New Dynamic Orderings for the Parallel One–Sided Block-Jacobi SVD Algorithm

2015 ◽  
Vol 25 (02) ◽  
pp. 1550003 ◽  
Author(s):  
Martin Bečka ◽  
Gabriel Okša ◽  
Marián Vajteršic
Keyword(s):  
Random Distribution ◽  
Singular Values ◽  
Parallel Execution ◽  
Jacobi Method ◽  
Stopping Criteria ◽  
The Matrix ◽  
Blocking Factors ◽  
Relative Errors ◽  
Parallel Iteration ◽  
Global And Local

Five variants of a new dynamic ordering are presented for the parallel one-sided block Jacobi SVD algorithm. Similarly to the two-sided algorithm, the dynamic ordering takes into account the actual status of a matrix—this time of its block columns with respect to their mutual orthogonality. Variants differ in the computational and communication complexities and in proposed global and local stopping criteria. Their performance is tested on a square random matrix of order 8192 with a random distribution of singular values using [Formula: see text], 32, 64, 96 and 128 processors. All variants of dynamic ordering are compared with a parallel cyclic ordering, two-sided block-Jacobi method with dynamic ordering and the ScaLAPACK routine PDGESVD with respect to the number of parallel iteration steps needed for the convergence and total parallel execution time. Moreover, the relative errors in the orthogonality of computed left singular vectors and in the matrix assembled from computed singular triplets are also discussed. It turns out that the variant 3, for which a local optimality in some precisely defined sense can be proved, and its combination with variant 2, are the most efficient ones. For relatively small blocking factors [Formula: see text], they outperform the ScaLAPACK procedure PDGESVD and are about 2 times faster.

scholarly journals PRECONDITIONED PARALLEL BLOCK–JACOBI SVD ALGORITHM

2006 ◽  
Vol 16 (03) ◽  
pp. 371-379 ◽  
Author(s):  
GABRIEL OKŠA ◽  
MARIÁN VAJTERŠIC
Keyword(s):  
Execution Time ◽  
Parallel Architecture ◽  
Singular Values ◽  
Singular Value ◽  
Parallel Execution ◽  
Qr Factorization ◽  
R Factor ◽  
Order Of Magnitude ◽  
Column Pivoting ◽  
Parallel Iteration

We show experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the R-factor, can lead to a substantial decrease of the number of outer parallel iteration steps in the parallel block-Jacobi SVD algorithm, whereby the details depend on the condition number and on the shape of spectrum, including the multiplicity of singular values. Best results were achieved for well-conditioned matrices with a multiple minimal singular value, where the number of parallel iteration steps has been reduced by two orders of magnitude. However, the gain in speed, as measured by the total parallel execution time, depends decisively on how efficient is the implementation of the distributed QR and LQ factorizations on a given parallel architecture. In general, the reduction of the total parallel execution time up to one order of magnitude has been achieved.

Organisation in the Structure of the Cytoplasmic Ground Substance

1978 ◽  
Vol 36 (3) ◽  
pp. 627-639
Author(s):  
K.R. Porter
Keyword(s):  
Endoplasmic Reticulum ◽  
Golgi Complex ◽  
Random Distribution ◽  
Ground Substance ◽  
The Matrix ◽  
Cell Processes ◽  
Cytoplasmic Ground Substance

Most types of cells are known from their structure and overall form to possess a characteristic organization. In some instances this is evident in the non-random disposition of organelles and such system subunits as cisternae of the endoplasmic reticulum or the Golgi complex. In others it appears in the distribution and orientation of cytoplasmic fibrils. And in yet others the organization finds expression in the non-random distribution and orientation of microtubules, especially as found in highly anisometric cells and cell processes. The impression is unavoidable that in none of these cases is the organization achieved without the involvement of the cytoplasmic ground substance (CGS) or matrix. This impression is based on the fact that a matrix is present and that in all instances these formed structures, whether membranelimited or filamentous, are suspended in it. In some well-known instances, as in arrays of microtubules which make up axonemes and axostyles, the matrix resolves itself into bridges (and spokes) between the microtubules, bridges which are in some cases very regularly disposed and uniform in size (Mcintosh, 1973; Bloodgood and Miller, 1974; Warner and Satir, 1974).

