Iterative Matrix Research Articles

Design of a Robust Decentralized Power System Stabilizer for Sudan National Grid

Power system stabilizers (PSS) are used to generate supplementary control signals for the excitation system in order to damp the low frequency, generator angle, speed variation, generator power, voltage magnitude and power flow oscillations. This paper presents robust decentralized power system stabilizer (PSS) design approaches for a power system consisting of 41 machines with 167 buses. This paper mainly focuses on developing robust decentralized control techniques for power systems, with special emphasis on problems that can be expressed as minimizing a linear objective function under linear matrix inequalities [LMI] in tandem with bilinear matrix inequalities [BMI] constraints. The design problem is considered the natural extension of the reduced order for decentralized dynamic output H2/H∞ -norm controller’s synthesis for power systems. The resulting optimization problem has a general bilinear matrix inequalities (BMIs) form which can be solved using an iterative linear matrix inequalities [LMIs] programming method. Simulations were carried out using loss of line without fault tests at transmission line on Sudan grid.

Sector-model subspace iteration for vibration of multi-stage, cyclically symmetric systems

This work develops a computationally efficient sector-model subspace iteration to obtain accurate eigensolutions for the vibration of multi-stage systems where each stage is cyclically symmetric. The method fully captures the inter-stage coupling. The component stages are cyclically symmetric but the full multi-stage model is generally asymmetric because the number of sectors can differ among the stages. The sector-model subspace iteration is equivalent to the full-model subspace iteration in its accuracy, but it is more computationally efficient. Different from the full-model subspace iteration that relies on a full multi-stage system model, the sector-model subspace iteration only uses matrices of single-stage sector models. Parallel computation applies to most steps in each iteration and many matrices do not need to be recomputed for each iteration. The improved computational efficiency of the sector-model subspace iteration is more significant when the full model has a large number of stages, a large number of sectors within each stage, and/or many degrees of freedom in each single-stage sector. The iteration converges rapidly and can start with any initial trial vectors. Faster convergence is achieved using high-quality trial vectors from reduced-order modeling methods, such as single-stage analyses, multi-stage cyclic symmetry reduction, and component mode synthesis. In contrast to reduced-order modeling methods, the iteration provides eigensolutions with controllable accuracy based on user-selected convergence criteria. The application and advantages of the sector-model subspace iteration are demonstrated using two example multi-stage bladed disks.

Accelerated convergence strategy of weighted least squares method for image reconstruction

The weighted least squares algorithm is effective in image reconstruction, in which the weight is determined by a covariance matrix of model error. To speed up the iterative convergence speed of the WLS method, we express the WLS iterative formula in a matrix form with an iterative relaxation coefficient and propose relaxation strategies based on minimizing the spectral radius of the iterative matrix. The numerical simulations demonstrate that the relaxation strategies proposed in this paper have the better reconstruction performances.

Genetic Algorithm-Based Adaptive Wiener Gain for Speech Enhancement Using an Iterative Posterior NMF

In this paper, we propose a genetic algorithm-based adaptive Wiener gain for speech enhancement using an iterative posterior non-negative matrix factorization (NMF). In the recent past, NMF-based Wiener filtering methods were used to improve the performance of speech enhancement, which has shown that they provide better performance when compared with conventional NMF methods. But performance degrades in non-stationary noise environments. Template-based approaches are more robust and perform better in non-stationary noise environments compared to statistical model-based approaches but are dependent on a priori information. Combining the approaches avoids the drawbacks of both. To improve the performance further, speech and noise bases are adapted simultaneously in the NMF approach. The usage of Super-Gaussian constraints in iterative NMF still improves the performance in non-stationary noise. The silence frame is a challenging task in the case of NMF; still there will be some amount of noise present in those frames. For further enhancement, we have combined with a genetic algorithm (GA)-based adaptive Wiener filter which performs well in denoising and also the GA search the adaptive [Formula: see text] allows us to control the trade-off between fitting the observed spectrogram of mixed speech and noise achieving high likelihood under our prior model. The proposed method outperforms other benchmark algorithms in terms of the source to distortion ratio (SDR), short-time objective intelligibility (STOI), and perceptual evaluation of speech quality (PESQ).

