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In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n × n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem. • The eigen-problems of reduced biquaternion matrices are further studied based on the complex representation form. • Propose new algebraic algorithms for finding the eigenvalues and the eigenvectors of a reduced biquaternion matrix. • An n × n reduced biquaternion matrix has infinite eigenvalues. • There are multiple eigenvalues corresponding to the same eigenvector of a reduced biquaternion matrix. • The proposed method is more comprehensive and can find more eigenvalues of a reduced biquaternion matrix. [ABSTRACT FROM AUTHOR]

This paper is intended as a companion to [12] and focuses on the numerical implementations of the method in the context of insurance problems with mixed controls: reinsurance acting as continuous control, and capital injection and dividend payment acting as singular ones. The aim is twofold. On the one hand, we provide a comparison of the output of our method in terms of the optimal value against two benchmarks. The first benchmark is provided by [4] in the no-reinsurance setting and when the claims are exponentially distributed. The second benchmark refers to [8] where capital injections are not permitted, but different claim distributions can be employed. On the other hand, we illustrate the improvement of the models in [8] when capital injection is allowed. We also look into the effect of reinsurance in the model [4]. Several cases for the equity cost parameter are discussed as are the boundary (or limit) situations. • We provide a validation of the Dual Linear Programming algorithm by showcasing its performance on two benchmarks. • With a relatively modest order of moments (4), the relative errors are of at most 3%. • Leveraging this algorithm reveals significant enhancements in the expected dividends, advocating for capital injections. [ABSTRACT FROM AUTHOR]

• The algorithm proposed in this paper is simpler and easier to understand compared to the algorithms in references [3] and [8]. • The paper focuses on the power domination problem on trees and presents a linear-time algorithm to solve it using the labeling method. In this paper, the power domination problem on trees is studied and we give a linear-time algorithm to it by using the labeling method. Our algorithm is simpler and easier to understand than those in [3,7]. [ABSTRACT FROM AUTHOR]

This Editorial paper provides an overview of a selection of contributions from researchers working in the fields of Subdivision, Applied Geometric and Algebraic Methods, Isogeometric Analysis, Refinability and Geometric Deep Learning, with special emphasis on numerical computation, advanced algorithms, their analysis, and their applications. [ABSTRACT FROM AUTHOR]

The n -dimensional HSDC with the logic graph H n is one of the most attractive server-centric data center networks for high incremental scalability. The routing design in network topology is very important for information transmission. In particular, the application of disjoint path covers can solve various problems such as program code optimization and mapping parallel programs to parallel structures. H n consists of 2 n cliques and each two vertices in a clique are adjacent. For any vertex sets { x , u } and { y , v } in W and B , respectively, and any two disjoint vertex sets A 1 , A 2 in some clique, where (W , B) is a bipartition of H n , in this paper, we prove that, when | A i | ≠ 1 and A i ∩ { x , y , u , v } = ∅ for i = 1 , 2 , if | A 1 ∪ A 2 | ≤ n − 3 , then there exists an (x , y) -path P 1 and a (u , v) -path P 2 in H n passing through A 1 and A 2 , respectively, such that P 1 and P 2 have no common vertices and P 1 ∪ P 2 contains all vertices of H n. Then P 1 ∪ P 2 is called the 2-disjoint path cover in H n and we propose the efficient algorithms for the path design. Furthermore, we prove that our main result is optimal. • In the problem of data harvest and information transmission, it is very important to design the disjoint path cover in the logic graph of Data Center Network. • We consider the construction of 2-disjoint path cover such that each path should contain all prescribe vertices which have low probability of failure. • The efficient algorithms have been given and we prove that the number of prescribed vertices is not more than n − 3 and explain that this upper bound is sharp. [ABSTRACT FROM AUTHOR]

Analogs of balls in n -dimensional Euclidean space are defined in the monoid of strings composed of letters of an alphabet if a metric is defined on the monoid. These analogs are not only balls in the monoid of strings but also regular languages. Since the Levenshtein distance, a reasonable metric between strings, was introduced in 1965, researchers have aimed to determine a formula for computing the volumes of balls of strings under this distance and their growth rate. However, this problem remains open. In this paper, we first derive a formula for the volumes of balls of strings under a metric obtained by extending the Hamming distance as a metric on the whole monoid and subsequently reveal the growth rate of balls of strings under the Levenshtein distance using this formula. Furthermore, we describe a conjecture on the formula for the volumes of spheres of strings under the Levenshtein distance. Next, we construct an efficient randomized algorithm that computes estimates of the volumes of the balls whose estimation errors are less than a given value with a probability greater than a specified value. Finally, we examine the validity of the proposed conjecture, using the (true) volumes obtained by exhaustive searches for small spheres and estimates of the volumes obtained by this algorithm for large ones. [ABSTRACT FROM AUTHOR]

This paper deals with the stability analysis of Linear Time-Invariant (LTI) systems in the presence of a bounded and time-invariant delay parameter. The paper extends the theory of characteristic quasi-polynomial to derive equivalent stability conditions. Based on the quasi-polynomial, a two-variate polynomial is defined and it is proved that system stability is equivalently guaranteed if all roots of the new polynomial are not allocated in a certain region of the right half plane (RHP). Moreover, the algorithm uses the proposed equivalent stability conditions to develop a set of implementable stability conditions based on the exposed edges theorem. The algorithm is applied to three simulation examples that admit some novel results for these systems. The first example shows the step-by-step procedure for the proposed algorithm. The second and the third ones compare various aspects of the proposed algorithm with some existing methods. Two hundred and fifty delay systems are randomly generated and the proposed algorithm is compared with them in terms of conservativeness and computational burden. The results reveal the superiority of the proposed method over the existing ones. [ABSTRACT FROM AUTHOR]

