Condition Numbers Of Matrices Research Articles

Detecting the causes of ill-conditioning in structural finite element models

In 2011, version 8.6 of the finite element-based structural analysis package Oasys GSA was released. A new feature in this release was the estimation of the 1-norm condition number κ1(K)=∥K∥1∥K-1∥1 of the stiffness matrix K of structural models by using a 1-norm estimation algorithm of Higham and Tisseur to estimate ∥K-1∥1. The condition estimate is reported as part of the information provided to engineers when they carry out linear/static analysis of models and a warning is raised if the condition number is found to be large. The inclusion of this feature prompted queries from users asking how the condition number impacted the analysis and, in cases where the software displayed an ill conditioning warning, how the ill conditioning could be “fixed”. We describe a method that we have developed and implemented in the software that enables engineers to detect sources of ill conditioning in their models and rectify them. We give the theoretical background and illustrate our discussion with real-life examples of structural models to which this tool has been applied and found useful. Typically, condition numbers of stiffness matrices reduce from O(1016) for erroneous models to O(108) or less for the corrected model.

A new frequency domain based approach for a nonlinear aeroelasticity analysis

In this paper, a new frequency-domain based approach for the investigation of aeroelasticity problems is introduced that is capable of handling both linear and nonlinear problems. This approach is based on coupling a conceptual method used in a structural dynamic analysis and an optimum equivalent linear frequency response function (OELF). Additionally, a new criterion for determining the flutter speed and the instability of nonlinear systems is introduced that is based on the condition number of the aeroelastic matrices. Due to the global nature of the condition number, the new criterion proves to be efficient and simple to use. To examine the efficiency of the new technique, a two-dimensional nonlinear airfoil with an unsteady aerodynamic model is considered.

Parallel Sparse Approximate Inverse Preconditioning on Graphic Processing Units

Accelerating numerical algorithms for solving sparse linear systems on parallel architectures has attracted the attention of many researchers due to their applicability to many engineering and scientific problems. The solution of sparse systems often dominates the overall execution time of such problems and is mainly solved by iterative methods. Preconditioners are used to accelerate the convergence rate of these solvers and reduce the total execution time. Sparse approximate inverse (SAI) preconditioners are a popular class of preconditioners designed to improve the condition number of large sparse matrices. We propose a GPU accelerated SAI preconditioning technique called GSAI, which parallelizes the computation of this preconditioner on NVIDIA graphic cards. The preconditioner is then used to enhance the convergence rate of the BiConjugate Gradient Stabilized (BiCGStab) iterative solver on the GPU. The SAI preconditioner is generated on average 28 and 23 times faster on the NVIDIA GTX480 and TESLA M2070 graphic cards, respectively, compared to ParaSails (a popular implementation of SAI preconditioners on CPU) single processor/core results. The proposed GSAI technique computes the SAI preconditioner in approximately the same time as ParaSails generates the same preconditioner on 16 AMD Opteron 252 processors.

Efficient preconditioning of laplacian matrices for computer graphics

We present a new multi-level preconditioning scheme for discrete Poisson equations that arise in various computer graphics applications such as colorization, edge-preserving decomposition for two-dimensional images, and geodesic distances and diffusion on three-dimensional meshes. Our approach interleaves the selection of fine-and coarse-level variables with the removal of weak connections between potential fine-level variables ( sparsification ) and the compensation for these changes by strengthening nearby connections. By applying these operations before each elimination step and repeating the procedure recursively on the resulting smaller systems, we obtain a highly efficient multi-level preconditioning scheme with linear time and memory requirements. Our experiments demonstrate that our new scheme outperforms or is comparable with other state-of-the-art methods, both in terms of operation count and wall-clock time. This speedup is achieved by the new method's ability to reduce the condition number of irregular Laplacian matrices as well as homogeneous systems. It can therefore be used for a wide variety of computational photography problems, as well as several 3D mesh processing tasks, without the need to carefully match the algorithm to the problem characteristics.