A matrix realignment method for recognizing entanglement

2003 ◽  
Vol 3 (3) ◽  
pp. 193-202
Author(s):  
K. Chen ◽  
L.-A. Wu
Keyword(s):  
Kronecker Product ◽  
Singular Values ◽  
Quantum Systems ◽  
Entangled States ◽  
Approximation Technique ◽  
Separable State ◽  
Simple Method ◽  
Bipartite Quantum Systems ◽  
The Matrix ◽  
Degree Of Entanglement

Motivated by the Kronecker product approximation technique, we have developed a very simple method to assess the inseparability of bipartite quantum systems, which is based on a realigned matrix constructed from the density matrix. For any separable state, the sum of the singular values of the matrix should be less than or equal to $1$. This condition provides a very simple, computable necessary criterion for separability, and shows powerful ability to identify most bound entangled states discussed in the literature. As a byproduct of the criterion, we give an estimate for the degree of entanglement of the quantum state.

2D Structural Acoustic Analysis Using the FEM/FMBEM with Different Coupled Element Types

2017 ◽  
Vol 42 (1) ◽  
pp. 37-48 ◽  
Author(s):  
Leilei Chen ◽  
Wenchang Zhao ◽  
Cheng Liu ◽  
Haibo Chen
Keyword(s):  
Acoustic Analysis ◽  
Interaction Analysis ◽  
Fast Multipole Method ◽  
Boundary Elements ◽  
Higher Order ◽  
Boundary Integral ◽  
The Matrix ◽  
Coupling Approach ◽  
Element Type ◽  
Relative Errors

Abstract A FEM-BEM coupling approach is used for acoustic fluid-structure interaction analysis. The FEM is used to model the structure and the BEM is used to model the exterior acoustic domain. The aim of this work is to improve the computational efficiency and accuracy of the conventional FEM-BEM coupling approach. The fast multipole method (FMM) is applied to accelerating the matrix-vector products in BEM. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The continuous higher order boundary elements and discontinuous higher order boundary elements for 2D problem are developed in this work to achieve higher accuracy in the coupling analysis. The performance for coupled element types is compared via a simple example with analytical solution, and the optimal element type is obtained. Numerical examples are presented to show the relative errors of different coupled element types.

scholarly journals Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach

Algorithms ◽  
2018 ◽  
Vol 11 (11) ◽  
pp. 184 ◽  
Author(s):  
Qing Li ◽  
Steven Liang
Keyword(s):  
Vibration Signal ◽  
Singular Values ◽  
Rolling Bearing ◽  
L1 Norm ◽  
Detection Approach ◽  
The Matrix ◽  
Noisy Observation ◽  
Weak Fault ◽  
Convex Regularization ◽  
Fault Information

Aimed at the issue of estimating the fault component from a noisy observation, a novel detection approach based on augmented Huber non-convex penalty regularization (AHNPR) is proposed. The core objectives of the proposed method are that (1) it estimates non-zero singular values (i.e., fault component) accurately and (2) it maintains the convexity of the proposed objective cost function (OCF) by restricting the parameters of the non-convex regularization. Specifically, the AHNPR model is expressed as the L1-norm minus a generalized Huber function, which avoids the underestimation weakness of the L1-norm regularization. Furthermore, the convexity of the proposed OCF is proved via the non-diagonal characteristic of the matrix BTB, meanwhile, the non-zero singular values of the OCF is solved by the forward–backward splitting (FBS) algorithm. Last, the proposed method is validated by the simulated signal and vibration signals of tapered bearing. The results demonstrate that the proposed approach can identify weak fault information from the raw vibration signal under severe background noise, that the non-convex penalty regularization can induce sparsity of the singular values more effectively than the typical convex penalty (e.g., L1-norm fused lasso optimization (LFLO) method), and that the issue of underestimating sparse coefficients can be improved.

