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Matrix calculations are often required for the analysis of any big-data cloud computing system. It is quite common to process big-data associated matrices possessing the sparsity and low-rank properties. In order to efficiently deal with big-data matrices, we propose a new divide-and-iterate framework, which can be invoked to solve an enormously large linear system of equations by taking advantage of factored matrices. The Kaczmarz algorithm (KA) is utilized here to design the parallel iterative algorithms which are capable of solving a large system of equations by iteratively updating the solution through the reduction into the factorized subsystems in parallel. The convergences of our proposed new iterative algorithms are justified by the rigorous proofs. Besides, the time- and memory-complexities are studied to demonstrate the resource efficiency of the proposed algorithms. Numerical experiments are also presented to illustrate the effectiveness of this proposed new framework.

Time-series interferograms acquired on a deep pit with a Ground-Based Synthetic Aperture Radar (GB-SAR) system showed that the atmospheric phase (AP) could be complexly space variant due to rapid changes of the weather conditions and steep topography. Conventional compensation methods that simulate the AP with typical parametrical models are no longer applicable. Based on the theoretical path integral model of the AP, a grid partition (GP) method is proposed. By dividing one interferogram into a certain number of small grids, the refraction variation inside each grid is assumed to be a constant. A system of linear equations is first built based on sufficient permanent scatterers (PSs). Then, bounds and inequality constraints are set to limit the refraction variation of each grid. A constrained linear least-square problem is solved with two-step process to estimate and compensate the AP. To fully validate the feasibility of the GP method, simulated phase interferograms based on four conventional AP models and with the consideration of deformation areas and noise phase are first processed. Then, four experimental interferograms with different types of AP components are processed and made comparisons with the conventional parametrical methods. The quantitative comparisons of the simulated and experimental data sets both proved that the GP method can well reduce the AP errors.

A novel time-domain (TD) integral-equation (IE) technique for analyzing the pulsed electromagnetic (EM) scattering by a 2-D metasurface, represented by a thin highly contrasting screen with combined dielectric and magnetic properties, is presented. The problem under consideration is formulated with the aid of the EM reciprocity theorem of the time-convolution time. Subsequently, upon discretizing the space–time solution domain, the reciprocity relation is cast into two independent matrix forms whose elements are derived analytically using the classic Cagniard–DeHoop (CdH) technique. The thus obtained convolution-type systems of equations are solved for induced electric and magnetic currents through a stable step-by-step updating procedure. The presented numerical method is finally validated using alternative TD solutions.

This article presents a new approach to construct reduced-order models for large nonlinear circuits excited by periodical or quasi-periodical sources. The key feature in the constructed reduced model is that it enables, through solving a much smaller system, tracing the variations in the periodical steady-state response of the original circuit with regards to variations in selected design parameters. The proposed approach is based on the notion of using discrete empirical interpolation to project the nonlinear operator of the circuit equation onto a smaller subspace. In addition, the original system equations are projected onto the subspace spanned by the first few derivatives of the solution with respect to the selected design parameters. Computational savings are made possible through solving a reduced system of equations instead of the full system. Numerical examples are presented to validate the accuracy and efficiency of the proposed approach.

Neural networks as well as the related neural dynamics have been widely exploited to conduct online computing operations for solving various problems in recent years. This article makes progress along this direction by proposing a saturation-allowed neural dynamics (SAND) model for solving the perturbed time-dependent system of linear equations with noise-tolerant capacity. Specifically, by elaborately constructing a new general framework enhanced by the error-integration information and nonlinear projection functions (PFs), a SAND model is proposed and investigated under various additive noises. In addition, theoretical analyses reveal that the proposed SAND model is of global convergence with zero theoretical error. Moreover, when the value of PF is strictly limited by bounds, i.e., a saturation function, the upper bound of the residual errors of the proposed SAND model is subject to the noises and the bounds of PF. Computer simulation results, as well as robot experiments, verify the superior property of the proposed SAND model for solving the perturbed time-dependent system of linear equations, compared with the state of the prior art.

In this letter, the homotopy continuation (HC) approach for a problem of coupling matrix (CM) extraction is presented. HC method is used to solve the system of polynomial equations that are formulated by comparing polynomial coefficients of given filter scattering parameters ${S_{11}}$ and ${S_{21}}$ and those being described based on well-known relations between scattering parameters and CM. HC requires an initial ( $t=0$ ) system of equations with a known solution. During the HC algorithm run, the system and solution are transformed to the final ( $t=1$ ) system and solution that is our goal. In our approach, the initial system ( $t=0$ ) and its solution are formulated by comparing polynomial coefficients of S-parameters obtained for CM including entries of any, even random, values. To validate the method a numerical experiment has been presented, i.e., CM extraction for measured sixth-order (two regular and two parasitic cross-couplings) filter. The experiment demonstrated excellent results obtained with the proposed method.

