Block Diagonal Matrix Research Articles

Parallel Algorithm for Transient Stability Simulation Based on Sherman-Morrison Formula

Parallel computing is an effective approach to transient stability real-time simulation of large-scale power system. In this paper, a new parallel algorithm for power system transient stability simulation is proposed by combining Gauss method and the Sherman-Morrison formula. The algorithm adopts the Gauss method to convert the differential-algebraic equations into a set of nonlinear algebraic equations by multi-stages discretization, while the algebraic system is solved using rigorous Newton method. On this basis, the whole Jacobian matrix involved in Newton method is split to a block diagonal matrix and a block constant coefficients matrix according to s time points, and then based on the block diagonal matrix, the computing tasks at s time points are fully decoupled through the extended Sherman-Morrison matrix inverse formula. The proposed algorithm preserves the good convergence of rigorous Newton method and meanwhile has a high degree of parallelism both in time and in space.

Spectral analysis of one class of the integro-differential operators

Let ℋ = L2[0,ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on the closed interval [0,ω]. In this paper, we study the spectral properties of a second-order differential operator which perturbed by an integral operator. The integral operator belong to the two-sided ideal of Hilbert-Schmidt operator in ℋ. We use similar operators method for studying this operator. The similar operators method was developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to the one with a diagonal and a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of an integro-differential operator are obtained.

Color Image and Multispectral Image Denoising Using Block Diagonal Representation.

Filtering images of more than one channel are challenging in terms of both efficiency and effectiveness. By grouping similar patches to utilize the self-similarity and sparse linear approximation of natural images, recent nonlocal and transform-domain methods have been widely used in color and multispectral image (MSI) denoising. Many related methods focus on the modeling of group level correlation to enhance sparsity, which often resorts to a recursive strategy with a large number of similar patches. The importance of the patch level representation is understated. In this paper, we mainly investigate the influence and potential of representation at patch level by considering a general formulation with a block diagonal matrix. We further show that by training a proper global patch basis, along with a local principal component analysis transform in the grouping dimension, a simple transform-threshold-inverse method could produce very competitive results. Fast implementation is also developed to reduce the computational complexity. The extensive experiments on both the simulated and real datasets demonstrate its robustness, effectiveness, and efficiency.

Robust Design of Connected Cruise Control Among Human-Driven Vehicles

This paper presents the robustness analysis for the head-to-tail string stability of connected cruise controllers that utilize motion information of human-driven vehicles ahead. In particular, we consider uncertainties arising from the feedback gains and reaction time delays of the human drivers. We utilize the linear fractional transformation and the $M$ - $\Delta$ uncertain interconnection structure to represent the uncertainties in the block-diagonal matrix $\Delta$ . The uncertain gains are directly incorporated in the uncertain interconnection structure, while the uncertain time delays are taken into account using the Rekasius substitution that preserves the tightness of the robustness bounds. This modeling framework scales are well for large-size connected vehicle systems. We demonstrate through two case studies how parameters in the connected cruise controller can be selected to ensure the robust string stability. Theoretical results are supported by the experiments that highlight the advantage of robust control designs.

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale $$H(\mathrm {div})$$ problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this $$H(\mathrm {div})$$ problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale $$H(\mathrm {div})$$ system is bounded by $$C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))$$ , where $$\hat{\delta }$$ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

Improving the Signal-to-Noise Ratio of Superresolution Imaging Based on Single-Pixel Camera

Based on the theories of single-pixel camera and compressed sensing image reconstruction, the sparse basis, the projection method of measurement matrix, and the signal reconstruction algorithm are optimized. First, for the sparse representation of image, the restraint matrix is designed combining the characteristics of wavelet sparse coefficients to enhance the sparse representation ability of the discrete wavelet base, which improves the quality of image reconstruction and imaging by single-pixel camera. Then, in the single-pixel camera system, the projection method is improved according to the characteristics of the digital mirror device micromirror, and a new bilateral projection method based on the block diagonal measurement matrix is proposed to realize the superresolution imaging of the outdoor scene, reducing the number of measurement and improving the imaging quality. Finally, for the algorithm of image reconstruction, combining the characteristics of convex optimization and nonconvex optimization algorithms, a new algorithm from convex optimization approximately to nonconvex optimization algorithm is proposed, and compared with the traditional image reconstruction algorithm, such as greedy pursuit algorithm, minimum norm algorithm, and image interpolation, the peak signal-to-noise ratio and imaging quality of reconstructed images are effectively improved. The feasibility of the proposed methods is demonstrated by simulation experiments and imaging experiments of single-pixel camera.

