System Of Linear Matrix Equations Research Articles

Gradient neural network model for the system of two linear matrix equations and applications

In this paper, the new type of Gradient Neural Network (GNN) model is proposed for the following linear system of matrix equations: AX=C,XB=D. The convergence analysis of given models is shown. The model is applied for the computation of the regular matrix inverse, as well as Moore-Penrose and Drazin generalized inverses. Some illustrative examples and simulations are given to verify theoretical results.

Gradient-based iterative approach for solving constrained systems of linear matrix equations

Gradient-based iterative approach for solving constrained systems of linear matrix equations

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.

Investigation of Micro-motion Kinematics of Continuum Robots for Volumetric OCT and OCT-guided Visual Servoing.

Continuum robots (CR) have been recently shown capable of micron-scale motion resolutions. Such motions are achieved through equilibrium modulation using indirect actuation for altering either internal preload forces or changing the cross-sectional stiffness along the length of a continuum robot. Previously reported, but unexplained, turning point behavior is modeled using two approaches. An energy minimization approach is first used to explain the source of this behavior. Subsequently, a kinematic model using internal constraints in multi-backbone CRs is used to replicate this turning point behavior. An approach for modeling the micro-motion differential kinematics is presented using experimental data based on the solution of a system of linear matrix equations. This approach provides a closed-form approximation of the empirical micro-motion kinematics and could be easily used for real-time control. A motivating application of image-based biopsy using 3D optical coherence tomography (OCT) is envisioned and demonstrated in this paper. A system integration for generating OCT volumes by sweeping a custom B-mode OCT probe is presented. Results showing high accuracy in obtaining 3D OCT measurements are shown using a commercial OCT probe. Qualitative results using a miniature probe integrated within the robot are also shown. Finally, closed-loop visual servoing using OCT data is demonstrated for guiding a needle into an agar channel. Results of this paper present what we believe is the first embodiment of a continuum robot capable of micro and macro motion control for 3D OCT imaging. This approach can support the development of new technologies for CRs capable of surgical intervention and micro-motion for ultra-precision tasks.

On the numerical solution of a class of systems of linear matrix equations

Abstract We consider the solution of systems of linear matrix equations in two or three unknown matrices. For dense problems we derive algorithms that determine the numerical solution by only involving matrices of the same size as those in the original problem, thus requiring low computational resources. For large and structured systems we show how the problem properties can be exploited to design effective algorithms with low memory and operation requirements. Numerical experiments illustrate the performance of the new methods.

Semi-analytical solution of three-dimensional steady state thermoelastic contact problem of multilayered material under friction heating

Multilayer coatings offer a possibility to design the surface according to the different requirements, which attracts intense attentions from engineering and research community. The coupled thermo-mechanical contact problem of a multilayered material is of great interest. This paper firstly derives the frequency response functions (FRF) of thermoelastic fields through thermoelastic governing equations. The unknown coefficients in the FRFs are assembled in a linear system of matrix equations according to the thermal and mechanical loadings on surface and continuity condition of heat flux, temperature, displacement and stresses at each interface; then the coefficients are solved and expressed recursively. Based on the closed-form solution of FRFs, a fast semi-analytical method (SAM) is developed to solve the three-dimensional thermoelastic contact problem involved in arbitrary multilayered materials. There are no limits on the number or the thickness of layers, and material parameters can be varied arbitrarily. The present model is verified by literature and FEM and shows a high robustness and efficiency. Thermoelastic contact of multilayered materials with different coating designs under friction heating is further studied and the thermal effect is explored.

Design of measurement difference autocovariance method for estimation of process and measurement noise covariances

The paper deals with the estimation of the process and measurement noise covariance matrices of a system described by the linear time-varying state–space model. In particular, the stress is laid on the correlation methods and a novel method, the measurement difference autocovariance method, is designed. The proposed method is based on the statistical analysis of an augmented measurement prediction error leading to a system of linear matrix equations for the elements of the noise covariance matrices. Compared to other correlation methods, the proposed method provides unbiased estimates even for a finite number of measurements. The theoretical results are discussed and illustrated in a numerical example.