The state estimation algorithm estimates the values of the state variables based on the measurement model described as the system of equations. Prior to applying the state estimation algorithm, the existence and uniqueness of the solution of the underlying system of equations is determined through the observability analysis. If a unique solution does not exist, the observability analysis defines observable islands and further defines an additional set of equations (measurements) needed to determine a unique solution. For the first time, we utilise factor graphs and Gaussian belief propagation algorithm to define a novel observability analysis approach. The observable islands and placement of measurements to restore observability are identified by following the evolution of variances across the iterations of the Gaussian belief propagation algorithm over the factor graph. Due to sparsity of the underlying power network, the resulting method has the linear computational complexity (assuming a constant number of iterations) making it particularly suitable for solving large-scale systems. The method can be flexibly matched to distributed computational resources, allowing for determination of observable islands and observability restoration in a distributed fashion. Finally, we discuss performances of the proposed observability analysis using power systems whose size ranges between 1354 and 70000 buses., Comment: 9 pages, 9 figure, version of the journal paper submitted for publication

Inferring the structures and the dynamics of the complex networked systems based on time series data is a challenging problem. The existing reconstruction methods often rely on the knowledge of the dynamics on networks. In many cases, a prior knowledge of the dynamics is unknown, so it is natural to ask: is it possible to reconstruct network and estimate the dynamical processes on complex networks only rely on the observed data? In this article, we develop a framework to reconstruct the structures of networks with binary-state dynamics, in which the knowledge of the original dynamical processes is unknown . Within the reconstruction framework, the transition probabilities of binary dynamical processes are described by the Sigmoid function in logistic regression, we then apply the mean-field approximation to enable maximum likelihood estimation (MLE), which gives rise to that the network structure can be inferred by solving the linear system of equations. Meanwhile, the original dynamical processes can be simulated by estimating the parameters in the Sigmoid function. Our framework has been validated by a variety of binary dynamical processes on synthetic and empirical networks, indicating that our method can not only reveal the network structures but also estimate the dynamical processes. Moreover, the high accuracy of our method is highlighted by comparing it with the existing methods.

The user-centric cloud radio access network (C-RAN) is promising for significantly reducing the channel training overhead because only the intra-cluster channel state information (CSI) is required. However, the inter-cluster interference may degrade the network performance. To address this problem, we present a novel framework for the uplink of user-centric C-RANs where the interference cancellation is posed as a system of bi-unknown vectors linear equations. To solve this unusual system of equations, we propose a quasi-least squares (QLS) algorithm and analyze its robustness by exploiting the random matrix theory. We reveal the fact that QLS is very sensitive to the channel estimation error due to involving the inverse of an ill-conditioned matrix. It is well known that truncated singular value decomposition (TSVD) is an effective regularization scheme that can mitigate this ill-conditioning effect. Accordingly, we employ TSVD to further improve the robustness of QLS against the imperfect channel estimation. In addition, since the performance of the TSVD based algorithm strongly depends on the truncation parameter, a parameter-choice method using the constant modulus (CM) feature is also provided. Finally, simulation results are presented to examine the effectiveness and robustness of the proposed method.

In our recent work on iterative computation in hardware, we showed that arbitrary-precision solvers can perform more favourably than their traditional arithmetic equivalents when the latter’s precisions are either under- or over-budgeted for the solution of the problem at hand. Significant proportions of these performance improvements stem from the ability to infer the existence of identical most-significant digits between iterations. This technique uses properties of algorithms operating on redundantly represented numbers to allow the generation of those digits to be skipped, increasing efficiency. It is unable, however, to guarantee that digits will stabilise, i.e. never change in any future iteration. In this article, we address this shortcoming, using interval and forward error analyses to prove that digits of high significance will become stable when computing the approximants of systems of linear equations using stationary iterative methods. We formalise the relationship between matrix conditioning and the rate of growth in most-significant digit stability, using this information to converge to our desired results more quickly. Versus our previous work, an exemplary hardware implementation of this new technique achieves an up-to 2.2x speedup in the solution of a set of variously conditioned systems using the Jacobi method.

We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOCCVI). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality (VI), which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Especially, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments, illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that, in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.

In this article, a temporal coupled-mode theory (CMT) formalism is proposed to design graphene-based metasurface (MTS) structures. Such structures may be nonuniform and synthesized by numerous graphene resonators. To obtain a desired functionality, such MTSs may be designed, by performing multiple parametric analyses of the structure’s geometric parameters. An accurate calculation of their response may be attained by a full-wave vectorial finite-element method (VFEM), by modeling graphene as a surface boundary condition; however, it requires high computational resources, especially for larger configurations. CMT offers an appealing time- and memory-efficient alternative, by solving a sparse system of equations, of particularly small size, fed by the results of simpler and smaller simulations, with the ability to take into account graphene dispersion. CMT may be used to design more complex MTS structures, by providing approximate and preliminary results, which shall be used as a starting point for the final FEM fine-tuning.

Although a variety of techniques have been developed to reduce the appearance of B-mode speckle, quantitative ultrasound (QUS) aims at extracting the hidden properties of the tissue. Herein, we propose two novel techniques to accurately and precisely estimate two important QUS parameters, namely, the average attenuation coefficient and the backscatter coefficient. Both the techniques optimize a cost function that incorporates data and continuity constraint terms, which we call AnaLytical Global rEgularized BackscatteR quAntitative ultrasound (ALGEBRA). We propose two versions of ALGEBRA, namely, 1-D- and 2-D-ALGEBRA. In 1-D-ALGEBRA, the regularized cost function is formulated in the axial direction, and the QUS parameters are calculated for one line of radio frequency (RF) echo data. In 2-D-ALGEBRA, the regularized cost function is formulated for the entire image, and the QUS parameters throughout the image are estimated simultaneously. This simultaneous optimization allows 2-D-ALGEBRA to "see" all the data before estimating the QUS parameters. In both the methods, we efficiently optimize the cost functions by casting it as a sparse linear system of equations. As a result of this efficient optimization, 1-D-ALGEBRA and 2-D-ALGEBRA are, respectively, 600 and 300 times faster than optimization using the dynamic programming (DP) method previously proposed by our group. In addition, the proposed technique has fewer input parameters that require manual tuning. Our results demonstrate that the proposed ALGEBRA methods substantially outperform least-square and DP methods in estimating the QUS parameters in phantom experiments.

