Sparse Block Matrix Research Articles

Real‐Valued Weighted Subspace Fitting Algorithm for DOA Estimation with Block Sparse Recovery

In this paper, the problem of direction‐of‐arrival (DOA) estimation for strictly noncircular sources under the condition of unknown mutual coupling is concerned, and then a robust real‐valued weighted subspace fitting (WSF) algorithm is proposed via block sparse recovery. Inspired by noncircularity, the real‐valued coupled extended array output with double array aperture is first structured via exploiting the real‐valued conversion. Then, an efficient real‐valued block extended sparse recovery model is constructed by performing the parameterized decoupling operation to avoid the unknown mutual coupling and noncircular phase effects. Thereafter, the WSF framework is investigated to recover the real‐valued block sparse matrix, where the spectrum of real‐valued NC MUSIC‐like is utilized to design a weighted matrix for strengthening the solutions sparsity. Eventually, DOA estimation is achieved based on the support set of the reconstructed block sparse matrix. Owing to the combination of noncircularity, parametrized decoupling thought, and reweighted strategy, the proposed method not only effectively achieves high‐precision estimation, but also efficiently reduces the computational complexity. Plenty of simulation results demonstrate the effectiveness and efficiency of the proposed method.

Role of conservation laws in the density matrix renormalization group

We explore matrix product state approximations to wave functions which have spontaneously broken symmetries or are critical. We are motivated by the fact that symmetries, and their associated conservation laws, lead to block-sparse matrix product states. Numerical calculations which take advantage of these symmetries run faster and require less memory. However, in symmetry-broken and critical phases the block-sparse ansatz yields less accurate energies. We characterize the role of conservation laws in matrix product states and determine when it is beneficial to make use of them.

Multi-UAV Cooperative Localization for Marine Targets Based on Weighted Subspace Fitting in SAGIN Environment

As an indispensable part of the Internet of Vehicles (IoV), unmanned aerial vehicles (UAVs) can be deployed for target positioning and navigation in the space–air–ground-integrated network (SAGIN) environment. Maritime target positioning is very important for the safe navigation of ships, hydrographic surveys, and marine resource exploration. Traditional methods typically exploit satellites to locate marine targets in the SAGIN environment, and the location accuracy does not satisfy the requirements of modern ocean observation missions. In order to localize the marine target, we develop a system architecture in this article, which contains UAVs integrated with monostatic multiple-input–multiple-output (MIMO) radars. The main thrust is to estimate the direction-of-arrival (DOA) via MIMO radar. Herein, we consider a general scenario that unknown mutual coupling exist and a novel sparse reconstruction algorithm is proposed. The mutual coupling matrix (MCM) is adopted with the help of its special structure, we formulate the data model as a sparse representation form. Then, two novel matrices, a weighted matrix, and a reduced-dimensional matrix are constructed to reduce the computational complexity and enhance the sparsity, respectively. Thereafter, a sparse constraint model is constructed using the concept of optimal weighted subspace fitting (WSF). Finally, the DOA estimation of maritime targets can be achieved by reconstructing the support of a block sparse matrix. Based on the DOA estimation results, multiple UAVs are used to cross-locate marine targets multiple times, and an accurate marine target position is achieved in the SAGIN environment. Numerical results are carried out, which demonstrates the effectiveness of the proposed DOA estimator, and the multi-UAV cooperative localization system can realize accurate target localization.

