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This paper addresses the problem of event-triggered $\mathcal {H}_{\infty }$ control for continuous Takagi-Sugeno fuzzy systems with repeated scalar nonlinearities. A feasible stability solution is first proposed based on the fuzzy-rule-dependent Lyapunov functional methods and positive definite diagonally dominant matrix techniques, which not only reduces the conservativeness of the resulting closed-loop dynamic system, but also ensures the concerned fuzzy system is asymptotically stable with a specified $\mathcal {H}_{\infty }$ disturbance attenuation performance. Then, sufficient conditions are presented for the existence of admissible state-feedback controller, and the cone complementarity linearization approach is employed to convert the non-convex feasibility problem into a sequential minimization one subject to linear matrix inequalities, which can be validly solved by employing standard numerical software. In the end, a numerical example and a Chua’s chaotic circuit system are provided to show the applicability of the proposed theories.

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

Strong H -tensors play an important role in the theories and applications of numerical linear algebra. It is necessary to identify whether a given tensor is a strong H -tensor or not. In this paper, we establish some iterative criteria for identifying strong H -tensors. These criteria depend on the elements of the tensors; therefore, they are easy to be verified. The results obtained in this paper extend the corresponding conclusions for strictly generalized diagonally dominant matrices. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are presented. Some numerical experiments show the feasibility and efficiency of the results which are obtained in this paper.

A real interval vector/matrix is an array whose entries are real intervals. In this paper, we consider the real linear interval equations \begin{document}$ \bf{Ax} = \bf{b} $\end{document} with \begin{document}$ {{\bf{A}} }$\end{document}, \begin{document}$ \bf{b} $\end{document} respectively, denote an interval matrix and an interval vector. The aim of the paper is to study the numerical solution of the linear interval equations for various classes of coefficient interval matrices. In particular, we study the convergence of interval AOR method when the coefficient interval matrix is either interval strictly diagonally dominant matrices, interval \begin{document}$ L $\end{document}-matrices, interval \begin{document}$ M $\end{document}-matrices, or interval \begin{document}$ H $\end{document}-matrices.

In the paper the author presents a novel kind of the WZ factorization algorithm, namely a block WZ factorization algorithm. The aim of this new algorithm is to utilize the computational power of contemporary computers with hierarchical memory. In the paper, some properties of the matrix Z are given and analyzed. Next, a version of the block WZ factorization is presented. The author shows that such a block WZ factorization exists for strictly diagonally dominant matrices. The computational cost of this block algorithm is presented. The time and the accuracy of proposed block WZ factorization algorithm for random dense square diagonally dominant matrices are reported. The block algorithm turned out to be faster even up to 300 times than the original WZ factorization.

Analysis of large-scale sequential data has been one of the most crucial tasks in areas such as bioinformatics, text, and audio mining. Existing string kernels, however, either (i) rely on local features of short substructures in the string, which hardly capture long discriminative patterns, (ii) sum over too many substructures, such as all possible subsequences, which leads to diagonal dominance of the kernel matrix, or (iii) rely on non-positive-definite similarity measures derived from the edit distance. Furthermore, while there have been works addressing the computational challenge with respect to the length of string, most of them still experience quadratic complexity in terms of the number of training samples when used in a kernel-based classifier. In this paper, we present a new class of global string kernels that aims to (i) discover global properties hidden in the strings through global alignments, (ii) maintain positive-definiteness of the kernel, without introducing a diagonal dominant kernel matrix, and (iii) have a training cost linear with respect to not only the length of the string but also the number of training string samples. To this end, the proposed kernels are explicitly defined through a series of different random feature maps, each corresponding to a distribution of random strings. We show that kernels defined this way are always positive-definite, and exhibit computational benefits as they always produce \emph{Random String Embeddings (RSE)} that can be directly used in any linear classification models. Our extensive experiments on nine benchmark datasets corroborate that RSE achieves better or comparable accuracy in comparison to state-of-the-art baselines, especially with the strings of longer lengths. In addition, we empirically show that RSE scales linearly with the increase of the number and the length of string., KDD'19 Oral Paper, Data and Code link available in the paper