Solving the large-scale linear system of equations is one of the most fundamental problems both in theory and practice. However this problem requires too much computational resource for most users to solve it. With the rapid development of cloud services, many users tend to outsource the expensive computing to the cloud server, which is regarded as an efficient way of solving such problem. Nevertheless, the cloud server can not protect the data privacy well, especially when the user’s linear system of equations contain private and sensitive data. There are many previous research works on secure outsourcing of systems of linear equations. In this paper we first analyze a privacy preserving CGM (conjugate gradient method) algorithm for secure outsourcing of large-scale systems of linear equations proposed in [1] . We find that the cloud server can recover the protected coefficient matrix of the linear system of equations from the message it receives, which makes the security method in this scheme fails. This is a serious problem, which makes the private and sensitive data of the user leak to the cloud server, and privacy preserving does not exist. To overcome this problem, we modified this algorithm to protect the message from leaking, which can protect the users’ privacy well. We also show the security of this new scheme and do experiments to show its efficiency.

As more and more oil fields enter the high water-cut stage, it is of great significance to develop low-cost multiphase flow meters (MPFMs) for high water-cut flows. Based on characteristic parameters of the interfacial wave, this article presents a cost-effective method for flowrate measurement of gas-water slug flow in the horizontal pipeline, which has a great potential to be developed as a key technique for gas-oil-water flow with high water cut. A series of experiments have been carried out on our laboratory flow loop in which the pipeline inner diameter is 50 mm. The superficial velocities of gas and water range from 0.97 to 5.73 m/s and 0.36 to 0.95 m/s respectively. The conductance sensors are used to extract the characteristic parameters of the interfacial wave including the average liquid film thickness and the wave velocity. The system of equations describing the relationships between the flowrates of different phases and the two characteristic parameters is established. By solving the system of equations, the flowrates of gas and water can be estimated. The experimental results show that the proposed method can give predictions for gas flowrate and water flowrate with the mean absolute percentage error (MAPE) in 2.95% and 8.95% respectively, which is promising for developing a cost-effective MPFM for high water-cut flows.

The inverse equivalent source problem related to near-field antenna measurements is typically ill-posed, i.e., the forward operator suffers from non-trivial null spaces. This issue is commonly tackled by pursuing a least-squares solution of the reconstructed near fields. We propose to find a solution of the normal error system of equations which minimizes the l2-norm of the source-coefficients reconstruction deviation. In the scope of near-field to far-field transformations (NFFFTs), advantages are found in a slightly better iterative solver convergence, a reduced number of unknowns, and—most importantly—a more convenient control of the stopping criterion of the iterative solution process. Since the residual of the normal-error solution equals the reconstruction deviation, the proposed formulation includes a meaningful stopping criterion based on the measurement error. All these claims are corroborated by NFFFTs of synthetic and real-world measurement data.

This brief proposes a robust distributed estimation algorithm in presence of impulsive noise. Impulsive noises are present both in the measurements and in the communication links in a sensor network. The proposed method is essentially a diffusion LMS algorithm with optimized variable coefficients in the adaptation and combination steps. The optimized coefficients are obtained based on the minimum disturbance principle. Moreover, it is shown that the optimized coefficients of the adaptation step are found by solving a linear system of equations, while the optimized coefficients of the combination step are calculated by an eigenvector of a particular matrix. Moreover, the minimum disturbances calculated theoretically and their upper bounds are derived mathematically. Simulation results show the better performance of the proposed minimum disturbance diffusion LMS algorithm over some state-of-the-art algorithms.

In this letter, we present a numerical model of magnetization dynamics that is based on the classical Stoner–Wohlfarth model and the Landau–Lifshitz–Gilbert equation. This approach uses a series of coefficients for the interpolation of the time needed by the magnetic moment to reach equilibrium and a linear system of equations for the variation in time of the angular components of the magnetization. The dynamic Stoner–Wohlfarth model will be used to accurately replicate the dynamic behavior of macrospins with significantly improved numerical efficiency when a constant or variable external magnetic field is applied.

Circular microphone arrays (CMAs) and concentric CMAs (CCMAs) have been used in a wide range of applications such as smartspeakers and teleconferencing systems because of their flexible steering ability. Although many efforts have been devoted to beamforming with CCMAs, most existing methods consider only the 2-dimensional (2D) case and assume that the sound sources of interest are in the same plane as the sensor array (generally the horizontal plane), which often does not hold true in practical applications. This paper deals with the problem of beamforming with uniform CCMAs (UCCMAs) in the 3-dimensional (3D) space to control the steering of the spatial response and meanwhile form frequency-invariant beampatterns for processing broadband acoustic and speech signals. The major contributions of this work are summarized as follows: 1) it presents an analysis based on the spherical harmonics decomposition about the $N$ th-order optimal and steerable directivity patterns; 2) a beamforming method is developed in which the beamformer's coefficients are identified by solving a linear system of equations formed by approximating the $N$ th-order optimal target beampattern with the beamformer's beampattern while the resulting beampattern can be steered flexibly in the 3D space; 3) the sufficient and necessary condition on the array geometry and sensors’ placement are given to ensure that the beamformer exists and is unique; and 4) the analytical forms of the directivity factor (DF) and white noise gain (WNG) of the resulting beamformer is given and discussion is presented on what conditions irregularities (deep nulls) in WNG and DF may occur. Simulations are provided to illustrate the property of the developed beamforming methods.