• Two new computational algorithms are proposed for estimating the Lyapunov exponents from time series data in order to test the null hypothesis of a chaotic behavior. • The results obtained provide robust evidence that the Jacobian indirect methods provide better estimates than traditional direct methods in all the experiments we have conducted. • We have shown empirically that the algorithms proposed are robust to the presence of (small) measurement errors because the results obtained are comparable to those which are noise free. • The empirical size of the algorithms proposed decreased and the empirical power increased as the sample size increased which means that our hypothesis tests are consistent and reliable. • This paper opens a new research line where new contributions may appear considering new machine learning methods and deep learning algorithms for estimating the Lyapunov exponents from time series data. Most of the existing methods and techniques for the detection of chaotic behaviour from empirical time series try to quantify the well-known sensitivity to initial conditions through the estimation of the so-called Lyapunov exponents corresponding to the data generating system, even if this system is unknown. Some of these methods are designed to operate in noise-free environments, such as those methods that directly quantify the separation rate of two initially close trajectories. As an alternative, this paper provides two nonlinear indirect regression methods for estimating the Lyapunov exponents on a noisy environment. We extend the global Jacobian method, by using local polynomial kernel regressions and local neural net kernel models. We apply such methods to several noise-contaminated time series coming from different data generating processes. The results show that in general, the Jacobian indirect methods provide better results than the traditional direct methods for both clean and noisy time series. Moreover, the local Jacobian indirect methods provide more robust and accurate fit than the global ones, with the methods using local networks obtaining more accurate results than those using local polynomials. [ABSTRACT FROM AUTHOR]

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cramér-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding the trajectories in an appropriate family of occupation measures, we provide the linearisation of such problems in which the continuous control is given by reinsurance policies and the singular one by dividends and capital injections. The linearisation translates into a dual dynamic programming (DDP) algorithm. An important part of the paper is dedicated to structural considerations allowing reasonable implementation. We also hint connections to methods relying on moment sum of squares and LMI (linear matrix inequality)-relaxations to approximate the optimal candidates. [ABSTRACT FROM AUTHOR]

• Algebra matrix method integrates the structural factor and the state information of social networks. • Monte Carlo simulation experiments verify the effectiveness of the algebra matrix evaluation. • The evaluation deviations of diffusion threshold are shown best performance comparing with three popular mean field methods. • The weighted average strategy is proposed as applications. Tacit knowledge is the knowledge existing in human brain which is not easy to be recorded or quantified, and often is learned in the face-to-face interactions. The tacit knowledge diffusion depends on the decision-making of tacit knowledge owners, and the expression of explicit knowledge carriers. However, the comprehensive influence of the tacit knowledge owners, explicit knowledge carriers and the relations of them were not attracted enough attention. In this paper, an algebra matrix method is used to integrate the multidimensional information of network structures and the nodes' states. By the algebra matrix method, the diffusion threshold of the tacit knowledge is calculated, which is called algebra matrix evaluation. This evaluation method is proven to be effective by comparing with Monte Carlo simulations on three types of artificial networks and five reals. With applications of the algebra matrix evaluation, we construct a co-author network according the data of the academic papers from 1980 to 2017 on Aminer platform, and define states of tacit knowledge owners and the explicit knowledge carriers by the scholar's career lengths and the paper's cited quantities respectively. It is found that the thresholds of tacit knowledge diffusion are decreasing with the expansions of the scale of the largest connected components, whether tacit knowledge diffuses in the co-author networks or in the largest connected components. And with the evolution of cumulative co-author network, the diffusion thresholds of tacit knowledge in the largest connected component decrease in ladder-like with unequal steps. Furthermore, it is find ignoring the state factor will lead to the deviation in the evaluation of tacit knowledge diffusion thresholds, which is 16.33% in the largest connected components and 45.07% in the whole network. [ABSTRACT FROM AUTHOR]

• A novel fractional gradient descent method with adaptive momentum is presented for training BP neural networks. • The convergence of the proposed algorithm is proved. • The proposed fractional algorithm can effectively improve the training performance and expand the selection range of learning rate. A novel fractional gradient descent method with adaptive momentum is presented in this paper to improve the convergence speed and stability for BP neural network training. The fractional Grünwald-Letnikov derivative is used for the fractional gradient. The coefficient of the momentum term is set as an adaptive variable, depending on the fractional gradient of the current step and the weight change of the previous step. We give a detailed convergence proof of the proposed method. Experiments on MNIST data sets and XOR problem demonstrate that the fractional gradient descent method with adaptive momentum term can effectively improve convergence speed, maintain stability of BP neural network training, help escape from local minimum points, and enlarge the selection range of the learning rate. [ABSTRACT FROM AUTHOR]

• A centrality model for urban street networks is proposed. • Non-local random walks are used to extend the Adapted PageRank Algorithm model. • The non-local movement of the random walker is more intuitive. In the urban street network domain, there is growing interest in extending conventional centrality measures to incorporate node-specific information (such as georeferenced socioeconomic data) in order to help identify important locations in an urban environment. One such centrality measure that is gaining attention is the Adapted PageRank Algorithm (APA) model. However, a fundamental concern with the APA model is the notion of teleportation because it means the random walker is equally likely to jump or 'teleport' to any intersection (node) in the street network, regardless of how far away it is. In this paper, we propose a centrality model that overcomes this counterintuitive idea. More specifically, we extend the APA model by modifying the jumping probabilities so that the random walker is more inclined to jump to a nearby intersection than a distant intersection. We accomplish this using non-local random walks which allow a random walker to jump to any node in the network with probabilities that depend on the distance separating the nodes. To demonstrate the differences between the two models, we present and discuss experimental results for a small ten node graph and a real-world urban street network. [ABSTRACT FROM AUTHOR]

• Deep insight on both sequential and parallel GMRES for linear systems. • Rich discussion on the convergence and acceleration of GMRES. • Brief account of other problems and block algorithms. This paper is about GMRES algorithms for the solution of nonsingular linear systems. We first consider basic algorithms and study their convergence. We then focus on acceleration strategies and parallel algorithms that are useful for solving challenging systems. We also briefly discuss other problems, such as systems with multiple right-hand sides, shifted systems, and singular systems. [ABSTRACT FROM AUTHOR]

In this paper we first write a proof of the perceptron convergence algorithm for the complex multivalued neural networks (CMVNNs). Our primary goal is to formulate and prove the perceptron convergence algorithm for the bicomplex multivalued neural networks (BMVNNs) and other important results in the theory of neural networks based on a bicomplex algebra. [ABSTRACT FROM AUTHOR]

