Linear Matrix Equations Research Articles

Iterative optimal solutions of linear matrix equations for hyperspectral and multispectral image fusing

For a linear matrix function f in X in {mathbb {R}}^{mtimes n} we consider inhomogeneous linear matrix equations f(X) = E for E ne 0 that have or do not have solutions. For such systems we compute optimal norm constrained solutions iteratively using the Conjugate Gradient and Lanczos’ methods in combination with the More–Sorensen optimizer. We build codes for ten linear matrix equations, of Sylvester, Lyapunov, Stein and structured types and their T-versions, that differ only in two five times repeated equation specific code lines. Numerical experiments with linear matrix equations are performed that illustrate universality and efficiency of our method for dense and small data matrices, as well as for sparse and certain structured input matrices. Specifically we show how to adapt our universal method for sparse inputs and for structured data such as encountered when fusing image data sets via a Sylvester equation algorithm to obtain an image of higher resolution.

Physics-Informed Supervised Residual Learning for Electromagnetic Modeling

In this study, physics-informed supervised residual learning (PhiSRL) is proposed to enable an effective, robust, and general deep learning framework for 2-D electromagnetic (EM) modeling. Based on the mathematical connection between the fixed-point iteration method and the residual neural network (ResNet), PhiSRL aims to solve a system of linear matrix equations. It applies convolutional neural networks (CNNs) to learn updates of the solution with respect to the residuals. Inspired by the stationary and nonstationary iterative scheme of the fixed-point iteration method, stationary and nonstationary iterative physics-informed ResNets (SiPhiResNet and NiPhiResNet) are designed to solve the volume integral equation (VIE) of EM scattering. The effectiveness and universality of PhiSRL are validated by solving VIE of lossless and lossy scatterers with the mean squared errors (MSEs) converging to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10^{-4}$ </tex-math></inline-formula> (SiPhiResNet) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10^{-7}$ </tex-math></inline-formula> (NiPhiResNet). Numerical results further verify the generalization ability of PhiSRL.

Analysis of the Truncated Conjugate Gradient Method for Linear Matrix Equations

The matrix-oriented version of the conjugate gradient (CG) method can be used to approximate the solution to certain linear matrix equations. To limit memory consumption, low-rank reduction of the factored iterates is often employed, possibly leading to disruption of the regular convergence behavior. We analyze the properties of the method in the matrix regime and identify the quantities that are responsible for early termination, usually stagnation, when truncation is in effect. Moreover, we illustrate relations between CG and a projection technique directly applied to the same matrix equation.

Towards non-linearly activated ZNN model for constrained manipulator trajectory tracking

As a powerful method for time-varying problems solving, the zeroing neural network (ZNN) is widely applied in many practical applications that can be modeled as time-varying linear matrix equations (TVLME). Generally, existing ZNN models solve these TVLME problems in the ideal no noise situation without inequality constraints, but the TVLME with noises and inequality constraints are rarely considered. Therefore, a non-linear activation function is designed, and based on the non-linear activation function, a non-linearly activated ZNN (NAZNN) model is proposed for solving constrained TVLME (CTVLME) problems. The convergence and robustness of the proposed NAZNN model are verified theoretically, and simulation results further demonstrate the effectiveness and superiority of the NAZNN model in dealing with CTVLME and the constrained robot manipulator trajectory tracking problems. In addition, the wheeled robot trajectory tracking fault problems with physical constraints are also analyzed theoretically, and the proposed NAZNN model is also applied to the manipulator trajectory tracking fault problem, and the experimental results prove that the NAZNN model also deal with the manipulator trajectory tracking fault problem effectively.

An anti-interference dynamic integral neural network for solving the time-varying linear matrix equation with periodic noises

In order to solve a time-varying linear matrix equation with periodic noises, an anti-interference dynamic integral neural network (AI-DINN) is proposed. Based on an indefinite unbounded vector/matrix-type error function, the proposed AI-DINN includes an integral structure, a recursive structure, and an adjustment module. It has the excellent ability to overcome the interference of periodic noises. This paper theoretically proves the convergence and robustness of the proposed AI-DINN for solving the time-varying linear matrix equation with the interference of periodic noises. Computer simulation results verify that the proposed AI-DINN method based on different activation functions can achieve convergence within limited time with the interferences of different periodic noises. In addition, the proposed AI-DINN with different activation functions has its own advantages with the interference of different types of periodic noises. Furthermore, comparative simulation experiments verify that the proposed AI-DINN has better convergence and anti-interference performance compared with state-of-the-art methods.