Partial derivatives of the eigen-triplet of the quadratic eigenvalue problem depending on several parameters

This paper concerns the partial derivatives of the eigen-triplet of the quadratic matrix polynomial Q(p,λ)=λ2M(p)+λC(p)+K(p), where M(p),C(p),K(p)∈Cn×n are complex analytic matrix valued functions, p∈Cm is a complex parameter vector. First, the analyticity theorem for simple eigenvalues and the corresponding eigenvectors is given. Second, a new method is proposed to compute partial derivatives of the eigen-triplet. The derivatives of the eigen-triplet can be obtained by solving algebraic linear equations of order (n-1), where it only requires the information of the eigen-triplet whose partial derivatives are to be computed, and what is more important, the condition numbers of the coefficient matrices are “better” than those of the nonsingular coefficient matrices arisen in the bordered matrix method [1] and Nelson’s method [5]. Numerical tests show the feasibility and efficiency of the new method. The results are better or at least comparable with current methods.

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

AbstractIn this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014

Direct trajectory optimization based on a mapped Chebyshev pseudospectral method

In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral (PS) method makes the points’ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method’s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.

The Exact Distribution of the Condition Number of Complex Random Matrices

Let G m×n (m ≥ n) be a complex random matrix and W = G m×n H G m×n which is the complex Wishart matrix. Let λ 1 > λ 2 > …>λ n > 0 and σ 1 > σ 2 > …>σ n > 0 denote the eigenvalues of the W and singular values of G m×n, respectively. The 2-norm condition number of G m×n is . In this paper, the exact distribution of the condition number of the complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials.

Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Mesh quality plays an essential role in finite element applications, since it affects the efficiency of the simulation with respect to solution accuracy and computational effort. Therefore, mesh smoothing techniques are often applied for improving mesh quality while preserving mesh topology. One of these methods is the recently proposed geometric element transformation method (GETMe), which is based on regularizing element transformations. It will be shown numerically that this smoothing method is particularly suitable, from an applicational point of view, since it leads to a significant reduction of discretization errors within the first few smoothing steps requiring only little computational effort. Furthermore, due to reduced condition numbers of the stiffness matrices the performance of iterative solvers of the resulting finite element equations is improved. This is demonstrated for the Poisson equation with a number of meshes of different complexity and type as well as for piecewise linear and quadratic finite element basis functions. Results are compared to those obtained by two variants of Laplacian smoothing and a state of the art global optimization-based approach.

Solution of the Pellet Equation by use of the Orthogonal Collocation and Least Squares Methods: Effects of Different Orthogonal Jacobi Polynomials

Two numerical methods in the family of weighted residual methods; the orthogonal collocation and least squares methods, are used within the spectral framework to solve a linear reaction-diffusion pellet problem with slab and spherical geometries. The node points are in this work taken as the roots of orthogonal polynomials in the Jacobi family. Two Jacobi polynomial parameters, alpha and beta, can be used to tune the distribution of the roots within the domain. Further, the internal points and the boundary points of the boundary-value problem can be given according to: i) Gauss-Lobatto-Jacobi points, or ii) Gauss-Jacobi points plus the boundary points. The objective of this paper is thus to investigate the influence of the distribution of the node points within the domain adopting the orthogonal collocation and least squares methods. Moreover, the results of the two numerical methods are compared to examine whether the methods show the same sensitivity and accuracy to the node point distribution. The notifying findings are as follows: i) The Legendre polynomial, i.e., alpha=beta=0, is a very robust Jacobi polynomial giving the better condition number of the coefficient matrix and the polynomial also give good behavior of the error as a function of polynomial order. This polynomial gives good results for small and large gradients within both slab and spherical pellet geometries. This trend is observed for both of the weighted residual methods applied. ii) Applying the least squares method the error decreases faster with increasing polynomial order than observed with the orthogonal collocation method. However, the orthogonal collocation method is not so sensitive to the choice of Jacobi polynomial and the method also obtains lower error values than the least squares method due to favorable lower condition numbers of the coefficient matrices. Thus, for this particular problem, the orthogonal collocation method is recommended above the least squares method. iii) The orthogonal collocation method show minor differences between Gauss-Lobatto-Jacobi points and Gauss-Jacobi plus boundary points.