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such $\boldsymbol {n}$ -neuron competitive neural networks can have $5^{\boldsymbol n}$ equilibria, among which $3^{\boldsymbol n}$ equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the $3^{\boldsymbol n}$ locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam's method with that of this paper. The method is applicable to all subdivision schemes with polynomial reproduction.It performs exact evaluation at rational parameters and approximate evaluation at other arbitrary parameters with tolerance control.It is efficient and robust for the presented schemes with corresponding coefficient matrix being strictly and diagonally dominant.It can also evaluate derivatives under the same framework.Extension of the method to surface cases is straightforward.

International audience; We investigate a mixed-integer variant of the alternating current optimal flow problem. The binary variables activate and deactivate power generators installed at a subset of nodes of the electrical grid. We propose some formulations and a mixed-integer semidefinite programming relaxation, from which we derive two mixed-integer diagonally dominant programming approximation (inner and outer, the latter providing a relaxation). We discuss dimensionality reduction methods to extract solution vectors from solution matrices, and present some computational results showing how both our approximations provide tight bounds.

The paper first introduces two-dimensional convection-diffusion equation with boundary value condition, later uses the finite difference method to discretize the equation and analyzes positive definite, diagonally dominant and symmetric properties of the discretization matrix. Finally, the paper uses fixed point methods and Krylov subspace methods to solve the linear system and compare the convergence speed of these two methods.

This paper suggests a unifying framework for matrix spectra localizations that originate from different generalizations of strictly diagonally dominant matrices. Although a lot of results of this kind have been published over the years, in many papers same properties were proven for every specific localization area using basically the same techniques. For that reason, here, we introduce a concept of DD-type classes of matrices and show how to construct eigenvalue localization sets. For such sets we then prove some general principles and obtain as corollaries many singular results that occur in the literature. Moreover, obtained principles can be used to construct and use novel Gersgorin-like localization areas. To illustrate this, we first prove a new nonsingularity result and then use established principles to obtain the corresponding localization set and its several properties. In addition, some new results on eigenvalue separation lines and upper bounds for spectral radius are obtained, too.

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach.

The convex hull of n+1 affinely independent vertices of the unit n-cube Cn is called a 0/1-simplex. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute 0/1-simplices in Cn can be described by nonsingular 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. The first part of this paper deals with giving a detailed description of how to efficiently compute, by means of a computer program, a representative from each orbit of an acute 0/1-simplex under the action of the hyperoctahedral group Bn of symmetries of Cn. A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature for n < 7. Using the computed cycle indices in combination with Polya's theory of enumeration shows that acute 0/1-simplices are extremely rare among all 0/1-simplices. In the second part of the paper, we study the 0/1-matrices that represent the acute 0/1-simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg 0/1-matrices of size n x n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler's Tree of Fractions that enumerates the rationals between 0 and 1. Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric matrices., 52 pages, 25 figures

In this paper, we generalize the projected Jacobi and the projected Gauss–Seidel methods to vertical linear complementarity problems (VLCPs) characterized by matrices with positive diagonal entries. First, we formulate the methods and show that the subproblems that must be solved at each iteration have an explicit solution, which is easy to compute. Then, we prove the convergence of the proposed procedures when the matrices of the problem satisfy some assumptions of strict or irreducible diagonal dominance. In this context, for simplicity, we first analyze the convergence in the special case of VLCPs of dimension $$2n\times n$$ , and we then generalize the results to VLCPs of an arbitrary dimension $$\ell n\times n$$ . Finally, we provide several numerical experiments (involving both full and sparse matrices) that show the effectiveness of the proposed approaches. In this context, our methods are compared with existing solution methods for VLCPs. A parallel implementation of the projected Jacobi method in CUDA is also presented and analyzed.