Application of Optimal Control to Inversion of Self-Potential Data: Theory and Synthetic Examples

Last decades, there has been an increased interest in the use of the self-potential (SP) method in hydrogeophysics. In response to this strong interest, we develop a novel approach to the inversion of SP data. Mathematically, the SP inverse problem is the source identification problem for the Poisson equation. Our approach substantially differs from the standard regularization approach, which explicitly includes the forward-problem operator into the cost functional. We formulated the inverse problem is as an optimal control problem and then translate it into a variational system. The system is approximated in suitable finite-element spaces giving rise to an algebraic problem with the saddle-point structure. In contrast to the standard approach, which leads to a dense linear system, our method results in a system with a sparse block matrix. It can be efficiently solved by either direct sparse solvers or preconditioned iterative solvers. In this paper, we present the formulation of the problem and its finite-element approximation. We discuss the iterative solution and preconditioning strategies. Our software implementation is based on an industrial finite-element package. We also present a numerical experiment with node-based linear finite elements on tetrahedral grids. Our results suggest that the proposed approach may serve as a rapid and reliable tool for large-scale SP inverse problems. Moreover, the same technique can easily be extended to a wide range of geophysical linear inverse problems, such as inversions of magnetic and gravity data.

Optimization of Precoder and Combiner in mmWave Hybrid Beamforming Systems for Multi-user Downlink Scenario

ABSTRACT Beamforming at millimeter wave (mmWave) band, promises to significantly support 5G networks in achieving their performance goals. The conventional digital beamforming uses a separate RF chain for each antenna element, while it leads to high cost and hardware complexity in mmWave massive MIMO antenna systems. Beamforming with multiple data streams called precoding improves the system’s spectral efficiency and one of its kind hybrid beamforming reduces the cost and overcomes the hardware limitation by using reduced number of RF chains. This work considers, transmit precoding, receive combining in mmWave hybrid beamforming systems and constructs a dictionary matrix containing array response vectors. This paper proposes an extended simultaneous orthogonal matching pursuit (ESOMP) algorithm to compute the block-sparse matrix. The nonzero rows of block-sparse matrix and dictionary matrix are further processed to achieve precoder/combiner optimization in multi-user downlink scenario. Simulation results reveal that the proposed method performs close to the ideal digital beamforming scheme while improving the spectral efficiency when compared to the state-of-the-art algorithm.

Sparse proportional viscous damping model for structures with large number of degrees of freedom

This paper proposes a proportional viscous damping model that is computationally very efficient for response history analysis of a large-scale structure with a large number of degrees of freedom. The proposed model is based on a bell-shaped basis function parameterized by the frequency and damping ratio at its peak. Any damping ratio curve in the frequency domain can be easily matched by adding several bell-shaped basis functions. Three methods based on least-squares curve-fitting are used to determine the model parameter values. This model allows for solving structural dynamic response using a sparse block matrix without approximation nor iterative refinement. The block matrix maintains the positive definiteness of the traditional effective stiffness matrix, allowing efficient solution methods such as Cholesky factorization and the conjugate gradient method for solutions. It also allows assigning different damping ratio curves to different structural zones, suitable for soil-structure interaction analysis. Its performance compared against existing damping models is particularly remarkable for large-scale structures with a large number of degrees of freedom in the order of thousands or even millions. The larger the number of degrees of freedom, the better the relative efficiency compared to existing damping models. Several examples are given to demonstrate this remarkable performance.

DOA Estimation of Strictly Noncircular Sources in Wireless Sensor Array Network via Block Sparse Representation

In this article, the direction of arrival estimation (DOA) of strictly noncircular sources with unknown mutual coupling is considered for wireless sensor array network (WSAN), and then a joint weighted block sparse recovery algorithm based on weighted subspace fitting (WSF) is proposed for DOA estimation.. In the proposed method, two block sparse representation models associated with the WSF principle are firstly constructed to remove the influence of unknown mutual coupling. Then, combining the advantages of noncircularity, a joint weighted block sparse recovery scheme is proposed to estimate DOA, in which the noncircular MUSIC-like (NC MUSIC-Like) spectrum function is utilized to form a weighted matrix for enhancing the solutions sparsity. Finally, the desired DOAs can be achieved with the help of the reconstructed block sparse matrix. Extensive experiments are simulated to verify that the proposed method can achieve superior estimation performance under the condition of unknown mutual coupling.