An Improvement of Convergence in Finite Element Analysis With Infinite Element Using Deflation

This paper presents a deflation technique to improve the convergence of finite element (FE) analyses with infinite elements. In FE analyses of electromagnetic fields, large air regions must be discretized into FE meshes. This leads to increases in computational time. The infinite element in which electromagnetic fields in air region are accurately expressed has been introduced in order to solve this problem. However, when using the infinite element, convergence of iterative liner solvers deteriorates because the condition number of FE matrices becomes large. In this paper, a deflation technique to improve convergence of iterative solvers is introduced. Numerical examples show that the proposed technique can improve convergence characteristics in a magnetostatic analysis with finite and infinite elements.

Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers

We show that the performance of frequency-domain solvers of Maxwell’s equations is greatly affected by the kind of the perfectly matched layer (PML) used. In particular, we demonstrate that using the stretched-coordinate PML (SC-PML) results in significantly faster convergence speed than using the uniaxial PML (UPML). Such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices. Additionally, we develop a diagonal preconditioning scheme that significantly improves solver performance when UPML is used.

Spectrum Sensing Algorithms via Finite Random Matrices

We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Specifically, we employ recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of uncorrelated and semi-correlated random dual central Wishart matrices of finite sizes in the design Hypothesis-Testing algorithms to detect the presence of PU signals. In particular, two algorithms are designed, with basis on the SCN distribution in the absence (H_0) and in the presence (H_1) of PU signals, respectively. Due to an inherent property of the SCN's, the H_0 test requires no estimation of SNR or any other information on the PU signal, while the H_1 test requires SNR only. Further attractive advantages of the new techniques are: a) due to the accuracy of the finite SCN distributions, superior performance is achieved under a finite number of samples, compared to asymptotic RMT-based alternatives; b) since expressions to model the SCN statistics both in the absence and presence of PU signal are used, the statistics of the spectrum sensing problem in question is completely characterized; and c) as a consequence of a) and b), accurate and simple analytical expressions for the receiver operating characteristic (ROC) — both in terms of the probability of detection as a function of the probability of false alarm (P_D versus P_F) and in terms of the probability of acquisition as a function of the probability of miss detection (P_A versus P_M) — are yielded. It is also shown that the proposed finite RMT-based algorithms outperform all similar alternatives currently known in the literature, at a substantially lower complexity. In the process, several new results on the distributions of eigenvalues and SCNs of random Wishart Matrices are offered, including a closed-form of the Marchenko-Pastur's Cumulative Density Function (CDF) and extensions of the latter, as well as variations of asymptotic the distributions of extreme eigenvalues (Tracy-Widom) and their ratio (Tracy-Widom-Curtiss), which are simpler than those obtained with the "spiked population model".

Bi-parameter incremental unknowns ADI iterative methods for elliptic problems

Bi-parameter incremental unknowns (IU) alternating directional implicit (ADI) iterative methods are proposed for solving elliptic problems. Condition numbers of the coefficient matrices for these iterative schemes are carefully estimated. Theoretical analysis shows that the condition numbers are reduced significantly by IU method, and the iterative sequences produced by the bi-parameter incremental unknowns ADI methods converge to the unique solution of the linear system if the two parameters belong to a given parameter region. Numerical examples are presented to illustrate the correctness of the theoretical analysis and the effectiveness of the bi-parameter incremental unknowns ADI methods.