A fixed-time distributed extended state observer for uncertain second-order nonlinear system

This paper investigates the fixed-time distributed estimation problem for a class of second-order nonlinear systems with uncertain input, unknown nonlinearity and matched perturbation. A fixed-time distributed extended state observer (FxTDESO) consisting of a group of local observer nodes under directed communication topology is proposed, and each node can reconstruct both the full state and unknown dynamics of the system. To achieve fixed-time stability, a Lyapunov function is elaborated, and based on this, sufficient conditions for the existence of the FxTDESO are established. Under time-invariant and time-varying disturbance, the observation errors can converge to the origin and a small region of the origin within a fixed time, respectively, where the upper bound of the settling time (UBST) is irrelevant to the initial conditions. Compared to the existing fixed-time distributed observers, the proposed observer can reconstruct both the unknown states and uncertain dynamics, and only the output of the leader and 1-dimensional output estimates from the neighboring nodes are needed in the observer design which effectively reduces the communication load. The paper also extends previous finite-time distributed extended state observer to the case of time-variant disturbance and eliminates the complex linear matrix equation assumption that guarantees the finite-time stability. Furthermore, the FxTDESO design for a class of high-order nonlinear systems is also discussed. Finally, simulation examples are conducted to demonstrate the effectiveness of the proposed observer.

Algebraic Characterizations of Relationships between Different Linear Matrix Functions

Let f(X1,X2,…,Xk) be a matrix function over the field of complex numbers, where X1,X2,…,Xk are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations.

Prescribed time convergence and robust zeroing neural network for solving time-varying linear matrix equation

Zeroing neural network offers a new solution method to solve the time-varying linear matrix equation. As an important component, the activation function directly affects the performance of zeroing neural network in solving time-varying linear matrix equation. Focusing on the unification of prescribed time convergence and strong robustness without changing the basic structure of zeroing neural network, a novel activation function is proposed for the first time in this paper. Compared with commonly used activation functions in previous work, the novel activation function has superior performance for zeroing neural network to solve the time-varying linear matrix equation. The first item is global asymptotic convergence, capable of converging from random initial states to the theoretical solution. The second item is the prescribed time convergence, i.e. the upper bound of convergence time is only related to the parameters of the novel activation function and zeroing neural network, which facilitates the prediction of the convergence process. The third item is strong robustness, which ensures that the solution converges in various noisy environments. Theoretical analysis and comparative simulation experiments verify that zeroing neural network with the novel activation function has these performances for both low- or high-dimensional time-varying linear matrix equation.

A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking

&lt;abstract&gt;&lt;p&gt;Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly.&lt;/p&gt;&lt;/abstract&gt;

Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images

&lt;abstract&gt;&lt;p&gt;The importance of quaternions in a variety of fields, such as physics, engineering and computer science, renders the effective solution of the time-varying quaternion matrix linear equation (TV-QLME) an equally important and interesting task. Zeroing neural networks (ZNN) have seen great success in solving TV problems in the real and complex domains, while quaternions and matrices of quaternions may be readily represented as either a complex or a real matrix, of magnified size. On that account, three new ZNN models are developed and the TV-QLME is solved directly in the quaternion domain as well as indirectly in the complex and real domains for matrices of arbitrary dimension. The models perform admirably in four simulation experiments and two practical applications concerning color restoration of images.&lt;/p&gt;&lt;/abstract&gt;

An effective treatment of adding-up restrictions in the inference of a general linear model

&lt;abstract&gt;&lt;p&gt;This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{\beta} + \pmb{\varepsilon} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{\beta} $ and a matrix equality for the covariance matrix of the error term $ \pmb{\varepsilon} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{\beta} $ using various analytical algebraic operations of the given vectors and matrices in the model.&lt;/p&gt;&lt;/abstract&gt;

Guaranteed root mean square estimates of linear matrix equations solutions under conditions of uncertainty

The article focuses on the linear estimation problems of unknown rectangular matrices, which are solutions of linear matrix equations with the right-hand sides belonging to bounded sets. The random errors of the observation vector have zero mathematical expectation, and the correlation matrix is unknown and belongs to one of two bounded sets. Explicit expressions of the guaranteed root mean square errors of estimates for linear operators acting from the space of rectangular matrices into some vector space are given. Guaranteed quasi-minimax root mean square errors of linear estimates are obtained. As the test examples, two options for solving the problem are considered, taking into account small perturbations of known observation matrices.

Zeroing Neural Networks Combined with Gradient for Solving Time-Varying Linear Matrix Equations in Finite Time with Noise Resistance

Due to the time delay and some unavoidable noise factors, obtaining a real-time solution of dynamic time-varying linear matrix equation (LME) problems is of great importance in the scientific and engineering fields. In this paper, based on the philosophy of zeroing neural networks (ZNN), we propose an integration-enhanced combined accelerating zeroing neural network (IEAZNN) model to solve LME problem accurately and efficiently. Different from most of the existing ZNNs research, there are two error functions combined in the IEAZNN model, among which the gradient of the energy function is the first design for the purpose of decreasing the norm-based error to zero and the second one is adding an integral term to resist additive noise. On the strength of novel combination in two error functions, the IEAZNN model is capable of converging in finite time and resisting noise at the same time. Moreover, theoretical proof and numerical verification results show that the IEAZNN model can achieve high accuracy and fast convergence speed in solving time-varying LME problems compared with the conventional ZNN (CZNN) and integration-enhanced ZNN (IEZNN) models, even in various kinds of noise environments.

