paper we present methods to analyze, restore and compose multi-sensor hyper (or multi)-spectra images. For the composite multi-sensor and/or hyper (or multi)-spectra image the Karhunen-Loeve (K-L) and Gram-Schmidt (G-S) orthogonalization techniques are used in combination with blur estimation and 3-D restoration methods. For the motion estimation or target detection in a set of two frames of the same spectral band the G-S orthogonalization is used. Results from real data from satellite images are presented.

Abstract: Segmented image coding segments an image into non-rectangular regions and approximates the texture in each region by a weighted sum of orthonormal base functions. These orthonormal base functions, which are region-specific, used to be generated by the Gram-Schmidt (GS) algorithm, which is unfortunately very time-consuming. This paper presents the polynomial recursive orthogonalization (PRO) algorithm which generates the same orthonormal base functions as GS, but which is faster than GS because it is based on a recurrence which has fewer terms than the corresponding GS equation. The paper presents theoretical and experimental results which show that PRO is two to three times faster in practice (depending on the number of computed base functions).

Abstract: What is the right way to measure the distance between two matrices? The question of distance between two matrices is directly connected with the notion of length of a vector

Abstract: This paper presents a computationally efficient, sequential method for attitude matrix estimation using gyro and vector measurements. The method is based on a recently introduced, minimal-parameter third-order method for solving the orthogonal matrix differential equation in R(sup n). In the three-dimensional case, these third-order attitude parameters can be interpreted as temporal-integrals of the body-frame angular velocity components. A nonlinear algorithm is developed, which uses this minimal set of three parameters to estimate the nine-parameter direction-cosine matrix. Having an extremely simple kinematic equation, these parameters render the resulting estimator highly computationally efficient. An orthogonalization procedure, incorporated into the measurement processing stage, enhances the accuracy and stability of the resulting algorithm, yet retains reasonable simplicity. The performance of the estimator is demonstrated via a Monte Carlo simulation study.

Abstract: Mathematica is introduced through examples: antisymmetric operators; Gram−Schmidt orthogonalization (vectors and orthogonal polynomials); contour integration (residue theorem) and line integrals; the Kepler equation (series methods); and spheroidal harmonics (eigenfunctions of partial differential equations). These examples highlight the power and versatility of Mathematica and indicate its application to teaching mathematics.

Abstract: For non-self-adjoint elliptic boundary value problems which are preconditioned by a substructuring method, i.e., nonoverlapping domain decomposition, we introduce and study the concept of subspace orthogonalization. In subspace orthogonalization variants of Krylov methods, the computation of inner products and vector updates, and the storage of basis elements is restricted to a (presumably small) subspace, in this case the edge and vertex unknowns with respect to the partitioning into subdomains. We discuss the convergence properties of these iteration schemes and compare them with Krylov methods applied to the full preconditioned system.

Abstract: First the functional description of a APM-modulator-demodulator-system (APM = amplitude phase modulation) using sequences of orthogonal Impulses for transmission is given. By means of a well known mathematical method for the orthogonalization of linear independent functions (Schmidtsches Orthogonalisierungsverfahren) the orthogonalization of linear independent functions is shown. Examples of sequences of orthogonalised impulses are presented which have better transmission characteristics as the well known sequences of Walsh-Functions or sequences of sine-cosine-functions. The orthogonality of several sequences of orthogonalized impulses after a bandlimiting is tested by computer-simulation and compared with corresponding sequences of Walsh-Functions and sine-cosine-functions. A general implementation scheme for an orthogonalization network is presented.

Abstract: In this paper two recently proposed single-vector Lanczos methods based on a simple restarting strategy are analysed and their suitability for the computation of closely clustered eigenvalues is evaluated. Both algorithms adopt an approach which yields a fixed k-step restarting scheme in which one eigenpair at a time is computed using a deflation technique in which each Lanczos vector generated is orthogonalized against all previously converged eigenvectors. In the first algorithm each newly generated Lanczos vector is also orthogonalised with respect to all of its predecessors; in the second, a selective orthogonalisation strategy permits re-orthogonalization between the Lanczos vectors to be almost completely eliminated. ‘Reverse communication’ implementations of the algorithms on an MPP Connection Machine CM-200 with 8K processors are discussed. Advantages of the algorithms include the ease with which they cope with genuinely multiple eigenvalues, their guaranteed convergence and their fixed storage requirements.

Abstract: We propose an optical-digital method and describe an experiment on finger-prints identification. The identification is based on comparing feature-vectors of small dimensionality obtained via expanding directions fields for each finger-print in terms of the Hadamard basis. The employment of the operation of 'partial' images orthogonalization in the course of directions field calculation has made it possible to enhance the reliability of identification.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.