Block-diagonal Research Articles

Inequalities for partial determinants of positive semidefinite block matrices

Inspired by a determinantal inequality for partial traces given by Lin [Can Math Bull 59 (2016)], we present some inequalities related to the partial determinants of positive semidefinite block matrices. Some interesting inequalities involving determinants are included.

CS-TSSOS: Correlative and Term Sparsity for Large-Scale Polynomial Optimization

This work proposes a new moment-SOS hierarchy, called CS-TSSOS , for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [ 40 ] while the latter was initially proposed by Wang et al. [ 42 ] and later exploited in the TSSOS hierarchy [ 46 , 47 ]. In doing so we obtain CS-TSSOS—a two-level hierarchy of semidefinite programming relaxations with (i) the crucial property to involve blocks of SDP matrices and (ii) the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints.

System asymptotic stability analysis of a kind of complex production processes based on multi-dimensional moving pattern

This paper presents a new method to study the system stability of a kind of complex production processes. The system dynamics is described by multi-dimensional pattern class variables instead of state variables or output variables. The system asymptotic stability is defined by system matrix and output class partition. Firstly, “pattern moving space” is constructed by a new maximum entropy clustering method. Multi-dimensional pattern class variables measured by pattern class centers are defined in this new space. Then the nonlinear state space model of the relevant control system is derived and established through two-step prediction model. On the basis of the original stability in Euclidean space, the multi-dimensional pattern moving state space is defined and Lyapunov stability is given in this space. Finally, experimental results show the effectiveness of the proposed conditions.

On positive and positive partial transpose matrices

A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.

Coherent Signal DOA Estimation for MIMO Radar under Composite Background of Strong Interference and Non-Uniform Noise.

To address the problems of low accuracy and low robustness of the conventional algorithm in estimating the direction of arrival (DOA) of coherent signals against a composite background of strong interference and non-uniform noise, in this paper, a coherent signal DOA estimation algorithm based on fixed projection blocking is proposed in conjunction with a multi-input multi-output (MIMO) radar. The covariance matrix of the received signal is first decomposed by eigenvalues, and a fixed projection matrix orthogonal to the interference guidance vector is constructed as the interference blocking matrix. Then, the received array signal is pre-processed to re-form the covariance matrix, and this matrix is rendered decoherent through a Toeplitz reconstruction. Finally, the reconstructed covariance matrix is estimated by DOA using the propagation operator algorithm to reduce the complexity. The simulation verifies that the proposed algorithm has a better robustness and higher accuracy than conventional algorithms for the DOA estimation of coherent signals in composite backgrounds.

Eigenvalue bounds for saddle-point systems with singular leading blocks

We derive bounds on the eigenvalues of saddle-point matrices with singular leading blocks. The technique of proof is based on augmenting the singular leading block to replace it with a positive definite matrix. Our bounds depend on the principal angles between the ranges or kernels of the matrix blocks. Numerical experiments validate our analytical findings.

The solution of inverse matrix by using unary polynomial ring theory

Matrix is in the core position in algebra, and the inverse matrix is particularly important. So far, there are adjoint matrix method, elementary transformation method, block matrix method and so on in solving the inverse matrix. In this paper, we will introduce a method of seeking the inverse matrix by using unary polynomial ring theory. This method can reverse the matrix equation into the algebra equation, cleverly avoiding the complex matrix equation.

Improvement to Gravity Drainage Recovery by Repressurization as a Criterion to Screen and Rank Naturally Fractured Reservoirs for Gas Injection