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

In this paper, we generalize and analyze the concepts of diagonal dominance and irreducibility in the framework of column and row representative matrices of a set. Our analysis includes the definition of particular sets of M- and H-matrices. We also analyze the form that the matrices of the introduced irreducible sets must have and the implications of the obtained results on the solution of vertical and horizontal linear complementarity problems. In this context, we prove that the projected Jacobi and the projected Gauss-Seidel methods for horizontal linear complementarity problems converge when the matrices of the problem satisfy one of the introduced generalizations of strict diagonal dominance or of irreducible diagonal dominance. This extends current convergence results.

In this paper, the question of when the subdirect sum of two doubly strictly diagonally dominant (DSDDs) matrices is addressed. Some sufficient conditions are given, and these sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

This paper studies the convergence properties of the well-known message-passing algorithm for convex optimisation. Under the assumption of pairwise separability and scaled diagonal dominance, asymptotic convergence is established and a simple bound for the convergence rate is provided for message-passing. In comparison with previous results, our results do not require the given convex program to have known convex pairwise components and that our bound for the convergence rate is tighter and simpler. When specialised to quadratic optimisation, we generalise known results by providing a very simple bound for the convergence rate.

Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

This paper is concerned with a multiobjective convex polynomial problem, where the objective and constraint functions are first-order scaled diagonally dominant sums-of-squares convex polynomials. We first establish necessary and sufficient optimality criteria in terms of second-order cone (SOC) conditions for (weak) efficiencies of the underlying multiobjective optimization problem. We then show that the obtained result provides us a way to find (weak) efficient solutions of the multiobjective program by solving a scalar second-order cone programming relaxation problem of a given weighted-sum optimization problem. In addition, we propose a dual multiobjective problem by means of SOC conditions to the multiobjective optimization problem and examine weak, strong and converse duality relations.

This paper presents a new fast power series solution method to solve the Hierarchal Method of Moment (MoM) matrix for a large complex, perfectly electric conducting (PEC) 3D structures. The proposed power series solution converges in just two (2) iterations which is faster than the conventional fast solver–based iterative solution. The method is purely algebraic in nature and, as such applicable to existing conventional methods. The method uses regular fast solver Hierarchal Matrix (H-Matrix) and can also be applied to Multilevel Fast Multipole Method Algorithm (MLFMA). In the proposed method, we use the scaling of the symmetric near-field matrix to develop a diagonally dominant overall matrix to enable a power series solution. Left and right block scaling coefficients are required for scaling near-field blocks to diagonal blocks using Schur’s complement method. However, only the right-hand scaling coefficients are computed for symmetric near-field matrix leading to saving of computation time and memory. Due to symmetric property, the left side-block scaling coefficients are just the transpose of the right-scaling blocks. Next, the near-field blocks are replaced by scaled near-field diagonal blocks. Now the scaled near-field blocks in combination with far-field and scaling coefficients are subjected to power series solution terminating after only two terms. As all the operations are performed on the near-field blocks, the complexity of scaling coefficient computation is retained as O(N)O⁢(N). The power series solution only involves the matrix-vector product of the far-field, scaling coefficients blocks, and inverse of scaled near-field blocks. Hence, the solution cost remains O(NlogN)O⁢(N⁢l⁢o⁢g⁢N). Several numerical results are presented to validate the efficiency and robustness of the proposed numerical method.

In this paper we consider distributed adaptive stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that uncertain multivariable linear systems are stabilizable by diagonal matrix high gains if the system matrix is an H-matrix with positive diagonal entries. Based on matrix measure and stability theory for diagonally dominant systems, we consider two classes of uncertain linear systems, and derive a threshold condition to ensure their exponential stability by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions using only local state information, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the adaptive distributed stabilization approach to adaptive synchronization control for large-scale complex networks consisting of nonlinear node dynamics and time-varying coupling weights. A unified framework for adaptive synchronization is proposed that includes several general design approaches for adaptive coupling weights to guarantee network synchronization., Comment: 16 Pages and 7 figures