• The variable elimination method is used to find the convex hull containing the concerned set of variables of interest. This generalizes the idea due to Madigan and Mosurski [1] to any undirected graphs. • We design a node absorbing algorithm based on variable elimination to find the convex hull containing the variables of interest in undirected graphs and give an analysis of the complexity of the proposed algorithm. • By analyzing the node absorbing algorithm, we propose a more efficient convex hull finding algorithm called inducing path absorbing algorithm, whose complexity is also analyzed. • We apply our proposed algorithm to a gene network and reduce the dimension from 31 to 6. • Our simulation demonstrates the proposed inducing path absorbing algorithm is efficient for realizing structural dimension reduction for complex networks with large scale. An undirected graphical model is a joint distribution family defined on an undirected graph, and the convex hull of a node set in the graph is the minimal convex subgraph containing it. It has been shown that a graphical model is collapsible onto the minimal local sub-model induced by the convex hull which contains variables of interest under Gaussian and multinomial distributions. This motivates many scholars to design algorithms for finding the unique convex hull containing nodes of interest in a graph. In this paper, we propose two algorithms called, respectively, the node absorption algorithm (NA) and the inducing path absorption algorithm (IPA), to find the minimal convex subgraph containing variables of interest in an undirected graph. These algorithms can be used as potential tools to find the minimal sub-model including variables of interest onto which a graphical model of large-scale can be collapsible. Experiments show that the proposed IPA significantly outperforms the NA and other existing algorithms. Furthermore, we apply the IPA to a gene network so as to collapse a large network onto a smaller network including the interested variables, and thus to achieve the aim of structural dimension reduction. [ABSTRACT FROM AUTHOR]

• The proposed Prox-NPGA can handle the non-smooth term in the objective function. • The convergences of CTA-Prox-NPGA and ATC-Prox-NPGA are proved. • The upper bounds of the step-sizes are given. • Prox-NPGA is not only an algorithm, but also an algorithmic framework. In this paper, we study a class of distributed constraint-coupled optimization problems, where each local function is composed of a smooth and strongly convex function and a convex but possibly non-smooth function. We design a novel proximal nested primal-dual gradient algorithm (Prox-NPGA), which is a generalized version of the exiting algorithm–NPGA. The convergence of Prox-NPGA is proved and the upper bounds of the step-sizes are given. Finally, numerical experiments are employed to verify the theoretical results and compare the convergence rates of different versions of Prox-NPGA. [ABSTRACT FROM AUTHOR]

• In Section 5 Preconditioned BiCOR and TCORS Algorithms are added, i.e., "PBiCOR" and "PTCORS", which are two main methods obtained by introducing the preconditioner based on the nearest Kronecker product in Loan and Pitsianis [28] to accelerate the proposed TBiCOR and TCORS methods in the previous version. • In "Numerical Experiments" section, we added "PBiCOR" and "PTCORS" and compared with other methods. Numerical examples show that PTCORS converges fastest among all methods listed, and PBiCOR converges a bit slower than PTCORS does. • We have added an acknowledgements part. Other sections, such as the title/ introduction/abstract/Numerical experiments/Conclusions section, are all updated. • We have checked the English usage of the revised paper carefully. In this paper, the preconditioned TBiCOR and TCORS methods are presented for solving the Sylvester tensor equation. A tensor Lanczos L -Biorthogonalization algorithm (TLB) is derived for solving the Sylvester tensor equation. Two improved TLB methods are presented. One is the biconjugate L -orthogonal residual algorithm in tensor form (TBiCOR), which implements the L U decomposition for the triangular coefficient matrix derived by the TLB method. The other is the conjugate L -orthogonal residual squared algorithm in tensor form (TCORS), which introduces a square operator to the residual of the TBiCOR algorithm. A preconditioner based on the nearest Kronecker product is used to accelerate the TBiCOR and TCORS algorithms, and we obtain the preconditioned TBiCOR algorithm (PTBiCOR) and preconditioned TCORS algorithm (PTCORS). The proposed algorithms are proved to be convergent within finite steps of iteration without roundoff errors. Several examples illustrate that the preconditioned TBiCOR and TCORS algorithms present excellent convergence. [ABSTRACT FROM AUTHOR]

• Active fault-tolerant strategy is introduced to distributed state estimation scheme. • Intermittent fault (IF) detector is designed and the detectability of IF is analyzed. • Estimation errors can remain the prescribed performance under the influence of IF. In this paper, the problem of distributed fault-tolerant state estimation is studied for stochastic systems over sensor networks with intermittent sensor faults. Compared with the traditional state estimation algorithms, the distinct advantage of fault-tolerant state estimation is that the estimator can keep good performance whether sensor faults occur or not. Different from the previous literature concerning with distributed fault diagnosis, the distributed fault diagnosis problem is investigated in this paper for intermittent faults, whose appearing time, disappearing time and magnitude are all nondeterministic. The distributed fault-tolerant state estimation scheme is constructed, in which the appearing time and disappearing time of intermittent faults are detected, intermittent faults are estimated and compensated. By means of the matrix inequality technique, the H ∞ performance of state estimation errors is guaranteed by properly choosing the estimator parameters. Finally, two examples are provided to demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

• Discrete time Consensus Based Optimization (CBO) algorithm on a convex set is introduced. • The error between optimal value and consensus value is rigorously calculated and further extended from previous paper. • Portfolio optimization problem was solved using our proposed constrained CBO algorithm. In this paper, we propose a predictor-corrector type Consensus Based Optimization(CBO) algorithm on a convex feasible set. Our proposed algorithm generalizes the CBO algorithm in [11] to tackle a constrained optimization problem for the global minima of the non-convex function defined on a convex domain. As a practical application of the proposed algorithm, we study the portfolio optimization problem in finance. In this application, we introduce an objective function to choose the optimal weight on each asset in an asset-bundle, which yields the maximal expected returns given a certain level of risks. Simulation results show that our proposed predictor-corrector type model is successful in finding the optimal value. [ABSTRACT FROM AUTHOR]