Noise Moment and Parameter Estimation of State-Space Model

The paper deals with the estimation of the moments and parameters of the state and measurement noises of a system described by a linear time-varying state-space model. In particular, a novel correlation measurement difference method is designed. The method is based on a statistical analysis of an augmented measurement prediction error leading to a system of linear matrix equations enabling higher-order noise moments estimation. The estimated moments, shown to converge to the true values, are then used for noise parameters computation. The theoretical results are discussed and illustrated in a numerical example.

Semi-analytic solution of three-dimensional temperature distribution in multilayered materials based on explicit frequency response functions

Multilayer coatings have been widely using in a wide range of applications, including industrial, biological and electrical areas, and the thermal distribution in a multilayered material is of great interest. In present paper, the frequency response functions (FRFs) of temperature field under unit point heat flux are derived through thermal conduction equation. The unknown coefficients in the FRFs are assembled in a linear system of matrix equations according to the heat input and continuity condition of heat flux and temperature at each interface; then the coefficients are solved and expressed recursively. Based on the closed-form solution of FRFs, a fast semi-analytical method (SAM) is developed to solve the three-dimensional steady state heat conduction in arbitrary multilayered materials, and there are no limits on the number or the thickness of layers. The temperature fields under different kinds of heat flux in multilayered coatings are studied. Moving heat flux and convection on the surface are also considered.

Noise Covariance Matrices Estimation for Systems with Time-Varying Availability of Sensors

The paper deals with the estimation of the noise covariance matrices of a time-varying system described by a state-space model with a time-varying set of sensors. In particular, the measurement difference autocovariance method, is proposed. The method is based on a statistical analysis of linearity transformed measurements leading to a system of linear matrix equations. The theoretical results are discussed and illustrated using a numerical study.

Analytical synthesis of functional observers for systems with signal perturbations

An algorithm for the analytical synthesis of functional observers is proposed which is based on a nondegenerate transformation of a state vector using the technology of matrix canonization and methods for solving linear matrix equations of arbitrary dimension. Conditions of the solvability of the synthesis problem and the invariance of the constructed observer to signal perturbations in the form of a system of linear matrix equations are formulated.

Convergence of ADGI methods for solving systems of linear matrix equations

Purpose – The linear matrix equations have wide applications in engineering, physics, economics and statistics. The purpose of this paper is to introduce iterative methods for solving the systems of linear matrix equations. Design/methodology/approach – According to the hierarchical identification principle, the authors construct alternating direction gradient-based iterative (ADGI) methods to solve systems of linear matrix equations. Findings – The authors propose efficient ADGI methods to solve the systems of linear matrix equations. It is proven that the ADGI methods consistently converge to the solution for any initial matrix. Moreover, the constructed methods are extended for finding the reflexive solution to the systems of linear matrix equations. Originality/value – This paper proposes efficient iterative methods without computing any matrix inverses, vec operator and Kronecker product for finding the solution of the systems of linear matrix equations.

Analytical synthesis of invariant reduced-order state observers

An algorithm for the analytical synthesis of reduced-order observers for dynamic systems with an output matrix of arbitrary form is proposed, and invariance conditions for the constructed observer with respect to external disturbances are formulated. Solvability conditions for the synthesis problem are obtained in the form of a system of linear matrix equations. The proposed algorithm is based on a nondegenerate transformation of the state vector using the matrix canonization technique and methods for solving linear matrix equations of arbitrary dimension.

Rank-one solutions for homogeneous linear matrix equations over the positive semidefinite cone

The problem of finding a rank-one solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a low-rank solution does not always exist. In this paper, we aim at developing some sufficient conditions for the existence of a rank-one solution to the system of homogeneous linear matrix equations (HLME) over the positive semidefinite cone. First, we prove that an existence condition of a rank-one solution can be established by a homotopy invariance theorem. The derived condition is closely related to the so-called P∅ property of the function defined by quadratic transformations. Second, we prove that the existence condition for a rank-one solution can be also established through the maximum rank of the (positive semidefinite) linear combination of given matrices. It is shown that an upper bound for the rank of the solution to a system of HLME over the positive semidefinite cone can be obtained efficiently by solving a semidefinite programming (SDP) problem. Moreover, a sufficient condition for the nonexistence of a rank-one solution to the system of HLME is also established in this paper.