Although kernel approximation methods have been widely applied to mitigate the O ( n 3 ) cost of the n × n kernel matrix inverse in Gaussian process methods, they still face computational challenges. The ‘residual’ matrix between the covariance matrix and the approximating component is often discarded as it prevents the computational cost reduction. In this paper, we propose a computationally efficient Gaussian process approach that achieves better computational efficiency, O ( m n 2 ) , compared with standard Gaussian process methods, when using m ≪ n data. The proposed approach incorporates the ‘residual’ matrix in its symmetric diagonally dominant form which can be further approximated by the Neumann series. We have validated and compared the approach with full Gaussian process approaches and kernel approximation based Gaussian process variants, both on synthetic and real air quality data.

In this paper, we propose a distributed Newton method for decenteralized optimization of large sums of convex functions. Our proposed method is based on creating a set of separable finite sum minimization problems by utilizing a decomposition technique known as Global Consensus that distributes the computation across nodes of a graph and enforces a consensus constraint among the separated variables. The key idea is to exploit the sparsity of the dual Hessian and recast the computation of the Newton step as one of the efficiently solving symmetric diagonally dominant linear equations. We show that our method outperforms the state-of-the-art algorithms, including ADMM. We validate our algorithm both theoretically and empirically. On the theory side, we demonstrate that our algorithm exhibits superlinear convergence within a neighborhood of optimality. Empirically, we show the superiority of this new method on a variety of large-scale optimization problems. The proposed approach is scalable to large problems and has a low communication overhead.

This paper is concerned with the multistability of switched neural networks with piecewise linear activation functions under state-dependent switching. Under some reasonable assumptions on the switching threshold and activation functions, by using the state-space decomposition method, contraction mapping theorem, and strictly diagonally dominant matrix theory, we can characterize the number of equilibria as well as analyze the stability/instability of the equilibria. More interesting, we can find that the switching threshold plays an important role for stable equilibria in the unsaturation regions of activation functions, and the number of stable equilibria of an $n$ -neuron switched neural network with state-dependent parameters increases to $3^{n}$ from $2^{n}$ in the conventional one. Furthermore, for two-neuron switched neural networks, the precise attraction basin of each stable equilibrium point can be figured out, and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines. Two simulation examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.

Consider the classical problem of solving a general linear system of equations $$Ax=b$$ . It is well known that the (successively over relaxed) Gauss–Seidel scheme and many of its variants may not converge when A is neither diagonally dominant nor symmetric positive definite. Can we have a linearly convergent G–S type algorithm that works for anyA? In this paper we answer this question affirmatively by proposing a doubly stochastic G–S algorithm that is provably linearly convergent (in the mean square error sense) for any feasible linear system of equations. The key in the algorithm design is to introduce a nonuniform double stochastic scheme for picking the equation and the variable in each update step as well as a stepsize rule. These techniques also generalize to certain iterative alternating projection algorithms for solving the linear feasibility problem $$A x\le b$$ with an arbitrary A, as well as high-dimensional minimization problems for training over-parameterized models in machine learning. Our results demonstrate that a carefully designed randomization scheme can make an otherwise divergent G–S algorithm converge.

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I^n can be represented by 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse of P is doubly stochastic and has the same support as P. The negated negative part C of P^-T is strictly row-substochastic and its support is complementary to that of D, showing that P^-T=D-C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex T in I^n having F as a facet. We call T the acute neighbor of S at F. If P represents a 0/1-simplex that is merely nonobtuse, P^-T can have entries equal to zero. Its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors at each of its facets. Next, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that such a simplex also has a block diagonal matrix representation with at least two diagonal blocks, and show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal fully indecomposable simplicial facets whose dimensions add up to n. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices., 26 pages, 17 figures