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

In this work, a new efficient solution algorithm named Combined Matrix Iterative Algorithm based on the Mini-Max Polynomial Approximation (MMPA) method is proposed for burnup calculations. As a new matrix exponential method in recent years, MMPA method has good numerical stability and computational accuracy, and all calculation are real number operations. As the same with Chebyshev Rational Approximation Method (CRAM) which is widely and successfully used in solving burnup equations, the method can also directly deal with rigid and sparse burnup matrices. Compared with the traditional solution algorithms based on MMPA method, the Combined Matrix Iterative Algorithm (MMPA-CMIA) proposed in this paper can effectively reduce the computational complexity and improve the computational efficiency after equivalently transforming the expression of the MMPA method. Then the MMPA-CMIA is used to theoretically demonstrate and analyze the solution efficiency of random matrices with different orders which has the similar distribution characteristics to burnup matrices. Compared with the 16-order CRAM method and the other two traditional solution algorithms based on MMPA method, the algorithm shows good acceleration effect and computational precision. Finally, the algorithm is used in the actual burnup benchmarks to verify and validate the accuracy and efficiency. The burnup matrices of different orders are constructed through detailed burnup chain and simplified burnup chain. The results show that the algorithm has good accuracy and numerical stability. And comparing with 16-order CRAM, the algorithm can save about 30% of computing time for solving burnup equations. It shows that the algorithm is a potentially powerful method for accelerating the solution of large-scale burnup calculations.

Takagi–Sugeno Fuzzy Realization of Stability Performance-Based Fault-Tolerant Control for Nonlinear Systems

This article is dedicated to studying realization issues of the stability performance-based nonlinear fault-tolerant control framework via Takagi–Sugeno (T–S)fuzzy models. To this end, the nonlinear fault-tolerant control strategy with an online fault detection system monitoring the system stability performance degradation induced by faults is first introduced by means of the stable image and kernel representations. On this basis, the T–S fuzzy models are applied to approximate the nonlinear system, and a design approach of the fuzzy observer-based controller is proposed for the system stabilization via the iterative linear matrix inequality method. With the controller gains, the fuzzy-model-based nominal stable image representation of the system is formulated, which leads to the generation of the input and output error signals. Then, with the reference signal and system input and output error signal data, a data-driven algorithm is given to online estimate the evaluation function defined in terms of system uncertainties and faults. By virtue of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> input–output stability of the controller stable kernel representation, a threshold calculation method is presented and, thus, the stability performance-based fault detection system based on fuzzy models is realized. Furthermore, for fault-tolerant purpose, the fault-tolerant controller design is discussed, which aims to retain the system stability. Two examples are provided in the end to illustrate the proposed results.

Research on Music Signal Processing Based on a Blind Source Separation Algorithm

The isolation of mixed music signals is beneficial to the extraction and identification of music signal features and to enhance music signal quality. This paper briefly introduced the mathematical model for separating blind source from mixed music signals and the traditional Independent Component Analysis (ICA) algorithm. The separation algorithm was optimized by the complex neural network. The traditional and optimized ICA algorithms were simulated in MATLAB software. It was found that the time-domain waveform of the signal isolated by the improved ICA-based separation algorithm was closer to the source signal. The similarity coefficient matrix, signal-to-interference ratio, performance index, and iteration time of the improved ICA-based algorithm was 62.3, 0.0011, and 0.87 s, respectively, which were all superior to the traditional ICA algorithm. The novelty of this paper is setting the initial iterative matrix of the ICA algorithm with the complex neural network.

Stability analysis for a class of switched systems with state constraints via time-varying Lyapunov functions

This article investigates the stability analysis for a class of continuous-time switched systems with state constraints under pre-specified dwell time switchings. The state variables of the studied system are constrained to a unit closed hypercube. Firstly, based on the definition of set coverage, the system state under saturation is confined to a convex polyhedron and the saturation problem is converted into convex hull. Then, sufficient conditions are derived by introducing a class of multiple time-varying Lyapunov functions in the framework of pre-specified dwell time switchings. Such a dwell time is an arbitrary pre-specified constant which is independent of any other parameters. In addition, the proposed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching instants. The feature of this paper is that the definition of set coverage is utilized to replace the restriction on the row diagonally dominant matrices with negative diagonal elements to analyze stability. The other feature of the constructed time-varying Lyapunov functions is that there are two time-varying functions. One of the two time-varying functions contains the jump rate, which will present a certain degree of freedom in designing the dwell time switching signal. An iterative linear matrix inequality (LMI) algorithm is presented to verify the sufficient conditions. Finally, two examples are presented to show the validity of the method.

