Inversion Of Matrices Research Articles

Direct methods onη‐Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

This paper provides two direct methods for solving the split quaternion matrix equation whereXis an unknown split quaternionη‐Hermitian matrix, andA, B, C, D, E, Fare known split quaternion matrices with suitable size. Our tools are the Kronecker product, Moore–Penrose generalized inverse, real representation, and complex representation of split quaternion matrices. Our main work is to find the necessary and sufficient conditions for the existence of a solution of the matrix equation mentioned above, derive the explicit solution representation, and provide four numerical algorithms and two numerical examples.

Explicit geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids

The geometric reinterpretation of the Finite Element Method (FEM) shows that Raviart–Thomas and Nédélec mass matrices map from degrees of freedoms (DoFs) attached to geometric elements of a tetrahedral grid to DoFs attached to the barycentric dual grid. The algebraic inverses of the mass matrices map DoFs attached to the barycentric dual grid back to DoFs attached to the corresponding primal tetrahedral grid, but they are of limited practical use since they are dense.In this paper we present a new geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids and possibly inhomogeneous and anisotropic materials, debunking the conventional wisdom that the barycentric dual grid prohibits a sparse representation for inverse mass matrices. In particular, we provide a unified framework for the construction of both edge and face mass matrices and their sparse inverses. Such a unifying principle relies on novel geometric reconstruction formulas, from which, according to a well-established design strategy, local mass matrices are constructed as the sum of a consistent and a stabilization part. A major difference with the approaches proposed so far is that the consistent part is defined geometrically and explicitly, that is, without the necessity of computing the inverses of local matrices. This provides a sensible speedup and an easier implementation. We use these new sparse inverse mass matrices to discretize a three-dimensional Poisson problem, providing the comparison between the results obtained by various formulations on a benchmark problem with analytical solution.

The Moore-Penrose inverse of symmetric matrices with nontrivial equitable partitions

In this paper we consider symmetric matrices that admit nontrivial equitable partitions. We determine some sufficient conditions for the quotient matrix of the Moore-Penrose inverse of the initial matrix to be equal to the Moore-Penrose inverse of its quotient matrix. We also study several particular cases when the computation of the Moore-Penrose inverse can be reduced significantly by establishing the formula for its computation based on the Moore-Penrose inverse of the quotient matrix. Among others we consider the adjacency matrix of a generalized weighted threshold graph.

Realtime parallel software implementation of a DS-CDMA Multiuser Detector

In this article the complexity and runtime performance of two Multiuser Detectors for Direct Sequence-Code Division Multiple Access were evaluated in two different hardware platforms. The innovation and aim is to take advantage of present parallel hardware to bring Multiuser technology to present and future Base Stations in order to increase the capacity of the overall system, to reduce the transmission power by the mobile stations and to reduce base station hardware requirements, in Universal Mobile Telecommunications System. The detectors are based on the Frequency Shift Canceller concatenated with a Parallel Interference Canceller. This detector implies the inversion of multiple identical size small matrices and because of that it is very scalable contrary to other solutions/detectors that only permit a sequential implementation despite their lower complexity. Implementations for the Time Division–Code Division Multiple Access, in two software platforms one in OpenMP and the other in CUDA were done taking into account the carrier and doppler frequency offsets (offset different for each user). The result shows that this deployment aware real-time implementation of the Multiuser Detectors is possible with a Graphics Processor Unit being three times faster than required.

Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series

Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths $$N=2048$$ , 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.

Numerically Stable Polynomially Coded Computing

We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1)We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes.2)We show that the condition number of every m ×m sub-matrix of an m ×n, n ≥ m Chebyshev-Vandermonde matrix, evaluated on the n-point Chebyshev grid, grows as O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2(n-m)</sup> ) for n > m.3)By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance.4)We propose a numerically stable specialization of Lagrange coded computing. Our approach involves the choice of evaluation points and a suitable decoding procedure. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods.

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity

A New Generalized Θ-Inverse vs. Moore-Penrose Structure: A Comparative Control-Oriented Practical Investigation

A new non-unique $\Theta $ -inverse of non-square polynomial matrices is presented in this paper. It is shown that the above inverse specializes to the unique Moore-Penrose one under several specific assumptions. Due to the existence of some degrees of freedom, the $\Theta $ -inverse outperforms the Moore-Penrose inverse in various inverse-related problems covering the multivariable control theory. In many scenarios the former inverse can stabilize a closed-loop control plant being unstable by application of the Moore-Penrose formula. Practical and theoretical simulation examples confirm the correctness of the proposed method.