Motivated by the growing successful use of fractional differential equations in the modeling of different important phenomena, in this paper we derive tools for practical analysis of the robust asymptotic stability of a (incommensurate) fractional order linear system. First, the concept of fractional pseudospectra is introduced. Second, driven by the simplicity and usefulness of spectral localizations in the analysis of various matrix properties, we introduce adequate localization techniques using the ideas that come from diagonally dominant matrices, in order to localize the fractional pseudospectra. In such way, many theoretical and practical applications of pseudospectra (robust stability, transient behavior, nonnormal dynamics, etc.) in fractional order differential systems can be linked to the specificity of the matrix entries, allowing one to understand certain phenomena in practice better. Third, we consider the fractional distance to instability in l ∞ , l 1 and l 2 norms, and determine efficient lower bounds. Finally, this novel approach is implemented on the realistic model of empirical food web to link the stability (that incorporates hereditary dynamics of living organisms) with the empirical data and their uncertainty limitations.

In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen–Cahn equation, and provide ample numerical results to verify our findings.

In this paper, we present refinement of multiparameters overrelaxation (RMPOR) method which is used to solve the linear system of equations. We investigate its convergence properties for different matrices such as strictly diagonally dominant matrix, symmetric positive definite matrix, and M-matrix. The proposed method minimizes the number of iterations as compared with the multiparameter overrelaxation method. Its spectral radius is also minimum. To show the efficiency of the proposed method, we prove some theorems and take some numerical examples.

Approximate matrix inversion based methods is widely used for linear massive multiple-input multiple-output (MIMO) received symbol vector detection. Such detectors typically utilize the diagonally dominant channel matrix of a massive MIMO system. Instead of diagonal matrix, a stair matrix can be utilized to improve the error-rate performance of a massive MIMO detector. In this paper, we present very large-scale integration (VLSI) architecture and field programmable gate array (FPGA) implementation of a stair matrix based iterative detection algorithm. The architecture supports a base station with 128 antennas, 8 users with single antenna, and 256 quadrature amplitude modulation (QAM). The stair matrix based detector can deliver a 142.34 Mbps data rate and reach a clock frequency of 258 MHz in a Xilinx Virtex -7FPGA. The detector provides superior error-rate performance and higher scaled throughput than most contemporary massive MIMO detectors.

Let A be a tridiagonal Toeplitz matrix denoted by A=Tritoep(β,α,γ). The matrix A is said to be: strictly diagonally dominant if |α|>|β|+|γ|, weakly diagonally dominant if |α|≥|β|+|γ|, subdiagonally dominant if |β|≥|α|+|γ|, and superdiagonally dominant if |γ|≥|α|+|β|. In this paper, we consider the solution of a tridiagonal Toeplitz system Ax=b, where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block 2×2 matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms, National Natural Science Foundation of China under Grant no. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, the research innovation program of Changsha University of Science and Technology for postgraduate students under Grant (CX2019SS34), and the Portuguese Funds through FCT-Fundação para a Ciência, within the Project UIDB/00013/2020 and UIDP/00013/2020

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author's best knowledge, the only numerical method in the literature is the heuristic one we put forward to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory., 33 pages, 9 figures, 1 table

The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume discretization of multi-group radiation diffusion equations, whose coefficient matrices can be rearranged into the $(G+2)\times(G+2)$ block form, where $G$ is the number of energy groups. The preconditioning techniques are based on the monolithic classical algebraic multigrid method, physical-variable based coarsening two-level algorithm and two types of block Schur complement preconditioners. The classical algebraic multigrid is applied to solve the subsystems that arise in the last three block preconditioners. The coupling strength and diagonal dominance are further explored to improve performance. We use representative one-group and twenty-group linear systems from capsule implosion simulations to test the robustness, efficiency, strong and weak parallel scaling properties of the proposed methods. Numerical results demonstrate that block preconditioners lead to mesh- and problem-independent convergence, and scale well both algorithmically and in parallel.