• In this paper, the nonlinear fault detection observer is constructed and the observer gain function is derivated by choosing the novel Lyapunov functions. The predetermined estimation accuracy-dependent nonnegative function is introducing in the last step of recursive algorithm. • The threshold design problem is solved by suppressing the residual within the predetermined output estimation accuracy range during the implementation process of observer. • The tight thresholds ϵ and ϵ are predetermined under strategic designing scheme. It is straightforward to implement by comparison with the existing empirical or calculation methods of threshold. This paper presents an fault detection (FD) method for a class of uncertain nonlinear systems with unmatched nonlinear fault functions and disturbances. A recursive FD observer is designed with predetermined and small output estimation error. The nonlinear observer gain function is achieved by introducing predetermined output estimation accuracy-dependent nonnegative functions. Combining Lyapunov functions, it is shown that the absolute value of the residual signal is equal or lesser than tight threshold before fault occurrence. The FD scheme is proposed following fault detectability analysis, simulation example indicates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

• Investigation of the performance of Boris-SDC as particle pusher in particle-in-cell code • Extension of Boris-SDC to relativistic regimes • Boris-SDC provides higher accuracy than Boris but computational cost is about the same The paper investigates two new use cases for the Boris Spectral Deferred Corrections (Boris-SDC) time integrator for plasma simulations. First, we show that using Boris-SDC as a particle pusher in an electrostatic particle-in-cell (PIC) code can, at least in the linear regime, improve simulation accuracy compared with the standard second order Boris method. In some instances, the higher order of Boris-SDC even allows a much larger time step, leading to modest computational gains. Second, we propose a modification of Boris-SDC for the relativistic regime. Based on an implementation of Boris-SDC in the RUNKO PIC code, we demonstrate for a relativistic Penning trap that Boris-SDC retains its high order of convergence for velocities ranging from 0.5 c to > 0.99 c. [ABSTRACT FROM AUTHOR]

• Firstly, a robust discrete-time non-smooth consensus protocol solves that consensus problem for the first-order leaderless multi-agent systems with external disturbance and discrete-time feedback. • Then, a discrete-time super-twisting observer is proposed to estimate the multi-agent systems disturbance. Combining the robust discrete-time non-smooth consensus protocol and discrete-time super-twisting observer, we get a new consensus protocol. The superiorities of this method contains two aspects, i.e., the disturbance rejection ability and a smaller consensus error. To the best authors' knowledge, the presented results in this paper are novel in the literature. In this paper, we firstly consider the consensus problems of first-order leaderless multi-agent systems with external disturbance. Then, we propose a robust discrete-time non-smooth (RDTNS) consensus protocol, which makes agents reach consensus within a region. Besides, we design a disturbance observer based on the discrete-time super-twisting algorithm (DTSTA), which can improve the disturbance rejection ability and decrease the consensus error for the multi-agent systems. At last, we get a robust discrete-time disturbance rejection control consensus protocol by combining the RDTNS consensus protocol with the discrete-time super-twisting observer (DTSTO). The simulation results verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• It is the first time to investigate the problem of decentralized finite-time tracking control with prescribed settling time for a class of p -normal form stochastic large-scale systems with output interconnections existing in both the drift and diffusion terms. • It is the first time to investigate the connective stability problem for stochastic large-scale systems when the system structure changes. • Differing from the existing results, this paper proposes a new mathematical treatment algorithm to deal with the interconnections of stochastic large-scale systems, such that the decentralized tracking control is achieved. • The proposed control strategy is more convenient for practical application than the general finite-time control. • Due to the complexity of the system mathematics model, this paper can better reflect the application of mathematics. This paper aims to solve the decentralized finite-time connective tracking control problem for p -normal form stochastic large-scale systems with output interconnections existing in both the drift and diffusion terms. By means of the stochastic system theory and a prescribed finite-time performance function (PFTPF), a novel design scheme is presented for the decentralized finite-time connective tracking controllers with an arbitrarily prescribed settling time. The connective stability problem of stochastic large-scale systems is investigated for the first time. In addition, a new solution for the decentralized tracking control problem of stochastic large-scale systems is presented via a novel mathematical treatment algorithm. The proposed controllers can ensure that the tracking errors converge to a predetermined region within an arbitrarily prescribed settling time and the controlled system is connectively bounded stable in probability. Three simulation examples are presented to exhibit the performance and the superiority of the new control strategy. [ABSTRACT FROM AUTHOR]

• A novel online mode-free integral reinforcement learning algorithm is proposed to solve the mutiplayer non-zero sum games. • The online learning is used to compute the corresponding N coupled algebraic Riccati equations. • The policy iterative algorithm is applied to solve the coupled algebraic Riccati equations corresponding to the multiplayer nonzero sum games. In this paper, a novel online mode-free integral reinforcement learning algorithm is proposed to solve the multiplayer non-zero sum games. We first collect and learn the subsystems information of states and inputs; then we use the online learning to compute the corresponding N coupled algebraic Riccati equations. The policy iterative algorithm proposed in this paper can solve the coupled algebraic Riccati equations corresponding to the multiplayer non-zero sum games. Finally, the effectiveness and feasibility of the design method of this paper is proved by simulation example with three players. [ABSTRACT FROM AUTHOR]

• The problem of system tracking a single ROS is considered in the first part. The necessary and sufficient condition for the output tracking problem of PBCNs to be solvable is obtained. Based on this condition, the event-based controller has been designed to solve the considered tracking problem. Compared with [18,34], the number of control switchings for the designed controller can be reduced effectively. • A necessary and sufficient condition that the TVROT is trackable in probability one starting from the given initial state is obtained. Moreover, a new strategy is offered to design the control sequence. Compared with [31], we extend the necessary and sufficient condition that for the given initial state, the system can realize output tracking of TVROT from BNs to PBNs. • An effective algorithm is provided to solve the maximum tracking probability for the given initial state which can not track TVROT with probability one. Compared with [26], the problem of the system tracking a TVROT rather than a fixed ROS is considered in this paper, and time-variant controller has been designed to make the state of the system track TVROT from the given initial state with maximum probability. This paper mainly concentrates on the issue of finite-time output tracking of probabilistic Boolean control networks (PBCNs). Two kinds of problems are considered: (1) the system tracks a fixed reference output signal (ROS); (2) the system tracks a time variant reference output trajectory (TVROT). For the first problem, we design an effective event-triggered controller to realize output tracking in probability one. An algorithm is offered to find the control invariant subset (CIS) of a set, and a triggering set sequence is constructed based on the CIS. Next, a necessary and sufficient condition is proposed to judge whether the system is trackable in probability one. For the second problem, a criterion is given to judge whether the initial state can generate TVROT with probability one and an algorithm is then provided to solve the corresponding control sequence. Furthermore, a profit function is established to solve the maximum tracking probability for the initial state which can not generate TVROT with probability one. Lastly, two examples are offered to explain the theoretical results. [ABSTRACT FROM AUTHOR]