Add–sub pivoting triangular factorization for symmetric matrix

Commonly used diagonal pivoting strategies for general symmetric matrix would bring about triangular factorization PAP’=LDL’, in which D is block diagonal matrix with either 1 × 1 or 2 × 2 diagonal blocks. This paper proposed a modified add–sub pivoting strategy resulting in D with no 2 × 2 diagonal block and a corresponding symmetric triangular factorization algorithm for solving symmetric linear system Ax=b. The factorization algorithm is based on symmetric Gaussian elimination; The pivoting strategy involves searching two suitable columns in the trailing submatrix A k by rook method, interchanging the first one of them with column k, and then adding the second one multiplied by a scalar γ to column k. This strategy is similar to an earlier one proposed by Dax; Its comparisons required are still O(n 2 ) ∼ O(n 3 ), but the bound on elements of L is lower, and especially the growth factor is bounded by 3. The results from numerical experiments with both ill-conditioned and good-conditioned symmetric indefinite linear systems show that the new algorithm is competitive with the B–K algorithm, the BBK algorithm, and the LU factorization with partial pivoting.

Computing the density of states for optical spectra of molecules by low-rank and QTT tensor approximation

In this paper, we introduce a new interpolation scheme to approximate the density of states (DOS) for a class of rank-structured matrices with application to the Tamm–Dancoff approximation (TDA) of the Bethe–Salpeter equation (BSE). The presented approach for approximating the DOS is based on two main techniques. First, we propose an economical method for calculating the traces of parametric matrix resolvents at interpolation points by taking advantage of the block-diagonal plus low-rank matrix structure described in [6,3] for the BSE/TDA problem. This allows us to overcome the computational difficulties of the traditional schemes since we avoid the construction of the matrix inverse and hence the need of stochastic sampling. Second, we show that a regularized or smoothed DOS discretized on a fine grid of size N can be accurately represented in a low rank quantized tensor train (QTT) format that can be determined through a least squares fitting procedure. The QTT tensor provides good approximation properties for strictly oscillating DOS functions with multiple gaps, in contrast to interpolation by problem independent functions like polynomials, trigonometric functions, etc. Moreover, the QTT approximant requires asymptotically much fewer (e.g., O(log⁡N)) functional calls compared with the full grid size N. Numerical tests indicate that the QTT approach yields accurate recovery of DOS associated with problems that contain relatively large spectral gaps. The QTT tensor rank only weakly depends on the size of a molecular system that paves the way for treatment of large-scale spectral problems.

Spectral-element method with arbitrary hexahedron meshes for time-domain 3D airborne electromagnetic forward modeling

Mainstream numerical methods for 3D time-domain airborne electromagnetic (AEM) modeling, such as the finite-difference (FDTD) or finite-element (FETD) methods, are quite mature. However, these methods have limitations in terms of their ability to handle complex geologic structures and their dependence on quality meshing of the earth model. We have developed a time-domain spectral-element (SETD) method based on the mixed-order spectral-element (SE) approach for space discretization and the backward Euler (BE) approach for time discretization. The mixed-order SE approach can contribute an accurate result by increasing the order of polynomials and suppress spurious solutions. The BE method is an unconditionally stable technique without limitations on time steps. To deal with the rapid variation of the fields close to the AEM transmitting loop, we separate a secondary field from the primary field and simulate the secondary field only, for which the primary field is calculated in advance. To obtain a block diagonal mass matrix and hence minimize the number of nonzero elements in the system of equations to be solved, we apply Gauss-Lobatto-Legendre integral techniques of reduced order. A direct solver is then adopted for the system of equations, which allows for efficient treatment of the multiple AEM sources. To check the accuracy of our SETD algorithm, we compare our results with the semianalytical solution for a layered earth model. Then, we analyze the modeling accuracy and efficiency for different 3D models using deformed physical meshes and compare them against results from 3D FETD codes, to further show the flexibility of SETD for AEM forward modeling.

Testing for independence of large dimensional vectors

In this paper new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose we study the weak convergence of linear spectral statistics of central and (conditionally) non-central Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) non-central Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.