In this paper, we concern the inverse problem of constructing a monic quadratic pencil which possesses the prescribed partial eigendata, and the damping matrix and stiffness matrix are symmetric tridiagonal. Furthermore, the stiffness matrix is positive semi-definite and weakly diagonally dominant, which has positive diagonal elements and negative off-diagonal elements. Based on the solution of the inverse eigenvalue problem, we apply the alternating direction method with multiplier to solve the finite element model updating problem for the serially linked mass-spring system. The positive semi-definiteness of stiffness matrix, nonnegativity of stiffness and the physical connectivity of the original model are preserved. Numerical results show that our proposed method works well.

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss–Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations.

Feature selection is a fundamental preprocess before performing actual learning; especially in unsupervised manner where the data are unlabeled. Essentially, when there are too many features in the problem, dimensionality reduction through discarding weak features is highly desirable. In this paper, we present a framework for unsupervised feature selection based on dependency maximization between the samples similarity matrices before and after deleting a feature. In this regard, a novel estimation of Hilbert–Schmidt independence criterion (HSIC), more appropriate for high-dimensional data with small sample size, is introduced. Its key idea is that by eliminating the redundant features and/or those have high inter-relevancy, the pairwise samples similarity is not affected seriously. Also, to handle the diagonally dominant matrices, a heuristic trick is used in order to reduce the dynamic range of matrix values. In order to speed up the proposed scheme, the gap statistic and k-means clustering methods are also employed. To assess the performance of our method, some experiments on benchmark datasets are conducted. The obtained results confirm the efficiency of our unsupervised feature selection scheme.

Signed Laplacian matrices generally fail to be diagonally dominant and may fail to be stable. For both undirected and directed graphs, in this paper we present conditions guaranteeing the stability of signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. Our conditions are necessary and sufficient for undirected graphs, but only sufficient for digraphs, the gap between necessity and sufficiency being filled by matrices who have this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). An exception is given by weight balanced signed digraphs, where eventual positivity corresponds to positive semidefinitness of the symmetric part of the Laplacian. Analogous conditions are obtained for signed stochastic matrices. Funding agencies: Swedish Research CouncilSwedish Research Council [2015-04390]

Smoothed particle hydrodynamics (SPH) is a meshfree Lagrangian particle method, and it has been applied to different areas in engineering and sciences. One concern of the conventional SPH is its low accuracy due to particle inconsistency, which hinders the further methodology development. The finite particle method (FPM) restores the particle consistency in the conventional SPH and thus significantly improves the computational accuracy. However, as pointwise corrective matrix inversion is necessary, FPM may encounter instability problems for highly disordered particle distribution. In this paper, through Taylor series analyses with integration approximation and assuming diagonal dominance of the resultant corrective matrix, a new meshfree particle approximation method, decoupled FPM (DFPM), is developed. DFPM is a corrective SPH method, and is flexible, cost-effective and easy in coding with better computational accuracy. It is very attractive for modeling problems with extremely disordered particle distribution as no matrix inversion is required. One- and two-dimensional numerical tests with different kernel functions, smoothing lengths and particle distributions are conducted. It is demonstrated that DFPM has much better accuracy than conventional SPH, while particle distribution and the selection of smoothing function and smoothing length have little influence on DFPM simulation results. DFPM is further applied to model incompressible flows including Poiseuille flow, Couette flow, shear cavity and liquid sloshing. It is shown that DFPM is as accurate as FPM while as flexible as SPH, and it is very attractive in modeling incompressible flows with possible free surfaces.

Radiation computation is very important for high energy density experiments design in the laser-driven Inertial Confinement Fusion. The view-factor based models are often used to calculate the radiation on the capsule inside a hohlraum. However, it usually takes much time to solve them when the number of equations is very large. In this paper, an efficient iteration approach GPU is presented. The core idea is: (1) guaranteed symmetry, strictly diagonally dominant, and positive definite properties underlying the models are described, (2) a preconditioned conjugate gradient iteration approach is presented to compute the radiation based on such guaranteed properties, and (3) such approach is then parallelized and implemented for GPU so that the large scale models, especially for the non-linear model, can be efficiently solved in reasonable time. Finally, two experimental targets for Shenguang laser facilities built in China are demonstrated and compared to validate the efficiency of the presented approach. The results show that, the models’ computation (1) can be speeded up with successive over-relax iteration method by eight times as compared with Cholesky factorization based direct approach, (2) can be accelerated more with the preconditioned conjugate gradient iteration approach by almost eight times, and (3) can be further accelerated about 2 to 4 times as it parallelized and run on the GPU, which enables the large scale models, can be efficiently solved in reasonable time on the usual desktop computers.