Low-rank local tangent space embedding for subspace clustering

Subspace techniques have gained much attention for their remarkable efficiency in representing high-dimensional data, in which sparse subspace clustering (SSC) and low-rank representation (LRR) are two commonly used prototypes in the fields of pattern recognition, computer vision and signal processing. Both of them aim at constructing a block sparse matrix via linearly representing data to make them be embedded into linear subspaces. However, few datasets satisfy the linear subspace assumption in the real world. In this paper, data are peered from viewpoint of manifold architecture under the framework of sparse representation. A globally low-rank representation with the Frobenius norm minimization is constructed under the constraint of local manifold embedding and a novel low-rank local embedding representation (LRLER) model for subspace clustering of datasets is proposed. In this model, the local as well as global manifold structures of a dataset are concerned. Clusters of a dataset are considered as sub-manifolds embedded in low-dimensional subspaces. To represent and segment samples with hybrid neighbors or interlaced manifold structures, the local tangent space analysis strategy is introduced to characterize the local structure of neighborhood of samples. The coefficients of locally linear embedding are rectified according to the relationship between local tangent spaces of samples and their neighbors. A local tangent space based low-rank local embedding representation model (LRLTSER) is built to deal with data with neighborhood aliasing distortion. Extensive experiments on synthetic datasets and real-world datasets are implemented and experimental results show superior performance of the proposed methods for subspace clustering compared to the state-of-the-art techniques.

Reweighted Regularized Sparse Recovery for DOA Estimation With Unknown Mutual Coupling

In this letter, the direction-of-arrival (DOA) estimation of uniform linear array under unknown mutual coupling is dealt with by proposing a reweighted regularized sparse recovery algorithm. The proposed method first formulates a block-sparse representation model without mutual coupling compensation and array aperture loss. Then, a reweighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization scheme is formulated to recover the block-sparse matrix, in which a weighted matrix is designed with a novel MUSIC-Like spectrum function for enhancing the sparsity of solution. Finally, the spatial spectrum of the recovered matrix is utilized to estimate DOAs. Due to using the whole array aperture and enhanced sparsity of solution, the proposed method can achieve superior performance than the existing regularized sparse recovery methods. Some simulation results are carried out to demonstrate the superiority of the proposed method.

Adaptive pixel-block based background subtraction using low-rank and block-sparse matrix decomposition

We present three stages of a novel backgrounds subtraction method in this paper: a new pixel-block based randomized arrangement is utilized to preprocess all the frame images, so that low-rank property of background and sparsity of foreground can be separated more easily; different foreground regions have different sparsity, we use a set of adaptive parameters for subtracting foregrounds according to the variances of frame pixels; finally, background model is built via an improved low-rank and block-sparse matrix decomposition based on the proposed adaptive pixel-block background subtraction. All these key measurements guarantee the considerable performance in background subtraction, which are also demonstrated in our experimental results.

Weighted Block Sparse Recovery Algorithm for High Resolution DOA Estimation with Unknown Mutual Coupling

Based on weighted block sparse recovery, a high resolution direction-of-arrival (DOA) estimation algorithm is proposed for data with unknown mutual coupling. In our proposed method, a new block representation model based on the array covariance vectors is firstly formulated to avoid the influence of unknown mutual coupling by utilizing the inherent structure of the steering vector. Then a weighted l 1 -norm penalty algorithm is proposed to recover the block sparse matrix, in which the weighted matrix is constructed based on the principle of a novel Capon space spectrum function for increasing the sparsity of solution. Finally, the DOAs can be obtained from the position of the non-zero blocks of the recovered sparse matrix. Due to the use of the whole received data of array and the enhanced sparsity of solution, the proposed method effectively avoids the loss of the array aperture to achieve a better estimation performance in the environment of unknown mutual coupling in terms of both spatial resolution and accuracy. Simulation experiments show the proposed method achieves better performance than other existing algorithms to minimize the effects of unknown mutual coupling.