Stability analysis and stabilisation of continuous-discrete fractional-order 2D Fornasini–Marchesini first model

This paper focuses on the structural stability and stabilisation of the continuous-discrete fractional-order two-dimensional Fornasini-Marchesini first model. The sufficient and necessary stability conditions are given in polynomial form at first. Secondly, to make the problem solvable, based on the property of Kronecker product and the Generalized–Kalman–Yakubovich–Popov Lemma, the new sufficient and necessary stability conditions are established in the form of linear matrix inequalities. Thirdly, with the help of the Projection Lemma, the stabilisation conditions are obtained after introducing a state feedback controller, which can be solved with an iterative linear matrix inequality algorithm. In the end, the effectiveness of the proposed results is verified by two numerical examples.

Research on Discrete Artificial Bee Colony Cache Strategy of UAV Edge Network

Unmanned aerial vehicle edge networks (UENs) can reduce the cache load of the core network and improve system performance to provide users with efficient content services. However, the time-varying characteristics of content popularity in UENs lead to a low accuracy of popularity prediction, and the capacity limitations of wireless channel conditions lead to a lower cache hit rate than the rates of traditional fiber-optic-based cache strategies. Therefore, this paper proposes the discrete artificial bee colony cache strategy of UENs (DABCCSU). First, the information–dynamics–dissemination model of UENs (IDDMU) is established to deduce the coupling relationship between the channel capacity and the service probability in IDDMU. The influence of the service probability change on the content dissemination process is discussed, and the content popularity in UENs is predicted by the state iteration matrix. Then, the discrete artificial bee colony cache (DABCC) optimization algorithm is proposed. The action function of the artificial bee colony is designed as a random action based on the historical cache strategy. The discrete cache strategy is used as an optimization variable, and the popularity prediction result obtained by IDDMU is used to maximize the cache hit rate. DABCC provides the optimal cache strategy for the UENs, and effectively improves the cache hit rate. The simulation result shows that the accuracy of DABCCSU in content popularity prediction is more than 90%, which achieves a good prediction effect. In terms of cache performance, the average cache hit rate of DABCCSU is 91.62%, which is better than the 51.09% of the Least Recently Used (LRU) strategy, 89.27% of the Greedy Algorithm (GA) and 54.26% of Binary Particle Swarm Optimization (BPSO). In addition, the cache hit rate of DABCCSU under different cache capacities is better than that of LRU, GA, and BPSO, showing a relatively stable performance. It shows that DABCCSU can achieve excellent content popularity prediction, and it can also maximize the cache hit rate under limited communication resources and cache resources to provide UENs with the optimal content cache strategy, and provides users with high-quality content services.

Accurate reliability analysis methods for approximate computing circuits

In recent years, Approximate Computing Circuits (ACCs) have been widely used in applications with intrinsic tolerance to errors. With the increased availability of approximate computing circuit approaches, reliability analysis methods for assessing their fault vulnerability have become highly necessary. In this study, two accurate reliability evaluation methods for approximate computing circuits are proposed. The reliability of approximate computing circuits is calculated on the basis of the iterative Probabilistic Transfer Matrix (PTM) model. During the calculation, the correlation coefficients are derived and combined to deal with the correlation problem caused by fanout reconvergence. The accuracy and scalability of the two methods are verified using three sets of approximate computing circuit instances and more circuits in EvoApprox8b, which is an approximate computing circuit open source library. Experimental results show that relative to the Monte Carlo simulation, the two methods achieve average error rates of 0.46% and 1.29% and time overheads of 0.002% and 0.1%. Different from the existing approaches to reliability estimation for approximate computing circuits based on the original PTM model, the proposed methods reduce the space overheads by nearly 50% and achieve time overheads of 1.78% and 2.19%.