• We in this paper propose the bi-conjugate gradient and bi-conjugate residual methods in their tensor forms for solving the Sylvester tensor equation. • BiCG-BTF and BiCR-BTF methods are superior to the modified conjugate gradient method in terms of both the number of iteration steps and CPU time. • From the nearest Kronecker product, the convergence rate of the preconditioned BiCG-BTF and BiCR-BTF methods is about twice that of the PGMRES-BTF method. This paper is concerned with solving high order Sylvester tensor equation arising in control theory. We propose the tensor forms of the bi-conjugate gradient and bi-conjugate residual methods for solving the tensor equation. To improve their performance, two preconditioned iterative algorithms based on the nearest Kronecker product are developed for finding its solution. We also prove that the proposed algorithms are convergent to an exact solution within finite iteration steps for any initial tensor in the absence of round-off errors. At last, some numerical examples are provided to illustrate the feasibility and validity of the algorithms proposed. [ABSTRACT FROM AUTHOR]

• CWENO, MUSCL and 1st-order upwind schemes combined in one framework. • Cascade adaptive solution admissibility criteria for MOOD technique. • The MOOD augmented methods further fortifies high-order methods. • Applied to 2D and 3D compressible multicomponent flow problems. In this paper the relaxed, high-order, Multidimensional Optimal Order Detection (MOOD) framework is extended to the simulation of compressible multicomponent flows on unstructured meshes. The diffuse interface methods (DIM) paradigm is used that employs a five-equation model. The implementation is performed in the open-source high-order unstructured compressible flow solver UCNS3D. The high-order CWENO spatial discretisation is selected due to its reduced computational footprint and improved non-oscillatory behaviour compared to the original WENO variant. Fortifying the CWENO method with the relaxed MOOD technique has been necessary to further improve the robustness of the CWENO method. A series of challenging 2-D and 3-D compressible multicomponent flow problems have been investigated, such as the interaction of a shock with a helium bubble, and a water droplet, and the shock-induced collapse of 2-D and 3-D bubbles arrays. Such problems are generally very stiff due to the strong gradients present, and it has been possible to tackle them using the extended MOOD-CWENO numerical framework. [ABSTRACT FROM AUTHOR]

A structured eigen-problem A x + F (x) = λ x is studied in this paper, where in applications A ∈ R n × n is an irreducible Stieltjes matrix. Under certain restrictions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of A , the Newton-like algorithm for this eigen-problem is well defined and converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method. [ABSTRACT FROM AUTHOR]

• A novel three-step stabilized algorithm for the Navier-Stokes type variational inequality is proposed. • The stabilized algorithm utilizes the lowest equal-order elements for the velocity and pressure. • Errors of the approximates solutions in H1 and L2 norms are derived. • Superiority of the three-step stabilized algorithm to both the one- and two-step stabilized algorithms is illustrated by numerical examples. In the light of two-grid finite element discretization, this paper is concerned with a three-step stabilized algorithm for the Navier-Stokes equations with nonlinear slip boundary conditions of friction type. The proposed algorithm proceeds three steps: the first step solves a stabilized nonlinear variational inequality problem on a coarse grid, the second step solves a stabilized linear variational inequality problem based on Newton iteration on a fine grid, and the third step solves the same stabilized linear problem with only different right-hand sides on a fine grid. We analyze the stability of the presented algorithm, and derive error estimate of the approximate solutions in L 2 norms of the velocity gradient and pressure. Finally, we demonstrate the effectiveness of the present algorithm by two numerical tests. [ABSTRACT FROM AUTHOR]

The number of subtrees, also referred to as the subtrees index, is a key parameter to measure graph structures such as networks. In this paper, we investigate the number of subtrees of planar two-tree networks. By "adding a virtual edge" and "edge orientation", we present a linear time algorithm for computing the number of subtrees of planar two-tree networks, as well as a family of planar two-connected networks. As applications, we provide the formulae for the number of subtrees of the famous small-world Farey network and GDURT network. We also discuss the relationship between the spanning subtree number and the subtree number of these networks. [ABSTRACT FROM AUTHOR]

• We propose a community detection algorithm based on a deep sparse autoencoder. • We combine the path weight matrix with the weighted adjacent paths of the node to obtain the similarity matrix. • The feature matrix has stronger ability to express the features of the network. • The proposed algorithm can more accurately identify community structures. At present, community detection methods are mostly focused on the investigation at unweighted networks. However, real-world networks are always complex, and unweighted networks are not sufficient to reflect the connections among real-world objects. Hence, this paper proposes a community detection algorithm based on a deep sparse autoencoder. First, the second-order neighbors of the nodes are identified, and we can obtain the path weight matrix for the second-order neighbors of the node. We combine the path weight matrix with the weighted adjacent paths of the node to obtain the similarity matrix, which can not only reflect the similarity relationships among connected nodes in the network topology but also the similarity relationships among nodes and second-order neighbors. Then, based on the unsupervised deep learning method, the feature matrix which has a stronger ability to express the features of the network can be obtained by constructing a deep sparse autoencoder. Finally, the K-means algorithm is adopted to cluster the low-dimensional feature matrix and obtain the community structure. The experimental results indicate that compared with 4 typical community detection algorithms, the algorithm proposed here can more accurately identify community structures. Additionally, the results of parameter experiments show that compared with the community structure found by the K-means algorithm, which directly uses the high-dimensional adjacency matrix, the community structure detected by the WCD algorithm in this paper is more accurate. [ABSTRACT FROM AUTHOR]