Robust Principal Component Pursuit for Fault Detection in a Blast Furnace Process

Since blast furnaces are generally controlled by operators, the minor faults regarded as disturbances might be contained in the collected data matrix. This can severely affect sample distributions, which leads to arbitrary fault detection results using traditional data-driven methods. In this paper, a novel fault detection method named robust principal component pursuit (PCP) to handle minor faults is proposed. The minor faults are separated from columns and rows, respectively, in the training matrix via two matrix norms. By applying the proposed robust PCP method, a low rank matrix containing important process information, as well as explicit variable relationships, and a block sparse matrix containing minor faults are derived. Moreover, the convergence of the proposed method is discussed. Hotelling’s T2 statistic is potentially useful for online process monitoring in the low rank matrix. Finally, to evaluate the decomposition capacity of the proposed method for a matrix containing minor faults, a compar...

Direction of Arrival Estimation for MIMO Radar via Unitary Nuclear Norm Minimization.

In this paper, we consider the direction of arrival (DOA) estimation issue of noncircular (NC) source in multiple-input multiple-output (MIMO) radar and propose a novel unitary nuclear norm minimization (UNNM) algorithm. In the proposed method, the noncircular properties of signals are used to double the virtual array aperture, and the real-valued data are obtained by utilizing unitary transformation. Then a real-valued block sparse model is established based on a novel over-complete dictionary, and a UNNM algorithm is formulated for recovering the block-sparse matrix. In addition, the real-valued NC-MUSIC spectrum is used to design a weight matrix for reweighting the nuclear norm minimization to achieve the enhanced sparsity of solutions. Finally, the DOA is estimated by searching the non-zero blocks of the recovered matrix. Because of using the noncircular properties of signals to extend the virtual array aperture and an additional real structure to suppress the noise, the proposed method provides better performance compared with the conventional sparse recovery based algorithms. Furthermore, the proposed method can handle the case of underdetermined DOA estimation. Simulation results show the effectiveness and advantages of the proposed method.

Nuclear norm minimization framework for DOA estimation in MIMO radar

In this paper, the direction of arrival (DOA) estimation for noncircular sources in multiple-input multiple-output (MIMO) radar is dealt with by a novel nuclear norm minimization (NNM) framework. The proposed method exploits the noncircular property of signals to extend the data model for doubling the array aperture. Then a block sparse model of the extended data is formulated without the influence of the unknown noncircularity phase, and a novel signal reconstruction algorithm based on nuclear norm minimization is proposed to recover the block-sparse matrix. In addition, a weight matrix based on the reduced dimensional noncircular Capon (RD NC-Capon) spectrum is designed to reweight the nuclear norm minimization for enhancing the sparsity of solution. Finally, the DOA is estimated from the non-zero blocks of the reconstructed matrix. Due to exploiting the extended array aperture and block-sparse information, the proposed method provides superior DOA estimation performance and higher angular resolution. Furthermore, the proposed method has a low sensitivity to the priori information on the number of sources. Simulation results are presented to verify the effectiveness and advantages of the proposed method.

Block Krylov Subspace Recycling for Shifted Systems with Unrelated Right-Hand Sides

Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. It has been shown that this renders them generally incompatible with techniques of subspace recycling [K. M. Soodhalter, D. B. Szyld, and F. Xue, Appl. Numer. Math., 81C (2014), pp. 105--118]. This problem, however, can be overcome. By interpreting a family of shifted systems as one Sylvester equation, we can take advantage of the known “shift invariance” of the Krylov subspace generated by the Sylvester operator. Thus we can simultaneously solve all systems over one block Krylov subspace using the full orthogonalization method or GMRES-type methods, even when they have unrelated right-hand sides. Because residual collinearity is no longer a requirement at restart, these methods are fully compatible with subspace recycling techniques. Furthermore, we realize the benefits of block sparse matrix operations which arise in the context of high-performance computing applications. In this paper, we discuss exploiting this Sylvester equation point of view which has yielded methods for shifted systems which are compatible with unrelated right-hand sides. From this, we propose a recycled GMRES method for simultaneous solution of shifted systems. Numerical experiments demonstrate the effectiveness of the methods.