In this paper, a new regularization term in the form of L1-norm based fractional gradient vector flow (LF-GGVF) is presented for the task of image denoising. A fractional order variational method is formulated, which is then utilized for estimating the proposed LF-GGVF. Overlapping group sparsity along with LF-GGVF is used as priors in image denoising optimization framework. The Riemann-Liouville derivative is used for approximating the fractional order derivatives present in the optimization framework. Its role in the framework helps in boosting the denoising performance. The numerical optimization is performed in an alternating manner using the well-known alternating direction method of multipliers (ADMM) and split Bregman techniques. The resulting system of linear equations is then solved using an efficient numerical scheme. A variety of simulated data that includes test images contaminated by additive white Gaussian noise are used for experimental validation. The results of numerical solutions obtained from experimental work demonstrate that the performance of the proposed approach in terms of noise suppression and edge preservation is better when compared with that of several other methods.

In this letter, the order recursive least squares (ORLS) is applied to the azimuth-only ULA to probe the DoA estimation under the unknown number of external sources. In the ORLS-based method, the iterative measured matrix equations are feasibly constructed by combining the two spatial modified Yule-Walker (MYW) systems of linear equations and are required to remain Hankel-block-matrix structure of augmented matrices after arrangement order of the unknowns and the maximum number of detectable sources are preset. Under the stationary assumption on source signals and noises, the remarkable advantage of the proposed method lies in that it can theoretically provide the zero LS error as iteration times is equal to the number of sources. The zero LS error is an evident mark to judge the number of sources, especially for the scenario of the unknown noise variances. The proposed method is free of uncorrelated and coherent signals and the corresponding various mixtures and remains computationally efficient in estimation owing to no eigenvalue decomposition (EVD) and matrix inversion operation. The effectiveness of the determination of the number and azimuth angles of sources versus the different SNRs and numbers of snapshots is numerically confirmed.

An efficient modal solution for scattering by an arbitrary isosceles triangular groove (ITG) in an infinite ground plane is suggested. The procedure consists of expanding the fields in two distinct regions and three different cylindrical coordinate systems, employing Graf’s addition theorem for convenience in imposing boundary conditions on an arc-shaped boundary, and determining the field expansion coefficients by solving a system of linear equations. The method is in good agreement with the time-consuming methods such as the finite-element method (FEM) and the method of moment (MoM) while it has an appropriate computational efficiency especially when the width of a groove increases. This method is utilized to investigate the effects of the problem parameters such as the size of ITG, the wave polarization, and the incidence angle on the scattering signature.

This article presents a novel approach for black-box modeling of 270 V-to-28 V dc–dc step-down converters used in more electric aircrafts. These converters normally feed constant power loads. The proposed deep learning approach, uses offline experimental data of the converter to find an accurate model that reproduces its behavior. It covers a broad range of loading conditions to build a model that replicates the whole behavior of the converter. This article compares the performance of the proposed method, which requires a very low computational burden once the model is trained, with that of a conventional recurrent neural network topology. Results presented in this article show the ability of the obtained solution to accurately emulate the behavior of the real step-down converter when the internal structure is unknown, with no knowledge of the internal parameters, thus preventing disclosure of manufacturer's confidential data. The modeling strategy presented in this article is validated with experimental data by using a step-down converter used in aircrafts. The approach is compared to existing modeling techniques to test its accuracy. This approach can also be applied to many power devices, including diverse types of power converters, power supplies, or filters among others.

We present a recursive version of the previously proposed fast algorithm for computation of the field scattered from arbitrary-shaped multilayer objects. As in the original version, the field at each layer is represented by a series of cylindrical functions with unknown coefficients. In order to determine these coefficients, a linear system of equations is obtained through a procedure based on the boundary conditions between the layers. Instead of solving this large system by applying conventional linear solvers as done in the original version, through an approach based on Thomas algorithm, we derive recursive expressions to compute the scattered field. While the accuracy of the results for both versions are almost the same, the recursive version has lower time and memory complexity. Hence, it supersedes the original version.

This article presents a new approach to construct parameterized reduced-order models for nonlinear circuits. The reduced model is obtained such that it matches the variations in the DC operating point of the original full circuit in response to variations in several of its key design parameters. The new approach leverages, through the use of circuit moments with respect to the design parameters, the discrete empirical interpolation approach developed in a model reduction in other domains and enables its efficient application to the problem of DC operating point in nonlinear circuits. Utilizing the idea of rooted trees, the proposed approach constructs orthogonal bases that are used in projecting the full equations of the large original nonlinear circuit onto a reduced system of nonlinear equations in a space with a much smaller dimension. The variations in the DC operating point of the full circuit are then obtained by solving the reduced system of equations, yielding significant computational savings. Numerical examples are presented to demonstrate the efficiency and accuracy of the reduced model in predicting the change in the DC operating point of the original circuit.