Simulations of Core Collapse Supernova Explosion on PEZY-SC Processors and GPUs

The core collapse supernovae are one of key phenomena to understand the history of the Universe and the origin of heavy elements. To understand their explosion mechanism, large scale numerical simulations are essential that require to solve a multi-physics system described by coupled equations of hydrodynamics and neutrino-radiation transfer in multidimensions. Since the neutrino transfer is governed by the Boltzmann equation in six-dimensional space, necessary computational resource rapidly increases as the number of grids in simulations grows. So far numerical studies have been performed mostly on massively parallel computers and only a few studies have been made using accelerator architectures, such as GPUs, despite their large potential. The PEZY-SC processors are novel many-core architecture that have tremendous potential for scientific high-performance computing. While they share typical features with GPUs, one needs to port and optimize an application considering the difference in their multi-level structure of cores and caches to fully make use of their computational performance. In this work, we apply the PEZY-SC processors to the numerical computations of neutrino radiation hydrodynamics under spherically symmetry as a prototype of the multi-dimensional supernova simulations. Adopting the implicit scheme for discretized time evolution, the bottlenecks of the simulations are as follows: (a) An iterative linear equation solver for the coefficient matrix in the evolution equation, (b) Computation of collision term in the Boltzmann equation, and (c) Inversion of a block diagonal matrix used for preconditioning. The most time-consuming part is typically (a), while (b) and (c) are non-negligible. The steps (a) and (c) concern a block tridiagonal matrix composed of O(1000) block matrices each having O(500) ranks. We offload these hot spots to PEZY-SC processors employing PEZY-CL (a variant of OpenCL) in our simulation code based on our previous studies to exploit the GPUs using NVIDIA CUDA framework. We measure the performance on Suiren Blue (wth PEZY-SC) and Suiren2 (PEZY-SC2) systems at KEK and compared with the result on a system with NVIDIA P100 processors. The achieved performance is sufficient for simulations with improved resolutions than the previous model size, which are not sufficient for recent observational progress. We also discuss the similarity and difference of these architectures and how to ease the porting effort as a prospect for application to the multi-dimension simulations.

Controllability of the linear switched dynamical systems of the special type

Switched systems is a special case of hybrid dynamical systems with discrete and continuous dynamics. They are widely applied when a real system cannot be described by one single model. In theoretical works on switched systems, switching signals and times can be random or given by some law. Stability depends both on vector fields and on the switching law. In the present paper, a different formulation of the problem is considered, that is the case, when switching signal is under our control. Namely, a switched system is called controllable if for any two points there exists a switching signal that allows to get from the first point to the second one. In the paper the controllability of linear switched systems of a special type is studied. More specifically, we consider a switch, that is carried out between two 2x2 matrices with purely imaginary eigenvalues of both matrices. In the first section we discuss the physical meaning of switched systems of this type. Namely, the problem of oscillation of a spring pendulum with a switchable stiffness coefficient is considered with the series and parallel connection of an additional spring to the system with one given spring. We prove that such a system is controllable, and propose the method of finding the controlling switching signal. In the second section we present the main result of the work. We formulate an algorithm that allows finding a set of switching signals for getting from any given initial point to any given end point. We present an example of such controlling switching signals, simulated in MATLAB. In the last section we propose a generalization of the obtained result and formulate the theorem that states the controllability of the special type switched system with a block-diagonal matrix of high dimension. The method presented in the paper can be generalized to study of controllability of linear switched systems of more general form.

English

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube In. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in In can be represented by 0/1-matrices P of size n × n whose Gramians G = P⊤P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse P−⊤ of P is doubly stochastic and has the same support as P. In fact, P has a fully indecomposable doubly stochastic pattern. The negative part C of P−⊤ is strictly row-substochastic and its support is complementary to that of D, showing that P−⊤ = D−C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex Ŝ in In having F as a facet. We call Ŝ the acute neighbor of S at F. If P represents a 0/1-simplex that is merely nonobtuse, the inverse of G = P⊤P is only weakly diagonally dominant and has nonpositive off-diagonal entries. These matrices play an important role in finite element approximation of elliptic and parabolic problems, since they guarantee discrete maximum and comparison principles. Consequently, P−⊤ can have entries equal to zero. We show that its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors Ŝ at each of its facets. In this paper, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that if S has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of S will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all n-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to n, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.

FPGA-based real-time compressed sensing of multichannel EEG signals for wireless body area networks

Abstract The purpose of this study is to solve the issues in reconstruction computation complexity of the current method in wireless body area network (WBAN) through developing compressed sensing (CS) for multichannel electroencephalogram, performing model optimization, designing a system for compressing and collecting electroencephalogram (EEG) signals, and implementing real time compression and collection of multichannel signals. Firstly, based on the distributed compressed sensing theory, we analyze the sparsity of EEG signal, screen digital sensing matrix models, design multichannel joint reconstruction algorithm, and perform optimization analysis as well as simulation verification at each step. Secondly, based on field programmable gate array (FPGA), we realize real time collection, storage, compression, and transmission of multichannel EEG by setting up a compression and collection system. Lastly, each system function module is inspected, and the performance of the compressed multichannel EEG system is evaluated from the perspective of computation complexity, reconstruction accuracy, instantaneity, etc. Evaluation results show that the improvement of real-time performance is contributed by the application of binary permutation block diagonal matrix (BPBD), which converts CS multiplications into additions with a simple circuit and reduces the computational time drastically. The average signal to noise distortion ratio for signal reconstruction reaches 21.74 dB under the compression ratio of 2, which also meets the requirement of WBAN. The proposed method has faster computation, better accuracy, and simpler coding, can be utilized in a variety of applications related to multichannel EEG, especially in situations where the system power consumption and real-time performance are critical.