Weight splitting iteration methods to solve quadratic nonlinear matrix equation [formula omitted

The quadratic nonlinear matrix equationQ(Y)=MY2+NY+P,occurs in many applications such as the Quasi-Birth-Death processes, pseudo-spectra for quadratic eigenvalue problems and the quadratic eigenvalue problems. In this work, we propose efficient parametric iterative methods for finding the solution of this quadratic matrix equation based on weight splitting (WS) on matrices M and N, separately. We show that the proposed methods converge to the solution of the quadratic nonlinear matrix equation under conditions. Every iteration in proposed methods requires the solution of one or two linear matrix equations. Finally, various numerical examples indicate that the obtained methods in terms of CPU time, accuracy and computational cost are superior to the famous methods.

Effective distributed algorithm for solving linear matrix equations

Effective distributed algorithm for solving linear matrix equations

An adaptive gradient-descent-based neural networks for the on-line solution of linear time variant equations and its applications

It is well-known that, the classical gradient-descent-based neural network (CGNN) model is used widely for the time-invariant problem solving. However, it is an extremely common problem for the time varying cases in the practical engineering applications, while the CGNN effective neural solver for the time-variant problems. For this reason, in this article, an adaptive GNN (AGNN) is presented for the linear time variant matrix equation (LTVME) on-line solving based on Lyapunov theory. Theoretical analysis already verified that the presented AGNN model could achieve the correct state solution effectively. The simulated experiment results further validate that the state solution of the AGNN model could be convergent to the correct solution of the solved time variant problems in theory.

Hybrid TDOA/AOA Hypocenter Localization Using the Constrained Least Squares Method with Deep Learning P-Onset Picking

In this study, we propose a hypocenter localization algorithm that uses the time difference of arrival (TDOA) and angle of arrival (AOA) as a hybrid model. The hypocenter measurements are detected by the accelerator sensors of the four separate observatories that are closest to the origin of an earthquake. The measurements are calibrated by the proposed deep learning P-onset picking system with short-time Fourier transform (STFT) signal analysis because the accurate detection of Primary waves (P-waves) is limited by seismic environmental noise. The revised measurements are used to calculate the precise distances between the observatories and hypocenters. The proposed hybrid TDOA/AOA is represented by a linear matrix equation that includes the unknowns of the precise distances, coordinates, and arrival angles to the observatories. We estimate a hypocenter using the constrained least squares method (CLS) under the constraints of the TDOA/AOA. The objective function with the constraints is optimized using the Lagrange function, and the asymptotic optimum is obtained by specifying the optimal Lagrange multipliers. Simulations show the performance of the proposed hypocenter localization method.

Global GPBiCGstab(L) method for solving linear matrix equations

Global Krylov subspace methods are effective iterative solvers for large linear matrix equations. Several Lanczos-type product methods (LTPMs) for solving standard linear systems of equations have been extended to their global versions. However, the GPBiCGstab(L) method, which unifies two well-known LTPMs (i.e., BiCGstab(L) and GPBiCG methods), has been developed recently, and it has been shown that this novel method has superior convergence when compared to the conventional LTPMs. In the present study, we therefore extend the GPBiCGstab(L) method to its global version. Herein, we present not only a naive extension of the original GPBiCGstab(L) algorithm but also its alternative implementation. This variant enables the preconditioning technique to be applied stably and efficiently. Numerical experiments were performed, and the results demonstrate the effectiveness of the proposed global GPBiCGstab(L) method.

Recurrent Neural Network Models Based on Optimization Methods

Many researchers have addressed problems involving time-varying (TV) general linear matrix equations (GLMEs) because of their importance in science and engineering. This research discusses and solves the topic of solving TV GLME using the zeroing neural network (ZNN) design. Five new ZNN models based on novel error functions arising from gradient-descent and Newton optimization methods are presented and compared to each other and to the standard ZNN design. Pseudoinversion is involved in four proposed ZNN models, while three of them are related to Newton’s optimization method. Heterogeneous numerical examples show that all models successfully solve TV GLMEs, although their effectiveness varies and depends on the input matrix.

Unknown Input Functional Observer Design for Discrete-Time Interval Type-2 Takagi–Sugeno Fuzzy Systems

This article proposes a novel unknown input functional observer design approach toward discrete-time interval type-2 Takagi–Sugeno fuzzy system models subject to measurable and unmeasurable premise variables. By constructing a new state vector that contains both the unknown inputs and the system states, functional observers are proposed for the cases with measurable and unmeasurable premise variables to estimate this new state vector for unknown input and/or state estimation. The observer design problem is converted into the solvability issue of a linear matrix equation involving observer gain matrices, and the existence conditions of the observers are explicitly obtained based on matrix rank analysis. Meanwhile, instead of solving the intricate Sylvester equation directly, the solution of the simplified matrix equation is employed to derive the observer gains. Moreover, the effectiveness and the superiority of the presented method are demonstrated via two illustrative examples.