Accurately estimating the parameters of a gray-box linear time-invariant state-space representation is still a challenging problem especially if the number of unknowns exceeds ten. The standard nonlinear optimization-based procedures often fail because the initial guesses are not in the domain of attraction of the user-defined cost function global minimum. Herein, a new solution is developed to give access to estimated parameters which are accurate estimates of the unknown parameters directly or, at worst, good initial guesses for any standard nonlinear optimization-based method. By assuming that 1) the gray-box model to identify is minimal and structurally identifiable, 2) the parameter dependency is affine, and 3) the available datasets have been transformed into a reliable and accurate black-box state-space model, our procedure consists in determining the unique similarity transformation between the black-box and gray-box representations of the system to identify. More specifically, we introduce a technique which first extracts a system of equations linear in terms of the similarity transformation; second, it solves this system by using standard linear algebra tools; finally, completes this two-step procedure by a numerical optimization solution (dedicated now to a very small number of unknowns) if the second step does not give access to all the similarity transformation parameters effectively. In this contribution, we first describe our new procedure, prove its capabilities of giving accurate estimates under specific practical constraints, and finally demonstrates its effectiveness with simulation examples.

In this paper, we study a class of binary nonlinear feedback shift register sequences generated by cascaded feedback registers, one is an LFSR and the other one generates a de Bruijn sequence. The cycle structure (in particular, the initial state of each cycle) is determined by solving a system of linear equations. As an application, we can generate de Bruijn sequences of large period algorithmically.

In this article, starting from full-wave partial differential equation (PDE)-based Maxwell’s equations where $\mathbf {E}$ and $\mathbf {H}$ are coupled, we derive a closed-form model of the inverse of Maxwell’s system of equations in the physical layout of an integrated circuit, package, and board. In this model, we decompose the inverse rigorously into $R$ , $C$ , $L$ , and full-wave components, with neither numerical computation nor approximation, and for an arbitrary physical layout. As a result, each component can be found independently and then superposed to obtain the total response of a layout to any circuit stimuli. The time marching and point-by-point frequency sweeping are also avoided for the $RC$ component as this component’s time and frequency dependencies are analytically revealed in the proposed model. Moreover, the full-wave component is efficiently represented by a small number of high-frequency modes. Using the proposed model, not only many accuracy issues related to existing layout modeling can be addressed but also we drastically speed up layout modeling and simulation and provide circuit designers with an effective model for layout automation. In addition, we develop fast and large-scale algorithms to find each component of the inverse rapidly, where many steps are made analytical, thus further saving CPU run time. The proposed work has been applied to large-scale layout extraction and analysis. Superior performance has been demonstrated in accuracy, efficiency, and capacity.

In this article, a rigorous temporal coupled-mode theory (CMT) formalism is developed, to analyze metasurface (MTS)-based structures. Such MTSs are generally rather complex, requiring extensive, full-wave simulations. To facilitate their design and minimize the associated computational demand, the CMT solves a small linear system of equations, fed by the results of certain much simpler and less time- and memory-consuming eigenvalue problems. The proposed method is versatile and offers an initial, valuable estimation of the structure's frequency response, which may be used as a guideline for the final MTS fine tuning. As proof of concept, split-ring resonator (SRR) MTSs coupled to microstrip lines are analyzed, and their response is compared with the full-wave finite-element method (FEM) results.

This work focuses on the robust optimization of a permanent magnet (PM) synchronous machine while considering a driving cycle. The robustification is obtained by considering uncertainties of different origins. Firstly, there are geometrical uncertainties caused by manufacturing inaccuracies. Secondly, there are uncertainties linked to different driving styles. The final set of uncertainties is linked to ambient parameters such as traffic and weather conditions. The optimization goal is to minimize the PM's volume while maintaining a desired machine performance measured by the energy efficiency over the driving cycle and the machine's maximal torque. The magnetic behavior of the machine is described by a partial differential equation (PDE) and is simulated by the finite-element method employing an affine decomposition to avoid reassembling of the system of equations due to the changing PM geometry. The Sequential Quadratic Programming algorithm is used for the optimization. Stochastic collocation is applied to compute moments of stochastic quantities. The robustness of the optimized configurations is validated by a Monte Carlo sampling. It is found that the uncertainties in driving style and road conditions have significant influence on the optimal PM configuration., 5 pages, 4 figures, 2 tables, Compumag 2019 conference

This article presents a new approach and preliminary modeling results of the temperature field nonuniformity reduction by using a multitorch radio frequency (RF) plasma system. This is also a way to achieve technical and economic feasibility for plasma systems with plume power over 200 kW, significantly reducing the development and production cycle. The mixing of plasma jets was numerically simulated by solving a system of equations, describing high-temperature turbulent plasma flows. The data obtained could be used for engineering of high-power plasma systems, including the ones for testing high-temperature thermal protection materials, waste to power, syntheses of new materials, and so on.