Iterative Solution to a System of Matrix Equations

An efficient iterative algorithm is presented to solve a system of linear matrix equations , with real matrices and . By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices and , a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions and to the given matrices and in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations , where , . The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices are large, our algorithm is efficient as well.

Fuzzy Approximate Solution of Positive Fully Fuzzy Linear Matrix Equations

The fuzzy matrix equationsA~⊗X~⊗B~=C~in whichA~,B~, andC~arem×m,n×n, andm×nnonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.

A finite iterative algorithm for solving the generalized [formula omitted]-reflexive solution of the linear systems of matrix equations

In this paper, we proposed an algorithm for solving the linear systems of matrix equations { ∑ i = 1 N A i ( 1 ) X i B i ( 1 ) = C ( 1 ) , ⋮ ∑ i = 1 N A i ( M ) X i B i ( M ) = C ( M ) . over the generalized ( P , Q ) -reflexive matrix X l ∈ R n × m ( A l ( i ) ∈ R p × n , B l ( i ) ∈ R m × q , C ( i ) ∈ R p × q , l = 1 , 2 , … , N , i = 1 , 2 , … , M ). According to the algorithm, the solvability of the problem can be determined automatically. When the problem is consistent over the generalized ( P , Q ) -reflexive matrix X l ( l = 1 , … , N ) , for any generalized ( P , Q ) -reflexive initial iterative matrices X l ( 0 ) ( l = 1 , … , N ) , the generalized ( P , Q ) -reflexive solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized ( P , Q ) -reflexive solution can also be derived when the appropriate initial iterative matrices are chosen. A sufficient and necessary condition for which the linear systems of matrix equations is inconsistent is given. Furthermore, the optimal approximate solution for a group of given matrices Y l ( l = 1 , … , N ) can be derived by finding the least-norm generalized ( P , Q ) -reflexive solution of a new corresponding linear system of matrix equations. Finally, we present a numerical example to verify the theoretical results of this paper.

Finite element solution of transient heat conduction using iterative solvers

PurposeTo provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.Design/methodology/approachThree‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.FindingsProvides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.Practical implicationsFortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.Originality/valueThis paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

Minimal realisation of recursive and nonrecursive three-dimensional systems

An algorithm is presented for minimal state-space realisation of recursive and nonrecursive three-dimensional systems. This algorithm reduces the problem of minimal realisation to that of solving a linear system of matrix equations. Necessary and sufficient conditions for minimal realisation are established and analytical expressions for the state-space system matrices are derived.

On a direct method for solving Helmholtz's type equations in 3-d rectangular regions

A direct solution method for solving elliptic pde's of the type k x ( z) · ∂ 2 ϕ/ ∂x 2 + k y ( z) · ∂ 2 ϕ/ ∂y 2 + k z · ∂ 2 ϕ/ ∂z 2 + σ( z) · ϕ = f( x, y, z) in 3D parallelepipeds with k z = const and k x ( z), k y ( z), σ( z) continuous functions of z, is presented. The spatial derivatives are approximated using the Hermite approach (Mchrstellen-verfahren) with O( h 6) truncation error for Dirichlet boundary conditions or for periodic solutions of the problem. For Neumann conditions, it seems that in order to retain the direct character of the numerical algorithm employed, one should approximate the first spatial derivatives on the boundary by means of conventional schemes having a truncation error of O( h 3) type rather than O( h 6) which accordingly reduce the overall accuracy of the results. Despite the substantial reduction of the overall accuracy for Neumann conditions, this case has not been exluded, because the structure of the difference equations remains invariant for problems in which instead of known values of first-order normal derivatives at the boundaries, these very boundaries constitute symmetry planes of the solution. This feature allows a direct solution method to be used for such a problem, whereas the O( h 6) truncation error of the difference schemes employed is retained. The given pde is discretised on a three-dimensional grid and the set of difference equations is formulated as a linear system of matrix equations whose solution is found by a suitable decomposition of unknows based on knowledge of the eigenvalues and eigenvectors of simple tridiagonal matrices. A hint for extending the applicability of the method—by means of a coordinate transformation—in cylindrical domains with an annular cross section, is also given.