• Multiple Optical Coherence Tomography (OCT) scanned sequences. • The procedure consists in a registration, an average and a filtering of the above OCT images (for this last step the Sampling Kantorovich (SK) algorithm has been used). • The retinal network has been detected by the application of an hybrid segmentation procedure (introduced in the present paper) based on classical Otsu and Phansalkar methods. The aim of this article is to provide an improvement in the reconstruction and visualization of retinal superficial capillary plexus and choriocapillaris images from healthy subjects. The implemented method uses multiple Optical Coherence Tomography (OCT) scanned sequences, performs a registration, an average and a filtering on them using for this last step the Sampling Kantorovich operator (SK). The resulting images are then analyzed with some well known methods in literature to quantitatively evaluate the improvement achieved by the proposed technique. Also fluoroangiography and histological images are introduced as reference to support the reliability and the goodness of the results. [ABSTRACT FROM AUTHOR]

Kernel ridge regression (KRR) is a popular scheme for non-linear non-parametric learning. However, existing implementations of KRR require that all the data is stored in the main memory, which severely limits the use of KRR in contexts where data size far exceeds the memory size. Such applications are increasingly common in data mining, bioinformatics, and control. A powerful paradigm for computing on data sets that are too large for memory is the streaming model of computation , where we process one data sample at a time, discarding each sample before moving on to the next one. In this paper, we propose StreaMRAK - a streaming version of KRR. StreaMRAK improves on existing KRR schemes by dividing the problem into several levels of resolution, which allows continual refinement to the predictions. The algorithm reduces the memory requirement by continuously and efficiently integrating new samples into the training model. With a novel sub-sampling scheme, StreaMRAK reduces memory and computational complexities by creating a sketch of the original data, where the sub-sampling density is adapted to the bandwidth of the kernel and the local dimensionality of the data. We present a showcase study on two synthetic problems and the prediction of the trajectory of a double pendulum. The results show that the proposed algorithm is fast and accurate. [ABSTRACT FROM AUTHOR]

• Two-dimensional distributed-order time-space fractional diffusion problem is considered and its finite difference discretization is studied. • The stability and convergence of the scheme are investigated. • The spatial second-order convergence and the temporal optimal convergence are obtained. • A fast and memory saving algorithm for solving DO time-space fractional diffusion equation is developed through Gauss quadrature formula, ESA method and PCG method. • Numerical experiments show strong effectiveness and efficiency of the method. In this paper, a fast algorithm is proposed for solving distributed-order time-space fractional diffusion equations. Integral terms in time and space directions are discretized by the Gauss-Legendre quadrature formula. The Caputo fractional derivatives are approximated by the exponential-sum-approximation method, and the finite difference method is applied for spatial approximation. The coefficient matrix of the discretized linear system is symmetric positive definite and possesses block-Toeplitz-Toeplitz-block structure. The preconditioned conjugate gradient method with a block-circulant-circulant-block preconditioner is employed to solve the linear system. Theoretically, the stability and convergence of the proposed scheme are discussed. Numerical experiments are carried out to demonstrate the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

• For optimal control, the stochastic information with current iterative results is used to solve the stochastic algebraic Riccati equation, which has stochastic characteristics. • Calculate the increment instead of the value itself to improve the calculation accuracy. • Newton's method with line search solve the potentially disastrous problem of the first Newton step. • The convergence and even quadratic convergence of the proposed incremental Newton's iterative algorithm have been discussed. In this paper, a novel incremental Newton's iterative algorithm for investigating the optimal control problem of Itô stochastic systems is presented. Newton's method is employed under the Fréchet derivative framework to iteratively solve a stochastic algebraic Riccati equation. Under moderate conditions, the convergence and even quadratic convergence of the proposed incremental Newton's iterative algorithm are discussed, respectively. In addition, the Newton's method is extended to the one with linear search. In the end, numerical results are given to demonstrate the effectiveness and superiority of the proposed algorithms. [ABSTRACT FROM AUTHOR]

• We consider the constrained stochastic convex optimization by projected subgradient algorithm. • A new convergence analysis approach is developed to prove the validity of the proposed algorithm. • Convergence rate is given without specifying the gains. This paper discusses a kind of convex optimization problem considering noises from information transmission in multi-agent systems. Different from previous works, we focus on the objective function which is a summation of strictly L 0 (F) -convex functions under random inner space. Our system is described by Itô formula, which leads to that it is hard to calculate second-order derivative when designing the projected subgradient algorithm. It is shown that all states in stochastic system will converge to the unique optimal state in the polyhedric set constraint by adopting projected subgradient algorithm and the convergence rate is also investigated. Numerical examples are provided to demonstrate the results. [ABSTRACT FROM AUTHOR]

• Unconditional stability and optimal error estimates of FEMs for a class of generalized nonlinear coupled Schrodinger equations with Caputo-type derivatives are mainly considered in this work. • FEM with nonuniform temporal meshes is constructed. • Fast algorithms based on sum-of-exponentials technique is built for the numerical schemes. The paper is concerned with the unconditional stability and optimal error estimates of Galerkin finite element methods (FEMs) for a class of generalized nonlinear coupled Schrödinger equations with Caputo-type derivatives. We improve the results in [1] to a higher order temporal scheme by using a type of new Grönwall inequality. By introducing a time-discrete system, the error is separated into two parts: the temporal error and the spatial error. As the result of τ -independent of the spatial error, we obtain the L ∞ -norm boundedness of the fully discrete solutions without any restrictions on the grid ratio. The unconditionally optimal L 2 -norm error estimate is then obtained naturally. Furthermore, in order to numerically solve the system with nonsmooth solutions, we construct another Galerkin FEM with nonuniform temporal meshes, and corresponding fast algorithm by using sum-of-exponentials technique is also built. Finally, numerical results are reported to show the accuracy and efficiency of the proposed FEMs and the corresponding fast algorithms. [ABSTRACT FROM AUTHOR]