A wavelet network (WN) is a feed-forward neural network that uses wavelets as activation functions for the neurons in its hidden layer. By predetermining the wavelet positions and dilations, the WN can turn into a linear regression model. The common approach for the construction of these WN families is to use least-squares type algorithms. In this letter, we propose a novel approach by formulating a WN as a sparse linear regression problem, which we call a sparse wavelet network (SWN). In this WN, the problem of calculating the unknown inner parameters of the network becomes that of finding the sparse solution of an under-determined system of linear equations. Our sparse solution algorithm is a non-convex sparse relaxation approach inspired by smoothed L0 (SL0), a distinguished sparse recovery algorithm. The proposed SWN can be applied as a tool for the prediction and identification of dynamical systems.

In this paper, firstly, a high accuracy one-step-ahead numerical differentiation formula with O(τ 4 ) pattern is proposed for discretization. Meanwhile, a high precision first-order derivative formula of the backward difference rule with O(τ 4 ) pattern error is given to approximate the derivative information. Then, two high accuracy discrete-time zeroing-type models (HADTZTM) with O(τ 4 ) pattern, i.e., HADTZTM with derivative information known (HADTZTM-K) and HADTZTM with derivative information unknown (HADTZTM-U), are developed, analyzed and investigated for online solving the dynamic system of linear equations (DSLEs). In addition, the 0-stability, consistency, and convergence of the HADTZTM-K and HADTZTM-U are verified for DSLEs. From a theoretical/numerical viewpoint, the classical models are revisited and analyzed for online solving DSLEs. Ultimately, simulation experiment including an application to the path-tracking of the four-link planar manipulator is conducted to demonstrate the efficiency and superiority of the HADTZTM-K and HADTZTM-U, where the HADTZTM-U overcomes the difficulty of derivative information unknown in practical applications.

Matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples. The problem of how to pre-select a subset of entries for sampling to maximize the reconstructed matrix fidelity is largely unaddressed. In this paper, we propose two sampling algorithms to tackle this problem: (i) a fast base sampling algorithm on general single graphs, and (ii) an extended sampling algorithm from the base algorithm for active matrix completion. Assuming the double graph smoothness prior where rows / columns of the matrix signal are smooth with respect to row / column factor graphs, we first show that a matrix signal can be recovered from partial observations by solving a system of linear equations ${\mathbf Q}{\mathbf x}={\mathbf y}$ , where the sought matrix is vectorized as ${\mathbf x}$ and interpreted as a signal residing on a single graph. On this single graph, to promote small reconstruction error and stability of the linear system, we maximize the smallest eigenvalue $\lambda _{\min }$ of ${\mathbf Q}$ by greedily selecting samples corresponding to the largest magnitude entries in the first eigenvectors of ${\mathbf Q}$ , based on a corollary of the Gershgorin circle theorem. Extending to the active matrix completion case, we alternately choose entries in the rows and columns corresponding to the largest magnitude entries in the first eigenvectors of the row / column coefficient matrices ${\mathbf Q}_r$ and ${\mathbf Q}_c$ . Our algorithm benefits computationally from warm start as the first eigenvectors are computed recurrently for more samples. Extensive experiments on both synthetic and real-world datasets show that our extended graph sampling algorithm outperforms existing sampling schemes for matrix completion, when combined with a range of modern matrix completion algorithms.

The prevalence of cloud computing greatly promotes the traditional computing paradigm. With the assistance of a cloud, the light-weight device can achieve computation-intensive tasks which may not be done on its own. While, computing the Hermite Normal Form (HNF), which is a standard form of integer matrices, not only is inescapable when solving the linear system of equations over integers, but also has lots of applications in many other fields, such as integer programming and lattice-based cryptography. However, the fast blowup phenomenon of intermediate numbers makes the HNF computation time-consuming. In this paper, we initialize the study of the cloud-assisted HNF computation and design an efficient outsourcing algorithm that enables the resource-constrained client to securely delegate this heavy computation to a resource-abundant yet maybe untrusted cloud server. The main idea involved in our algorithm is a novel matrix encryption method based on random permutation, unimodular matrix transformation and triangular matrix transformation, which makes our algorithm protect the client's input/output information with the one-way notion and enable the client to detect the cloud's deception with the optimal probability 1. Besides, rigorous theoretical analysis and extensive experimental evaluation validate the efficiency and the practical performance of our design, and the substantial client-side savings are remarkable as the problem size increases.

In this letter, we present qpSWIFT, a real-time quadratic program (QP) solver. Motivated by the need for a robust embedded QP solver in robotic applications, qpSWIFT employs standard primal-dual interior-point method, along with Mehrotra predictor–corrector steps and Nesterov–Todd scaling. The sparse structure of the resulting Karush–Kuhn–Tucker linear system in the QP formulation is exploited, and sparse direct methods are utilized to solve the linear system of equations. To further accelerate the factorization process, we only modify the corresponding rows of the matrix factors that change during iterations and cache the nonzero Cholesky pattern. qpSWIFT is library free, written in ANSI-C and its performance is benchmarked through standard problems that could be cast as QP. Numerical results show that qpSWIFT outperforms state-of-the-art solvers for small scale problems. To evaluate the performance of the solver, a real-time implementation of the solver in the model predictive control framework through experiments on a quadrupedal robot are presented.

Classical numerical techniques remain very computationally expensive to model the magnetization of twisted multifilamentary superconductors in three dimensions because they require solving a large system of equations. In this paper, a reduced model is proposed to approximate the ac losses of these types of wires. It is established through the projection of the physical problem in a helical Frenet basis and the use of periodic boundary conditions on a slice of the wire. The reduced model on the slice leads to systems of more reasonable size leading to smaller and faster simulations than when the whole wire is modeled. Simulation results show that the proposed method leads to computed losses similar to those obtained on a whole wire. Several simulation cases are proposed to prove the validity of the proposed model, and a comparison to some ac loss measurements is proposed to show the effectiveness of the proposed approach. The implementation relies on the H-formulation and is solved with the open-source solver, Getdp.