This paper is concerned with the problem of partially mode-dependent l 2 − l ∞ filtering for discrete-time nonhomogeneous Markov jump systems (MJSs) with repeated scalar nonlinearities. Our main purpose is to design a partially mode-dependent l 2 − l ∞ filter such that the filtering error system is stochastically stable with l 2 − l ∞ performance. The time-varying transition probabilities (TPs) are depicted as a polytope. By adopting the diagonally dominant Lyapunov functional and based on linear matrix inequalities (LMIs), sufficient conditions are presented for the existence of the filter. Three examples are given to validate the effectiveness of our results.

This paper focuses on the state feedback stabilization problem for a class of discrete-time singular Markov jump systems with repeated scalar nonlinearities. First, on the basis of the implicit function theorem and the diagonally dominant Lyapunov approach, a sufficient condition is obtained, which ensures the regularity, causality, uniqueness of solution in the neighbourhood of the origin, and stochastic stability for the system under consideration. Moreover, by employing some lemmas and matrix inequalities, the sufficient condition is changed into a set of linear matrix inequalities. Then, the procedures of designing the state feedback controller are given. Eventually, three examples are presented to show the validness of the proposed approach.

In this paper, we prove that the minimum eigenvalue of a strictly diagonally dominant Z-tensor with positive diagonal entries lies between the smallest and the largest row sums. The novelty comes from the upper bound. Moreover, we show that a similar upper bound does not hold for the minimum eigenvalue of a strictly diagonally dominant tensor with positive diagonal entries but with arbitrary off-diagonal entries. Furthermore, other new bounds for the minimum eigenvalue of nonsingular M-tensors are obtained.

Large-scale tridiagonal matrix solvers based on heterogeneous systems currently cannot balance computational efficiency and numerical stability when solving a non-diagonally dominant matrix. A tridiagonal solver combined central processing unit with graphics processing unit is proposed, based on SPIKE2 as a solver framework, a simplified SPIKE algorithm as a central processing unit component, and a diagonal pivot algorithm as a graphics processing unit component. The solver performance is further improved by using a data-layout-transformation mechanism to obtain continuous addresses, reducing memory communication using constant memory to store unchanged data in the calculation process, and employing a kernel-fusion mechanism to reduce power consumption of graphics processing unit. For a diagonally dominant matrix, extended Thomas algorithms and cycle reduction to replace the graphics processing unit component are proposed in the solver. Experimental results show that the tridiagonal matrix solver in this paper can effectively consider both numerical stability and computational efficiency, and reduce total power consumption while improving memory efficiency.

This paper proposes a simple and low numerical complexity technique for the simultaneous achievement of open and closed loop diagonal dominance. The method is based on the introduction of a static high gain inner feedback which modifies the plant to attain a diagonal dominant open loop compensated plant over an entire bandwidth of frequencies. The technique can be also applied to uncertain plants. The sufficient conditions derived for the open loop diagonal dominance also assure the simultaneous achievement of closed loop diagonal dominance under weak assumptions on the diagonal elements of the decoupled controller robustly stabilizing the plant. These assumptions agree with the usual closed-loop performance specifications, so that closed-loop diagonal dominance is achieved without any extra complication on the controller design procedure. The main merits of the present method are: applicability to uncertain plants, ease of implementation and low computational cost. Two examples taken from the relevant literature, are given to show the effectiveness of the proposed approach.