• A ROM-accelerated parallel-in-time preconditioner is developed for solving evolutionary PDEs. • Low dimensional approximation is introduced to the ParaDIAG preconditioning procedure. • More than an order of magnitude speed-up has been achieved by the new algorithm. In this paper we propose a model reduction technique to speed up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at each iteration. Therefore, several heuristic acceleration techniques are introduced in the greedy basis generation algorithm, that is built upon a residual-based error indicator, to further boost up its computational efficiency. Several numerical experiments are conducted, which illustrate the favorable computational efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in comparison with the multigrid-based one. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm based on the Shultz iterative method and divided differences to find the roots of nonlinear equations. Using the new method, we present a rapid and powerful algorithm to compute an approximate inverse of an invertible tensor. Analysis of the convergence error shows that the convergence order of the method is a linear combination of the Fibonacci sequence and also is rapid and powerful in finding and keeping sparsity of the obtained approximate inverse of the sparse tensors. The algorithm is extended for computing the Moore-Penrose inverse of a tensor. As an application, we use the iterates obtained by the algorithm as a preconditioner for the tensorized Krylov subspace method, e.g., LSQR based tensor form to solve the multilinear system A ★ N X = B. Several examples are also provided to show the efficiency of the proposed method. Finally, some concluding remarks are given. [ABSTRACT FROM AUTHOR]

• A new kind of data-based fault detection method is proposed, which combines adaptive dynamic programming and multiple-modeling. • A stable working state is established by using ADP and history data. Once the system runs out of the stable state, it's known for sure that fault occurs. • A whole stable working space of the system can be built with the method of multi-modeling and data of all working modes, as long as system states remain in the working space, system is working normally, exclude the disturbance of modes switching, and improve the accuracy of the fault detection algorithm. • The fault can be isolated, if the direction and magnitude of the deviate vector are solved, which can be achieved by approximation and iteration using the method of ADP. This paper presents a adaptive dynamic programming-based fault detection and isolation (FDI) scheme to detect and isolate faults in an aircraft jet engine. To this end, the weights in Actor-Critic neural networks are first tuned to learn the input-output map of the jet engine considering its multiple working modes. The convergences of the trainings in Critic-Actor neural networks are strictly proved without knowing the drift dynamics and the input dynamics in the presence of unknown nonlinearities and approximation errors. Using the residuals that are generated by measuring the difference of each network output and the measured engine output, various criteria are established for accomplishing the fault diagnosis task, that addresses the problem of fault detection and isolation of the system components. A number of simulation studies are carried out for combustion chamber of a single-spool jet engine to demonstrate and illustrate the advantages, capabilities, and performance of our proposed fault diagnosis scheme. [ABSTRACT FROM AUTHOR]

• The original identification model is decomposed into several sub models according to the dimension of output and different forms of parameters. • To solve the unmeasurable noise terms in the information matrices, we construct some auxiliary models based on the obtained parameter estimates. • To cut down the redundant estimates and solve the associate terms , a partially coupled nonlinear parameter optimization algorithm is proposed. A key to the analysis and design of a dynamic system is to establish a suitable mathematical model of the system. This paper investigates the parameter optimization problem of a class of radial basis function-based multivariate hybrid models. Taking into account the high dimensions of the models and different forms of the parameters, the original identification model is separated into several regressive sub-identification models according to the characteristics of model outputs. Some auxiliary models are constructed to solve the unmeasurable noise terms in the information matrices. For the purpose of eliminating the redundant computation and to deal with the associate terms caused by the model decomposition, inspired by the coupling concept, a partially-coupled nonlinear parameter optimization algorithm is proposed for the multivariate hybrid models. Through the computational efficiency analysis and numerical simulation verification, it is shown that the proposed algorithm has low computational complexity and high parameter estimation accuracy. [ABSTRACT FROM AUTHOR]

• We propose a model of cartoon and texture decomposition in the opponent color space. • Perception of color is usually better represented in opponent color space than the RGB space. • We generalize the L 1 norm and L ∞ norm for the vector-valued vector. • We introduce the TV norm and G-norm in the opponent color space and the primal-dual method is applied to find the solution. • Numerical results are shown the performance of the proposed model. The Meyer model has been successfully applied to decompose cartoon component and texture component for the gray scale image, where the total variation (TV) norm and the G-norm are respectively modeled to capture the cartoon component and the texture component in an energy minimization method. In this paper, we extend this model to the color image in the opponent color space, which is closer to human perception than the RGB space. It is important to extend the TV norm and the G-norm correspondingly because the color image is viewed as a vector-valued vector. We introduce the definition of the L 1 norm and L ∞ norm for the vector-valued vector and accordingly define the TV norm and the G-norm for the color image. In order to handle the numerical difficulty caused by the non-differentiability of the TV norm and G-norm, the dual formulations are used to represent these norm. Then the decomposition problem is reformulated into a minimax problem. A first-order primal-dual algorithm is readily applied to compute the saddle point of the minimax problem. Numerical results are shown the performance of the proposed model. [ABSTRACT FROM AUTHOR]

• Finite-difference approximation of the Sturm-Liouville problem with frozen argument is studied. • Theory of inverse spectral problems for the finite-difference approximation is developed. • Uniqueness theorems are proved and reconstruction algorithms are obtained for the degenerate and non-degenerate cases. • Numerical algorithm for solving the Sturm-Liouville inverse problem with frozen argument is developed. • Effectiveness of the algorithm is illustrated by numerical examples. This paper deals with the discrete system being the finite-difference approximation of the Sturm–Liouville problem with frozen argument. The inverse problem theory is developed for this discrete system. We describe the two principal cases: degenerate and non-degenerate. For these two cases, appropriate inverse problems statements are provided, uniqueness theorems are proved, and reconstruction algorithms are obtained. Moreover, the relationship between the eigenvalues of the continuous problem and its finite-difference approximation is investigated. We obtain the "correction terms" for approximation of the discrete problem eigenvalues by using the eigenvalues of the continuous problem. Relying on these results, we develop a numerical algorithm for recovering the potential of the Sturm–Liouville operator with frozen argument from a finite set of eigenvalues. The effectiveness of this algorithm is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