A Fast Sparse Azimuth Super-Resolution Imaging Method of Real Aperture Radar Based on Iterative Reweighted Least Squares With Linear Sketching

It is greatly significant to achieve radar forward-looking region imaging. Due to the limitation of phase ambiguity and small Doppler gradient in forward-looking region, synthetic aperture radar and Doppler beam sharpening cannot work for forward-looking imaging, while real aperture radar (RAR) has arbitrary imaging geometry. Nevertheless, restricted by the antenna aperture, azimuth resolution of RAR is coarse, super-resolution technology is required to improve its azimuth resolution. Exploiting the sparse prior information of the target, the super-resolution problem can be transformed into an $L_1$ norm minimization problem mathematically. Iterative reweighted algorithm can effectively solve the $L_1$ norm minimization problem by replacing $L_1$ norm with reweighted $L_2$ norm and computing the weight in each iteration. However, it suffers from a large computational load due to the repeated multiplications and inversions of large matrices. In this article, a fast azimuth super-resolution imaging method of RAR based on iterative reweighted least squares (IRLS) with linear sketching (LS) was proposed to achieve fast super-resolution imaging of RAR. The LS theory is employed to compress echo matrix and antenna measurement matrix into much smaller matrices via multiplying them by an embedded matrix. Then, the IRLS solver was utilized to address the reconstructed objective function. Much of the expensive computation can then be performed on the smaller matrices, thereby accelerating the algorithm. Simulations and experimental data prove that the proposed algorithm can offer a time complexity reduction without loss of imaging performance.

The g-Drazin inverses of special operator matrices

The g-Drazin inverses of special operator matrices

Characterizations and representations for the Drazin inverse of anti-triangular block operator matrices with index less than two

Characterizations and representations for the Drazin inverse of anti-triangular block operator matrices with index less than two

TOWARDS OVERCOMING THE CURSE OF DIMENSIONALITY IN PREDICTIVE MODELLING AND UNCERTAINTY QUANTIFICATION

This invited presentation summarizes new methodologies developed by the author for performing high-order sensitivity analysis, uncertainty quantification and predictive modeling. The presentation commences by summarizing the newly developed 3rd-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for linear systems, which overcomes the “curse of dimensionality” for sensitivity analysis and uncertainty quantification of a large variety of model responses of interest in reactor physics systems. The use of the exact expressions of the 2nd-, and 3rd-order sensitivities computed using the 3rd-ASAM is subsequently illustrated by presenting 3rd-order formulas for the first three cumulants of the response distribution, for quantifying response uncertainties (covariance, skewness) stemming from model parameter uncertainties. The use of the 1st-, 2nd-, and 3rd-order sensitivities together with the formulas for the first three cumulants of the response distribution are subsequently used in the newly developed 2nd/3rd-BERRU-PM (“Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”), which aims at overcoming the curse of dimensionality in predictive modeling. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing a subjective user-defined “cost functional quantifying the discrepancies between measurements and computations.” By utilizing the 1st-, 2nd- and 3rd-order response sensitivities to combine experimental and computational information in the joint phase-space of responses and model parameters, the 2nd/3rd-BERRU-PM generalizes the current data adjustment/assimilation methodologies. Even though all of the 2nd- and 3rd-order are comprised in the mathematical framework of the 2nd/3rd-BERRU-PM formalism, the computations underlying the 2nd/3rd-BERRU-PM require the inversion of a single matrix of dimensions equal to the number of considered responses, thus overcoming the curse of dimensionality which would affect the inversion of hessian and higher-order matrices in the parameter space.

TOWARDS OVERCOMING THE CURSE OF DIMENSIONALITY IN PREDICTIVE MODELLING AND UNCERTAINTY QUANTIFICATION

This invited presentation summarizes new methodologies developed by the author for performing high-order sensitivity analysis, uncertainty quantification and predictive modeling. The presentation commences by summarizing the newly developed 3rd-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for linear systems, which overcomes the “curse of dimensionality” for sensitivity analysis and uncertainty quantification of a large variety of model responses of interest in reactor physics systems. The use of the exact expressions of the 2nd-, and 3rd-order sensitivities computed using the 3rd-ASAM is subsequently illustrated by presenting 3rd-order formulas for the first three cumulants of the response distribution, for quantifying response uncertainties (covariance, skewness) stemming from model parameter uncertainties. The use of the 1st-, 2nd-, and 3rd-order sensitivities together with the formulas for the first three cumulants of the response distribution are subsequently used in the newly developed 2nd/3rd-BERRU-PM (“Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”), which aims at overcoming the curse of dimensionality in predictive modeling. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing a subjective user-defined “cost functional quantifying the discrepancies between measurements and computations.” By utilizing the 1st-, 2nd- and 3rd-order response sensitivities to combine experimental and computational information in the joint phase-space of responses and model parameters, the 2nd/3rd-BERRU-PM generalizes the current data adjustment/assimilation methodologies. Even though all of the 2nd- and 3rd-order are comprised in the mathematical framework of the 2nd/3rd-BERRU-PM formalism, the computations underlying the 2nd/3rd-BERRU-PM require the inversion of a single matrix of dimensions equal to the number of considered responses, thus overcoming the curse of dimensionality which would affect the inversion of hessian and higher-order matrices in the parameter space.