BLOCK MATRIX PRECONDITIONER METHOD FOR THE ELECTRIC FIELD INTEGRAL EQUATION (EFIE) FORMULATION BASED ON LOOP-STAR BASIS FUNCTIONS

In this paper, electromagnetic scattering problems are analyzed using an electric fleld integral equation (EFIE) formulation that is based on loop-star basis functions so as to avoid low-frequency instability problems. Moreover, to improve the convergence rate of iterative methods, a block matrix preconditioner (BMP) is applied to the EFIE formulation based on loop star-basis functions. Because the matrix system resulting from the conventional method of moments is a dense matrix, a sparse matrix version of each block matrix is constructed, followed by the inversion of the resultant block sparse matrix using incomplete factorization. Numerical results show that the proposed BMP is e-cient in terms of computation time and memory usage.

Fast Conversion Algorithm for the Dolby Digital (Plus) AC-3 Audio Coding Standards

The Dolby Digital (AC-3) and the Dolby Digital Plus or Enhanced AC-3 (E-AC-3) systems are currently the key enabling technologies for high-quality compression of digital audio signals. For the time-to-frequency transformation of an audio data block and vice versa, both systems have adopted a long transform being the modified discrete cosine transform (MDCT). The AC-3 additionally defines two variants of cosine-modulated filter banks called the first and second short transforms. A fast conversion algorithm is presented to convert the frequency coefficients of the long (MDCT) transform to those of two short transforms and vice versa, directly in the frequency domain. It is based on a block sparse matrix factorization of a conversion matrix. The fast conversion algorithm is efficient in terms of the structural simplicity, arithmetic complexity and memory requirements compared to the obvious conversion methods. Moreover, the existing AC-3 fast computational modules may be simply re-used in the conversion procedures. Consequently, the E-AC-3 to AC-3 bit stream conversion and the AC-3 to E-AC-3 bit stream transcoding can be realized in a simplified and efficient way, thus minimizing the amount of partial decoding/encoding and memory requirements during the conversion and transcoding processes.

Wavelet-Based Time-Domain Solution of Multiconductor Transmission Lines With Skin and Proximity Effect

A time-domain wavelet-based method for the analysis of transmission lines (TLs) characterized by the presence of skin and proximity effects is presented. The method is based on the time domain TL equations, expressed in terms of transient resistance and convolution operator. A formulation of the convolution in the wavelet domain is presented here, and is used to expand the TL equations in the wavelet domain. In this way, the TL equations, properly discretized in the space variable, are reduced to an algebraic system characterized by a sparse block matrix whose blocks are easily calculated. The solution of the system is conveniently performed using standard techniques, avoiding step by step procedures. The method can be applied to linear and nonlinear problems, characterized by conductors of different shapes. The results obtained are compared with standard techniques, showing both accuracy and CPU time reduction.

A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax= G( x), where A∈ R n×n is a large sparse block matrix and G: R n→ R n is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that A∈ R n×n is a block H-matrix of different types and G: R n→ R n is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.

Solving Large and Sparse Linear Equations in Analog Circuit Simulation on a Cluster of Workstations

For developing analog circuits at Siemens AG, the method of Harmonic Balance was integrated into the circuit simulator TITAN. The equations of Harmonic Balance are solved by Newton's method. The corresponding Jacobian is a very large sparse block matrix. Hitherto this kind of problem was computed either on a single workstation or on a vector computer. In this paper, we present an algorithm that makes it possible to simulate analog circuits on a cluster of workstations. Since the bulk of data is stored in a distributed way, even very large problems which often cannot be processed on vector computers due to insufficient memory capacity may now be solved.