Efficient Stochastic Galerkin Spectral Methods for Optimal Control Problems Constrained by Fractional PDEs with Uncertain Inputs

This paper is devoted to designing fast solvers and efficient preconditioners for the optimal control problems (OCPs) constrained by stochastic fractional elliptic equations. We first prove the existence and uniqueness of the stochastic optimal control solution and then derive the stochastic optimality system. For the numerical approximation, we use the stochastic Galerkin spectral methods, which apply the stochastic Galerkin method for the discretization of random variables and employ the spectral-Galerkin approach for the approximation of spatial variables. To solve the large coupled saddle-point system resulted from discretization, we adopt the most commonly used MINRES method and a more effective PPCG method in low-rank matrix iteration format. Specially, we develop the efficient preconditioners based on the matrix decomposition method and the virtual variable method. We also study the eigenvalue distribution of the corresponding preconditioned matrix. Besides, for the approximation of the discretized state equation, we use the mean-based approximation and the Ullmann approximation to handle different values of the variance of the random inputs. Finally, we present numerical experiments to demonstrate the effectiveness of our solvers and preconditioners.

Unsupervised feature selection method based on iterative similarity graph factorization and clustering by modularity

Feature selection is an important research area aimed at eliminating unwanted features from high-dimensional datasets. Feature selection methods are categorized according to the availability of labels. Supervised methods usually select features that have high correlations with the data labels, however, this cannot be done by unsupervised methods, since the labels are not available, making the development of them even more challenging. Some unsupervised methods proposed in the literature get around this problem by generating “pseudo-labels”, using clustering techniques, and then making feature selection as in the supervised scenario. There are also methods in which the selection is not guided by a clustering criterion, generally performing some kind of reconstruction of the original dataset in low dimensionality. However, in both approaches a local structure is generated, usually starting from a similarity graph built by using the entire set of features. This may compromise the results of the methods when there are many irrelevant and noisy features, which makes it difficult to reveal patterns or an initial representation of the data. Another drawback of some current methods is the large amount of hyper-parameters, or the computational time required for a single execution, which can render such methods unfeasible for some large datasets. In order to address these problems, it is proposed in this work a new unsupervised feature selection method, called KNMFS, which performs the scoring of the features in low computational time using a three-step procedure: (1) a similarity graph learning step based on non-negative iterative matrix factorization together with a Gaussian filter to mitigate the noise effects caused by irrelevant features; (2) a clustering step from the learned graph using a modularity optimization strategy and (3) a random forest algorithm is applied to compute the scores based on the labels generated by the clustering step. To verify the effectiveness of the proposed method, experiments in real and synthetic datasets were conducted. In both cases, the obtained results showed that the KNMFS method, compared to other state-of-the-art methods, obtained good results according to the metrics of Accuracy, ARI and NMI. Friedman’s statistical tests were also performed to give stronger evidence to the reported results.

Partial randomized benchmarking

In randomized benchmarking of quantum logical gates, partial twirling can be used for simpler implementation, better scaling, and higher accuracy and reliability. For instance, for two-qubit gates, single-qubit twirling is easier to realize than full averaging. We analyze such simplified, partial twirling and demonstrate that, unlike for the standard randomized benchmarking, the measured decay of fidelity is a linear combination of exponentials with different decay rates (3 for two qubits and single-bit twirling). The evolution with the sequence length is governed by an iteration matrix, whose spectrum gives the decay rates. For generic two-qubit gates one slowest exponential dominates and characterizes gate errors in three channels. Its decay rate is close, but different from that in the standard randomized benchmarking, and we find the leading correction. Using relations to the local invariants of two-qubit gates we identify all exceptional gates with several slow exponentials and analyze possibilities to extract their decay rates from the measured curves.

An efficient matrix iteration family for finding the generalized outer inverse

This paper presents a new iterative family for computing the generalized outer inverse with a prescribed range and null space of a given complex matrix. It is proved that the proposed methods achieve at least ninth-order of convergence. In general, the improved formulation of scheme uses only seven matrix multiplications at each iteration, but for the specific parameters, it uses only five matrix multiplications. The theoretical discussion on computational efficiency index is presented. Further, numerical results obtained are compared with existing robust methods to verify the theoretical analysis and higher computational efficiency. Several numerical examples, including rectangular, rank-deficient, large sparse, and non-singular matrices from the matrix computation tool-box (mctoolbox) are included. Further, the performance of methods is measured on randomly generated singular matrices and boundary value problem. It is demonstrated that the presented scheme gives improved results than the existing Schulz-type iterative methods for calculating the generalized inverse.