scholarly journals Computing mu-values for Representations of Symmetric Groups in Engineering Systems

2018 ◽  
Vol 11 (3) ◽  
pp. 774-792
Author(s):  
Mutti-Ur Rehman ◽  
M. Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Symmetric Groups ◽  
Singular Values ◽  
Linear Control ◽  
Linear Control Systems ◽  
Complex Numbers ◽  
Engineering Systems ◽  
Matrix Representations ◽  
The Matrix

In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv.

scholarly journals REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

2019 ◽  
Vol 7 ◽  
Author(s):  
ANIRBAN BASAK ◽  
ELLIOT PAQUETTE ◽  
OFER ZEITOUNI
Keyword(s):  
Toeplitz Matrices ◽  
Random Variable ◽  
Singular Values ◽  
Equivalence Theorem ◽  
Empirical Measure ◽  
Uniform Random Variable ◽  
Normal Matrices ◽  
Nilpotent Matrix ◽  
The Matrix ◽  
Triangular Toeplitz Matrix

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let$M_{N}$be a deterministic$N\times N$matrix, and let$G_{N}$be a complex Ginibre matrix. We consider the matrix${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where$\unicode[STIX]{x1D6FE}>1/2$. With$L_{N}$the empirical measure of eigenvalues of${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties$L_{N}$to the singular values of$z-M_{N}$, with$z\in \mathbb{C}$. We then compute the limit of$L_{N}$when$M_{N}$is an upper-triangular Toeplitz matrix of finite symbol: if$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$where$\mathfrak{d}$is fixed,$a_{i}\in \mathbb{C}$are deterministic scalars and$J$is the nilpotent matrix$J(i,j)=\mathbf{1}_{j=i+1}$, then$L_{N}$converges, as$N\rightarrow \infty$, to the law of$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$where$U$is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when$\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in$M_{N}$.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

scholarly journals UNITARY LOCAL INVARIANCE

2005 ◽  
Vol 03 (04) ◽  
pp. 655-659 ◽  
Author(s):  
MATTEO G. A. PARIS
Keyword(s):  
Singular Values ◽  
Complete Characterization ◽  
Bipartite State ◽  
Local Invariance ◽  
The Matrix ◽  
Pure States

We address unitary local (UL) invariance of bipartite pure states. Given a bipartite state |Ψ〉〉 = ∑ij ψij |i〉1 ⊗ |j〉2 the complete characterization of the class of local unitaries U1 ⊗ U2 for which U1 ⊗ U2|Ψ〉〉 = |Ψ〉〉 is obtained. The two relevant parameters are the rank of the matrix Ψ, [Ψ]ij = ψij, and the number of its equal singular values, i.e. the degeneracy of the eigenvalues of the partial traces of |Ψ〉〉.

The Empirical Distribution of the Singular Values of a Random Hankel Matrix

2015 ◽  
Vol 14 (03) ◽  
pp. 1550027 ◽  
Author(s):  
Mansi Ghodsi ◽  
Nader Alharbi ◽  
Hossein Hassani
Keyword(s):  
Normal Distribution ◽  
Random Matrix ◽  
Empirical Distribution ◽  
Hankel Matrix ◽  
Singular Values ◽  
Shape Parameters ◽  
The Matrix ◽  
Non Parametric

The empirical distribution of the eigenvalues of the matrix HHT divided by its trace is considered, where H is a Hankel random matrix. The normal distribution with different parameters are considered and the effect of scale and shape parameters are evaluated. The correlation among eigenvalues are assessed using parametric and non-parametric association criteria.

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