S3 flavored left-right symmetric model of quarks

We construct a model based on the electroweak gauge group $\rm SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ augmented by an $S_3$ symmetry. We assign nontrivial $S_3$ transformation properties to the quarks and consequently we need two scalar bidoublets. Despite the extra bidoublet we have only six Yukawa couplings thanks to the discrete symmetry. Diagonalization of the quark mass matrices shows that at the leading order only the first two generations mix, resulting in a block diagonal CKM matrix, and the first generation quarks are massless. Inclusion of subleading terms produce an acceptable CKM matrix up to corrections of order $\lambda^4$. As for the first generation quark masses, we obtain satisfactory value of $m_u/m_d$. The masses themselves, though being in the same ballpark, turn out to be somewhat smaller than the accepted range.

The Tropical Matrix Groups with Symmetric Idempotents

In this paper we study the semigroup Mn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.

Cospectral graphs, GM-switching and regular rational orthogonal matrices of level p

The GM-switching, invented by Godsil and McKay, is a powerful method to construct cospectral graphs. In its simplest version, the method can be roughly described as follows: let A(G) be the adjacency matrix of a graph G on n vertices, let Q=diag(R,In−2m) be a block diagonal matrix, where R=1mJ−I2m is an orthogonal matrix of order 2m, J is the all-one matrix and Ik denotes the identity matrix of order k. If G satisfies certain conditions, then QTA(G)Q is an adjacency matrix of a graph G′ which is cospectral with G (with cospectral complements).In this paper, we consider the problem of replacing the above R with another rational orthogonal matrix U of order 2p (p an odd prime). We characterize all graphs G admitting a GM-switching corresponding to Q=diag(U,In−2p), i.e., such that QTA(G)Q is an adjacency matrix of a graph G′, and hence giving a counterpart of the GM-switching method for generating cospectral graphs (with cospectral complements), which can be used to construct pairs of non-isomorphic cospectral graphs that are not obtainable by the original GM-switching method.Next, we consider the problem of how to determine whether a given graph has a non-trivial GM-switching associated with some U. A simple algebraic condition is presented which partially answers this question.

A Fuzzy Co-Clustering Algorithm via Modularity Maximization

In this paper we propose a fuzzy co-clustering algorithm via modularity maximization, named MMFCC. In its objective function, we use the modularity measure as the criterion for co-clustering object-feature matrices. After converting into a constrained optimization problem, it is solved by an iterative alternative optimization procedure via modularity maximization. This algorithm offers some advantages such as directly producing a block diagonal matrix and interpretable description of resulting co-clusters, automatically determining the appropriate number of final co-clusters. The experimental studies on several benchmark datasets demonstrate that this algorithm can yield higher quality co-clusters than such competitors as some fuzzy co-clustering algorithms and crisp block-diagonal co-clustering algorithms, in terms of accuracy.

Thermal Entanglement Between Two Two-Level Atoms in a Two-Photon Jaynes-Cummings Model with an Added Kerr Medium

In this paper, we consider a Hamiltonian model that includes interaction of two coupled two-level atoms with a single-mode quantized electromagnetic field in a cavity via the degenerate two-photon transition. The cavity is filled with a Kerr-like medium and is held at a temperature T. The free field Hamiltonian possesses the su(1,1) symmetry which realized by either even or odd photon-number states. The total number of excitation as a constant of motion, provides a decomposition of the Hilbert space of system into direct sums of invariant subspaces. As a results, the representation of the Hamiltonian becomes block-diagonal matrix with three blocks. After diagonalizing each block, we obtain thermal state of system in the whole Hilbert space and within its excitation subspaces. Finally, the effect of temperature, atom-atom and Kerr-type couplings on the degree of thermal entanglement between the atoms are investigated. Our results show that within the single-excitation subspace spanned with odd photon-number states, the entanglement between the atoms is thermally robust.

Alternative over-identifying restriction test in the GMM estimation of panel data models

A new over-identifying restriction test in the generalized method of moments (GMM) estimation of panel data models is proposed. In contrast to the conventional over-identifying restriction test, where the sample covariance matrix of the moment conditions is used in the weighting matrix, the proposed test uses a block diagonal weighting matrix constructed from the efficient optimal weighting matrix. It is shown that the proposed test statistic asymptotically follows the weighted sum of the chi-square distribution with one degree of freedom. A detailed local power analysis is provided for dynamic panel data models, and it is demonstrated that the new test has a comparable power to the conventional J test in many cases. The Monte Carlo simulations reveal that the proposed test has a substantially better size property than the conventional test does.