In this technical note, a distributed algorithm is proposed for multiagent networks to achieve a least squares solution of a system of linear equations, in which each agent only knows part of the overall equations and communicates only with its nearby neighbors. The proposed algorithm is discrete time but does not involve small or time-varying step sizes. Given that the network is fixed, connected, and undirected, the proposed algorithm enables all agents in the network to achieve exponentially fast the same least squares solution; this is validated by simulations.

A fast nested cross approximation (NCA) algorithm is developed in this paper for solving large-scale electromagnetic problems. Different from the existing NCA, the proposed method does not rely on the projection of the basis functions onto the dummy interpolation points to select pivots of each cluster. Instead, a purely algebraic and kernel-independent algorithm is developed to find the row and column pivots of all clusters in $\mathcal {O}(N\log {}N)$ complexity for constant-rank cases with controlled accuracy. This algorithm is then further extended to an $\mathcal {O}(N)$ NCA algorithm, which includes a bottom-up tree traversal for finding the local pivots of each cluster, followed by a top-down procedure to take into account the far field of each cluster. The proposed method has a reduced complexity compared to that reported in the mathematical literature. The resultant nested representation constitutes an $\mathcal {H}^{2}$ -matrix representation of the original dense system of equations, whose solution can be obtained in linear complexity in both iterative and direct solvers. The method is also applicable to variable rank cases, but the complexity therein depends on the rank’s relationship with $N$ . Various numerical experiments have demonstrated the accuracy and computational performance of the proposed algorithms.

This paper presents a closed-form and non-iterative solution for the long-studied SCADA-based State Etimation (SE) problem, where unsynchronized traditional measurements from Remote Terminal Units (RTUs) are used. In this regard, a novel reformulation of the problem is introduced where unknowns are expressed as complex variables in terms of unsynchronized SCADA measurements. To this end, it is assumed that bus voltage amplitudes as well as current amplitudes and active and reactive power flows/injections are available. The resulting system of equations is solved by the classic linear weighted leastsquares method. In contrast to the traditional approaches, several drawbacks such as initialization and issues with convergence (especially for large-scale systems) are resolved. Moreover, the proposed approach does not use synchrophasors. The method is validated on a 3-bus test network and applied to the IEEE 118-bus test system, 1341-bus and 9241-bus European high-voltage transmission networks.

In this paper, we present a highly efficient analysis of double-layered stacked gratings that are rotated from each other by any arbitrary angle using the asymptotic corrugations boundary conditions (ACBC) along with classical vector potentials. By casting the resultant system of linear equations into a homogeneous matrix equation, the eigenvalue problem can then be solved, after which the surface-wave dispersion diagram may be generated. In addition to the solutions for modal resonances under sourceless conditions, reflection and transmission analyses of plane-wave incidences constituting source-driven problems are also presented for both normal and oblique incidences. The properties of copolarization and cross-polarization levels in those scenarios are also discussed. The results computed by the code written based on this proposed modal method are compared and validated with those simulated using a commercial full-wave solver. A prototype of the structure was also manufactured and measured.

Solving linear Diophantine systems of equations is applied in discrete-event systems, model checking, formal languages and automata, logic programming, cryptography, networking, signal processing, and chemistry. For modeling discrete systems with Petri nets, a solution in non-negative integer numbers is required, which represents an intractable problem. For this reason, solving such kinds of tasks with significant speedup is highly appreciated. In this paper we design a new solver of linear Diophantine systems based on the parallel-sequential composition of the system clans. The solver is studied and implemented to run on parallel architectures using a two-level parallelization concept based on MPI and OpenMP. A decomposable system is usually represented by a sparse matrix; a minimal clan size of the decomposition restricts the granulation of the technique. MPI is applied for solving systems for clans using a parallel-sequential composition on distributed-memory computing nodes, while OpenMP is applied in solving a single indecomposable system on a single node using multiple cores. A dynamic task-dispatching subsystem is developed for distributing systems on nodes in the process of compositional solution. Computational speedups are obtained on a series of test examples, e.g., illustrating that the best value constitutes up to 45 times speedup obtained on 5 nodes with 20 cores each.

We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous $C^{p-1}$Cp-1 basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous $C^{p-1}$Cp-1 IGA spaces by introducing low-continuity $C^0$C0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, we propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA versus IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of $p^2$p2 when considering rIGA discretizations rather than highly-continuous IGA spaces.

In this paper, the surface-wave modal analysis of the transverse strip-grated dielectric-coated metal rod is done by using the asymptotic strip boundary conditions (ASBC) along with classical vector potentials in the cylindrical coordinate system. The yielded solvable system of linear equations is then cast into a homogeneous matrix equation, from which the characteristic equation as well as field distributions in explicit functional forms may be obtained. Modal dispersion diagrams and field distributions generated by this method are compared and validated with those simulated by an independent commercial full wave solver. The time cost by the ASBC method is also compared with that of the simulation software, upon which the vast superiority in speed of the proposed technique over the benchmark tool is demonstrated, but yet preserving high accuracy. Parametric studies that portray the variations of dispersion properties with various geometric attributes are carried out. Comparisons with its counterpart transverse corrugated rod are also carried out, and studies of dielectric losses are presented as well. A prototype was fabricated and measured, the experimental results of which concur well with theoretical expectations.