Effects of Jacobi polynomials on the numerical solution of the pellet equation using the orthogonal collocation, Galerkin, tau and least squares methods

A number of different numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau and least squares (LSQ) methods, are used within the spectral framework to solve a linear reaction–diffusion pellet problem with slab and spherical geometries. The node points are in this work taken as the roots of orthogonal polynomials in the Jacobi family. Two Jacobi polynomial parameters, α and β, can be used to tune the distribution of the roots within the domain. The objective of this paper is thus to investigate the influence of the node point distribution within the domain adopting the weighted residual methods mentioned above. Moreover, the results obtained with the different weighted residual methods are compared to examine whether the numerical approaches show the same sensitivity to the node point distribution. The notifying findings are as follows:(i)Considering the condition number of the coefficient matrices, the different weighted residual methods do not show the same sensitivity to the roots of the polynomials in the Jacobi family. On the other hand, the simulated error obtained adopting the Galerkin, tau and orthogonal collocation methods for different α, β-combinations differ insignificantly. The condition number of the LSQ coefficient matrix is relatively large compared to the other numerical methods, hence preventing the simulation error to approach the machine accuracy.(ii)The Legendre polynomial, i.e., α=β=0, is a very robust Jacobi polynomial giving on average the lowest condition number of the coefficient matrices and the polynomial also give among the best behaviors of the error as a function of polynomial order. This polynomial gives good results for small and large gradients within both slab and spherical pellet geometries.(iii)Adopting the Legendre polynomial, the Galerkin and tau methods obtain favorable lower condition numbers than the orthogonal collocation and LSQ methods.

Condition number of a square matrix with i.i.d. columns drawn from a convex body

We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly dis- tributed in an isotropic convex body. We deduce that the condition number of such matrices is of the order of the size of the matrix and give an estimate on its tail behavior.

Distribution of the Demmel Condition Number of Wishart Matrices

This paper studies the Demmel condition number of Wishart matrices, a quantity which has numerous applications to wireless communications, such as adaptive switching between beamforming and diversity coding, link adaptation, and spectrum sensing. For complex Wishart matrices, we give an exact analytical expression for the probability density function (p.d.f.) of the Demmel condition number, and also derive simplified expressions for the high tail regime. These results indicate that the condition of complex Wishart matrices is dominantly decided by the difference between the matrix dimension and degree of freedom (DoF), i.e., the probability of drawing a highly ill conditioned matrix decreases considerably when the difference between the matrix dimension and DoF increases. We further investigate real Wishart matrices, and derive new expressions for the p.d.f. of the smallest eigenvalue, when the difference between the matrix dimension and DoF is odd. Based on these results, we succeed to obtain an exact p.d.f. expression for the Demmel condition number, and simplified expressions for the high tail regime.

Exact Demmel Condition Number Distribution of Complex Wishart Matrices via the Mellin Transform

In this letter we study the distribution of the Demmel condition number of complex Wishart matrices, which has various applications in the performance analysis of communication systems. New exact expressions are derived for the probability density function and cumulative distribution function, which are very simple and valid for any matrix dimensions. These results are obtained by taking advantage of properties of the Mellin transform for products of independent random variables.

Tie-line constrained distributed state estimation

This paper presents an implementable distributed state estimation method for online analysis of power systems having multiple, geographically separated areas. Distributed state estimation apart from giving a faster solution, also improves the condition number of the resultant gain matrices. The method proposed here uses the conventional WLS estimator (Gauss–Newton method) with equality constraints forcing the tie line flows as calculated by the adjacent areas to be equal. Based on the topology of the network, the system is partitioned into multiple areas and a processor is assigned to each of these areas for solving the local state estimation problem. Simulations carried out on the IEEE 14, 30 and 118 bus systems show good convergence properties and improvement on the condition number of the gain matrices when compared with centralized algorithms.

Riesz bases of wavelets and applications to numerical solutions of elliptic equations

We investigate Riesz bases of wavelets in Sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order. First, we study bi-cubic splines on the unit square with homogeneous boundary conditions. The approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation. Second, we develop a fairly general theory for Riesz bases of Hilbert spaces equipped with induced norms. Under the guidance of the general theory, we are able to construct wavelet bases for Sobolev spaces on the unit square. The condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded. Third, we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient. Finally, we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case.