Further characterizations of the weak group inverse of matrices and the weak group matrix

<abstract> This paper is devoted to consider some new characteristics and properties of the weak group inverse and the weak group matrix. First, we characterize the weak group inverse of a square matrix based on its range space and null space. Also several different characterizations of the weak group inverse are presented by projection and the Bott-Duffin inverse. Then by using the core-EP decomposition, we investigate the relationships between weak group inverse and other generalized inverses. And some new characterizations of weak group matrix are obtained. </abstract>

Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product

&lt;abstract&gt;&lt;p&gt;Reverse order laws for generalized inverses of products of matrices are a class of algebraic matrix equalities that are composed of matrices and their generalized inverses, which can be used to describe the links between products of matrix and their generalized inverses and have been widely used to deal with various computational and applied problems in matrix analysis and applications. ROLs have been proposed and studied since 1950s and have thrown up many interesting but challenging problems concerning the establishment and characterization of various algebraic equalities in the theory of generalized inverses of matrices and the setting of non-commutative algebras. The aim of this paper is to provide a family of carefully thought-out research problems regarding reverse order laws for generalized inverses of a triple matrix product $ ABC $ of appropriate sizes, including the preparation of lots of useful formulas and facts on generalized inverses of matrices, presentation of known groups of results concerning nested reverse order laws for generalized inverses of the product $ AB $, and the derivation of several groups of equivalent facts regarding various nested reverse order laws and matrix equalities. The main results of the paper and their proofs are established by means of the matrix rank method, the matrix range method, and the block matrix method, so that they are easy to understand within the scope of traditional matrix algebra and can be taken as prototypes of various complicated reverse order laws for generalized inverses of products of multiple matrices.&lt;/p&gt;&lt;/abstract&gt;

Algorithms for solving scattering problems for the Manakov model of nonlinear Schrödinger equations

Abstract We introduce numerical algorithms for solving the inverse and direct scattering problems for the Manakov model of vector nonlinear Schrödinger equation. We have found an algebraic group of 4-block matrices with off-diagonal blocks consisting of special vector-like matrices for generalizing the scalar problem’s efficient numerical algorithms to the vector case. The inversion of block matrices of the discretized system of Gelfand–Levitan–Marchenko integral equations solves the inverse scattering problem using the vector variant the Toeplitz Inner Bordering algorithm of Levinson’s type. The reversal of steps of the inverse problem algorithm gives the solution of the direct scattering problem. Numerical tests confirm the proposed vector algorithms’ efficiency and stability. We also present an example of the algorithms’ application to simulate the Manakov vector solitons’ collision.

Dual generalized inverses and their use in solving systems of linear dual equations

This paper deals with properties of dual generalized inverses and uses them in solving linear dual equations. The first part of this paper addresses several properties of dual generalized inverses of dual matrices and provides new theoretical and computational insights. In the second part necessary and sufficient conditions for a dual matrix to have a {1}-dual generalized inverse are obtained. A dual matrix with a rank deficient primal part generically (almost always) is shown to have no {1}-dual generalized inverse and hence, neither a {1,3}-dual generalized inverse or a Moore-Penrose dual generalized inverse. The necessary and sufficient condition to have a solution to a system of linear dual equations is obtained. An explicit expression for the solution, when one exists, is obtained and it is shown not to be unique, in general. The dual analog of the real least-squares problem is obtained through the {1,3}-dual generalized inverse; as with real matrices, it is shown that this least-squares dual solution is not unique, in general. The dual analog of the real minimum-length least-squares solution is obtained through the development of the Moore-Penrose dual generalized inverse. Numerical examples are provided throughout, mainly from kinematics, to illustrate the results obtained.

On the coincidence of the Drazin inverse and the Drazin-Moore-Penrose inverses

ABSTRACT The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given.

The group inverse of some circulant matrices

In a previous work the authors presented the necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three real parameters and, moreover, we obtained a closed formula for their inverse. The techniques we used allowed us to reduce the computational cost of evaluating the inverse. In this work we complete the theoretical analysis of the mentioned circulant matrices by considering complex parameters and also the singular case. In addition, we explicitely compute the group inverse of a specific kind of this class of matrices using the same techniques that the authors developed for the non singular case; that is, solving first order linear difference equations.

Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods

Abstract We discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from 𝒪 ⁢ ( k 3 ⁢ d ) {\mathcal{O}(k^{3d})} to 𝒪 ⁢ ( d ⁢ k d + 1 ) {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.