Due to peripheral nerve stimulation, the magnetic particle imaging (MPI) method is limited in the maximum applicable excitation-field amplitude. This in turn leads to a limitation of the size of the covered field of view (FoV) to few millimeters. In order to still capture a larger FoV, MPI is capable to rapidly acquire volumes in a multi-patch fashion. To this end, the small excitation volume is shifted through space using the magnetic focus fields. Recently, it has been shown that the individual patches are preferably reconstructed in a joint fashion by solving a single linear system of equations taking the coupling between individual patches into account. While this improves the image quality, it is computationally and memory demanding since the size of the linear system increases in the best case quadratically with the number of patches. In this paper, we will develop a reconstruction algorithm for MPI multi-patch data exploiting the sparsity of the joint system matrix. A highly efficient implicit matrix format allows for rapid on-the-fly calculations of linear algebra operations involving the system matrix. Using this approach, the computational effort can be reduced to a linear dependence on the number of used patches. The algorithm is validated on 3-D multi-patch phantom data sets and shown to reconstruct large data sets with 15 patches in less than 22 s.

Error floor is one major reason for limited use of low-density parity-check (LDPC) codes in applications requiring very low bit error rate. In this brief, an efficient erasure searching post-processor (ESPP) is proposed to lower the error floor of LDPC codes. Here, when a decoding failure is detected, the most unreliable symbols in the decoder output vector are erased, then their values are re-calculated by solving a system of linear equations with a sparse coefficient matrix. Compared to the state-of-the-art post processors, the ESPP has much lower computation complexity. Simulation results show that the proposed method significantly improves the decoding performance of LDPC codes in the error-floor region. Additionally, a well-optimized hardware architecture is developed for the proposed post-processor. Algorithmic transformation and architecture optimization are well explored to reduce the hardware complexity and latency. Synthesis results demonstrate the effectiveness of the proposed architecture.

In near-field scanning, the model parameters for the antenna under test can be determined from measured probe outputs by solving a linear system of equations in the least-squares sense. The model parameters are far-field pattern values, cylindrical or spherical-wave expansion coefficients, or equivalent surface-source values. The normal matrix of this linear system of equations is sometimes extremely ill-conditioned. This occurs when certain sets of model parameters lie outside the spatial bandwidth of the operator that computes the probe output. One remedy is to restrict the sets of model parameters allowed and perform upsampling if needed to achieve the desired accuracy. These ideas are illustrated through analysis and examples that involve both 2-D and 3-D scanning geometries.

Selective harmonic elimination - pulse width modulation (SHE-PWM) can be utilized to improve the efficiency of multilevel voltage source converters due to its ability to provide low switching frequency and tight control of low-order harmonics. In addition, SHE-PWM with half-wave (HW) symmetry provides a higher number of solutions than quarter-wave (QW) symmetry and therefore, the waveform design can be improved. This work proposes a universal formulation, which can be utilized with HW symmetry, that provides a unique system of equations valid for any possible multilevel waveform. Thereby, without using predefined waveforms, this formulation provides the ability to search simultaneously both the firing angles and the switching patterns, simplifying significantly the search process and providing a high number of solutions. With the aim of selecting the optimum sets of firing angles, the solutions provided by HW and QW symmetries are compared, based on several metrics of harmonic performance, for particular test cases. Experimental results also validate the universal formulation with HW symmetry. Eusko Jaurlaritza; Secretaria de Estado de Investigacion Desarrollo e Innovacion

It has been found that there exists an essential similarity between solving equations and controlling dynamic systems: Both errors are expected to decrease to zero (or an acceptably tiny value) as soon as possible. By exploiting such a similarity, researchers have presented and investigated continuous-time recurrent neural network models for solving time-varying problems. To be compatible with digital computers, it is desirable to develop discrete-time neural algorithms from the control perspective for performance improvement. In this paper, a discrete-time zeroing neural algorithm is proposed for the solving system of linear equations with the aid of control techniques. To lay a basis for theoretical analyses, the proposed zeroing neural algorithm with nonlinearity is converted into a second-order linear system plus a residual term, and then, analyzed using the control theory. Theoretical results and numerical experiments are provided, which illustrate that the proposed neural algorithm possesses an improved performance compared to the existing solutions.

A new AE location method using tri-variate kernel density estimator is developed in this paper. Firstly, combinations of every six arrivals are obtained from a multi-sensor location system, and the preliminary location results are obtained by solving the systems of linear equations constructed by these arrival combinations. Secondly, the tri-variate scaled kernel functions at each AE source coordinate are established. The tri-variate kernel density estimator is constructed by adding and normalizing these scaled kernel functions. Finally, the extreme value of the density function is calculated and the coordinate corresponding to the extreme value is extracted as the final location result. Pencil-lead break experiments were carried out. The results verified that the proposed method was more accurate and effective than traditional methods in the location performance. Moreover, the influence of outlier scales and proportions of the proposed method were investigated by simulation tests. Results showed that the location performance of the proposed method was higher than that of traditional methods under different outlier scales and proportions.