Urban Landscape Information Construction and Visual Communication Design Based on Digital Image Matrix Reconstruction

This paper adopts the matrix reconstruction method to analyze the digital image of urban landscape information in-depth and uses it to study and design the visual communication of its construction. Based on the special structure and properties of Toeplitz matrices, a stepwise structured Augmented Lagrange Multiplier (SALM) algorithm for Toeplitz matrix filling is proposed by introducing structured operators. The main idea of the algorithm is to structure the iteration matrix at each step, that is, to reassign the elements on each diagonal of the matrix by the operator. In this way, the approximation matrix always maintains the Toeplitz structure during the iterative process, and the fast singular value decomposition of the Toeplitz. A matrix can be exploited, thus saving time. The convergence theory of the new algorithm is further discussed. A lightweight progressive feature fusion module is designed to improve network learning efficiency, which consists of two components: progressive local connectivity and feature attention. Specifically, progressive local connectivity extracts multilevel features for fusion by layer-by-layer separation and local splicing. Feature attention evaluates the importance of features using both channel and spatial dual attention modules. The Oculus Rift virtual reality device system is used, and the OSG graphics rendering engine is used as the basis for the scene data transfer of the overall 3D cityscape towards the virtual reality headset display device system. With the support of the Oculus SDK, the secondary rendering of the overall 3D cityscape in the immersive virtual reality module is carried out, and the corresponding OSG camera browsing interface is constructed to realize the two-eye immersive virtual display of the 3D cityscape in a virtual reality headset display device, assisting in connecting the virtual cityscape to reality.

Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation

Abstract In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the ϑ-scheme. The simultaneous treatment of all time instances leads to a global system of linear equations which provides the potential for a higher degree of parallelization of multigrid solvers due to the increased number of degrees of freedom per spatial unknown. It is shown that the all-at-once system based on an equidistant discretization in space and time stays well conditioned even if the number of blocked time-steps grows arbitrarily. Furthermore, mesh-independent convergence rates of the considered two-grid algorithm are proved by adopting classical Fourier arguments in space without assuming periodic boundary conditions. The rate of convergence with respect to the Euclidean norm does not deteriorate arbitrarily if the number of blocked time steps increases and, hence, underlines the potential of the solution algorithm under investigation. Numerical studies demonstrate why minimizing the spectral norm of the iteration matrix may be practically more relevant than improving the asymptotic rate of convergence.

Deep solution operators for variational inequalities via proximal neural networks

Following Bauschke and Combettes (Convex analysis and monotone operator theory in Hilbert spaces, Springer, Cham, 2017), we introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in real, separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g., in valuation of American-style contracts in Black–Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as projected Jacobi and projected SOR methods.

An iterative transfer matrix approach for estimating the sound speed and attenuation constant of air in a standing wave tube.

In this work, an iterative method based on the four-microphone transfer matrix approach was developed for evaluating the sound speed and attenuation constant of air within a standing wave tube. When the air inside the standing wave tube is treated as the material under test, i.e., as if it were a sample of porous material, the transfer matrix approach can be used to identify the air's acoustic properties. The wavenumber within the tube is complex owing to the formation of a visco-thermal boundary layer on the inner circumference of the tube. Starting from an assumed knowledge of the air properties, an iterative method can be applied in the post-processing stage to estimate the complex wavenumber. Experimental results presented here show that although the results are sensitive to ambient temperature, a semi-empirical formula previously proposed by Temkin [(1981). Elements of Acoustics (John Wiley & Sons)] matches closely with the measured sound speed and attenuation constant, as does a theoretical formulation proposed by Lahiri et al. [(2014). J. Sound Vib. 333(15), 3440-3458]. Further, it is shown that the Temkin [(1981). Elements of Acoustics (John Wiley & Sons)] and Lahiri et al. [(2014). J. Sound Vib. 333(15), 3440-3458] predictions accurately represent the variation of sound speed with frequency, in contrast to the formula recommended in the ASTM E1050 standard [(2019). American Society for Testing and Materials], in which the sound speed is assumed to be independent of frequency.

Randomized Monte Carlo algorithms for matrix iterations and solving large systems of linear equations

Abstract Randomized scalable vector algorithms for calculation of matrix iterations and solving extremely large linear algebraic equations are developed. Among applications presented in this paper are randomized iterative methods for large linear systems of algebraic equations governed by M-matrices. The crucial idea of the randomized method is that the iterations are performed by sampling random columns only, thus avoiding not only matrix-matrix but also matrix-vector multiplications. The suggested vector randomized methods are highly efficient for solving linear equations of high dimension, the computational cost depends only linearly on the dimension.