• We prove that a highly scalable data center network (HSDC) is vertex-symmetric. • We investigate the independent spanning trees (IST) problem on HSDC. • We propose an algorithm to construct n ISTs on the n -dimensional HSDC, denoted as H S n. • The algorithm can be parallelized to run in O (n) time using N = 2 n processors. • As a by-product, we obtain the connectivity of H S n to be n. An emerging datacenter network (DCN) with high scalability called HSDC is a server-centric DCN that can help cloud computing in supporting many inherent cloud services. For example, a server-centric DCN can initiate routing for data transmission. This paper investigates the construction of independent spanning trees (ISTs for short), a set of the rooted spanning trees associated with the disjoint-path property, in HSDC. Regarding multiple spanning trees as routing protocol, ISTs have applications in data transmission, e.g., fault-tolerant broadcasting and secure message distribution. We first establish the vertex-symmetry of HSDC. Then, by the structure that n -dimensional HSDC is a compound graph of an n -dimensional hypercube Q n and n -clique K n , we amend the algorithm constructing ISTs for Q n to obtain the algorithm required by HSDC. Unlike most algorithms of recursively constructing tree structures, our algorithm can find every node's parent in each spanning tree directly via an easy computation relied upon only the node address and tree index. Consequently, we can implement the algorithm for constructing n ISTs in O (n N) time, where N = n 2 n is the number of vertices of n -dimensional HSDC; or parallelize the algorithm in O (n) time using N processors. Remarkably, the diameter of the constructed ISTs is about twice the diameter of Q n. [ABSTRACT FROM AUTHOR]

• In order to solve the problem of unbalanced sample, a new Tradaboost algorithm is designed. • An industrial control anomaly detection system based on transfer learning is constructed. • The identification performance of both source and target domains is improved. • The experimental results show that the improved algorithm has higher performance. In industrial control systems, industrial infrastructure is often attacked by hackers. Due to the serious sample imbalance in industrial control data, the traditional machine learning method has poor performance in anomaly detection. In this paper, TrAdaboost algorithm is applied to industrial control anomaly detection. The samples that are easy to classify are taken as the source domain data, and the samples with poor classification effect are taken as the target domain. The source domain data is used to guide the target domain data training. Then, we improve the traditional TrAdaboost algorithm from two aspects of initial weight and final classifier, and apply it to industrial control anomaly detection. Finally, the performance of the algorithm on two different industrial control data sets is verified. And the improved algorithm is compared with other traditional algorithms. The experimental results show that the improved TrAdaboost algorithm has a significant advantage in predicting categories with a small sample size. This algorithm can accurately identify a few abnormal samples. Moreover, the F1 value, recall and precision value of the improved TrAdaboost algorithm on the two data sets have been significantly improved. This indicates that the improved TrAdaboost algorithm greatly improves the overall prediction accuracy of the model. [ABSTRACT FROM AUTHOR]

• A new yet general positive switched neural network model is established to fill the academic gap. • The influence of sensor faults and actuator saturation on models is considered simultaneously. Besides, with interval uncertainties taken into consideration, a more comprehensive interval observer is constructed. • The proposed method in this paper can be applied to the general nonlinear hybrid systems. Application of some novel techniques from the theory of nonlinear network cluster to positive uncertain switched systems yields a deep insight into the nonlinear dynamics under investigation. This article focuses on the robust exponential stabilization of positive uncertain switched neural networks subject to actuator saturation and sensor faults. Given the existence of interval uncertainty and the constraint concerning positivity of the original system, a new positive state-bounding observer is constructed to guarantee the coinstantaneous estimation of system state and sensor faults. To deal with actuator saturation, the convex hull scheme is employed. By designing the state-feedback controller and utilizing the multiple time-varying linear co-positive Lyapunov function, sufficient conditions for the robust exponential stability on the studied system are established under dwell-time switching. Furthermore, for optimizing the observer matrix, an iterative algorithm is developed. Eventually, a numerical example is exploited to illuminate the feasibility and effectiveness of both the deduced results and the proposed approaches. [ABSTRACT FROM AUTHOR]

Link prediction is a fundamental challenge in network science. Among various methods, similarity-based algorithms are popular for their simplicity, interpretability, high efficiency and good performance. In this paper, we show that the most elementary local similarity index Common Neighbor (CN) can be linearly decomposed by eigenvectors of the adjacency matrix of the target network, with each eigenvector's contribution being proportional to the square of the corresponding eigenvalue. As in many real networks, there is a huge gap between the largest eigenvalue and the second largest eigenvalue, the CN index is thus dominated by the leading eigenvector and much useful information contained in other eigenvectors may be overlooked. Accordingly, we propose a parameter-free algorithm that ensures the contributions of the leading eigenvector and the secondary eigenvector the same. Extensive experiments on real networks demonstrate that the prediction performance of the proposed algorithm is remarkably better than well-performed local similarity indices in the literature. A further proposed algorithm that can adjust the contribution of leading eigenvector shows the superiority over state-of-the-art algorithms with tunable parameters for its competitive accuracy and lower computational complexity. [ABSTRACT FROM AUTHOR]

Given a convex n -gon, we can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and Pérez-Lantero (Discrete Mathematics, 2020), proved to be planar and Hamiltonian. In this paper, we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an O (n) -time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in O (n) time. We further study the independence number, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex n -gon we can select at least ⌈ n / 4 ⌉ pairwise disjoint disks, and we prove that for every n ≥ 3 there exist convex n -gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs. [ABSTRACT FROM AUTHOR]

• In this paper, we proposed an algorithm to solve mixed solutions of the matrix Equation ∑ i = 1 t A i X i B i = E with sub-matrix constraints. We also prove that the iterative solution sequence generated by the algorithm is convergent. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case, for both small-scale and large-scale matrices. The work belongs to the field of numerical algebra, and has been widely concerned. We put forward and analyze in details an iterative method to find the mixed solutions of a matrix equation with sub-matrix constraints. The convergence of the approximated solution sequence generated by the iterative method is investigated, showing that if the constrained matrix equation is consistent, the mixed solution group can be obtained after a finite number of iterations. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case for both small-scale and large-scale matrices. [ABSTRACT FROM AUTHOR]

In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for highly overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical experiments show that, for the same accuracy, our method behaves better in computing time compared with the maximal weighted residual Kaczmarz algorithm. [ABSTRACT FROM AUTHOR]