Summary Many of the naturally fractured carbonate reservoirs of the Middle East exhibit low natural-depletion recoveries. The reason is that most of their oil reserves are stored in the low-permeability host rocks and are left behind by the fast advancing gas/oil contact (GOC) and water/oil contact (WOC) in fractures. Producing the remaining oil in the large gas-invaded zone of these reservoirs has been a crucial reservoir management issue. We show in this study using experimental observations, analytical calculations, and numerical investigations that repressurizing naturally fractured reservoirs (NFRs) by crestal immiscible gas injection has the potential to produce a large portion of this remaining oil by improving gravity drainage (GD) through two main mechanisms. One is that at higher pressures, the gas-oil interfacial tension (IFT) and hence the capillary forces that control recovery by GD are lessened, allowing additional recovery. This mechanism is aided by the other one, which is swelling of the oil at higher pressures. In this way, repressurization is thought to be not only a means for pressure maintenance but also a methodology for enhanced-oil recovery (EOR). This is confirmed by both laboratory studies and field performance of large-scale gas injection projects. Despite the desire for implementation of projects of repressurization, gas availability and cost of these projects are important concerns, requiring a cost-benefit analysis. Screening and ranking methodologies have been previously presented for some EOR techniques but not for repressurization by gas in NFRs. Evaluating the performance of gas injection in NFRs is often done using methodologies such as numerical simulations, which are in-depth, costly, and tedious. The methodology developed here is simple, requiring spreadsheet calculations. To develop the methodology, we first obtain simple relations to calculate additional GD recovery by considering the interplay of capillary and gravity forces in a matrix block subjected to pressurization by equilibrium gas injection and then use experimental data from literature to show that these relations can predict primary and secondary GD recoveries to a good approximation. We also show by mechanistic studies using a history-matched numerical model that IFT reduction and oil swelling are the main mechanisms contributing to additional oil recovery. Then, we propose a methodology to screen and rank candidate NFRs for gas injection that uses commonly available reservoir data and is based upon two criteria, these being additional oil recovered from a matrix block by pressurization and required volume of gas to produce an additional barrel of oil. We then implement this methodology to more than 20 Iranian NFRs and identify six reservoirs with potential for additional recovery of more than 20%. By quantifying and including the uncertainties associated with the reservoir data, we illustrate that for the reservoirs under study, capillary pressure parameters along with matrix-block height are the main parameters affecting GD recovery and should be characterized more accurately.

Dealing with nonlattice spatially variable data contaminated by white noise using Kronecker-product formulation

Gaussian random field is often used to model the spatial variability in geotechnical engineering. In order to characterize the spatial variability of a site of interest, it may be necessary to estimate the autocovariance parameters based on the site investigation data. When the site investigation data consist of multiple soundings, the computational cost for the autocovariance parameter estimation can be high. The Kronecker-product (KP) formulation is a useful tool for reducing this computational cost. However, there are two implementation limitations for the KP formulation. One is the consideration of the observation noise, and another is its implementation to nonlattice data. This paper proposes a method to overcome these two limitations while maintaining the KP formulation and the rigor of the solution. Because the proposed method provides analytical solutions, the computed results are in full agreement with those without the KP formula. For the observation noise, the eigen-decomposition of the autocovariance matrix is adopted. For nonlattice data, they are linked to lattice data by subtracting/adding data points from/to lattice data, and the block matrix inversion is used to derive the KP formulation. The reduction of the computational cost due to the use of the KP formulation is confirmed via numerical simulations.

Scalable Graph Convolutional Network Training on Distributed-Memory Systems

Graph Convolutional Networks (GCNs) are extensively utilized for deep learning on graphs. The large data sizes of graphs and their vertex features make scalable training algorithms and distributed memory systems necessary. Since the convolution operation on graphs induces irregular memory access patterns, designing a memory- and communication-efficient parallel algorithm for GCN training poses unique challenges. We propose a highly parallel training algorithm that scales to large processor counts. In our solution, the large adjacency and vertex-feature matrices are partitioned among processors. We exploit the vertex-partitioning of the graph to use non-blocking point-to-point communication operations between processors for better scalability. To further minimize the parallelization overheads, we introduce a sparse matrix partitioning scheme based on a hypergraph partitioning model for full-batch training. We also propose a novel stochastic hypergraph model to encode the expected communication volume in mini-batch training. We show the merits of the hypergraph model, previously unexplored for GCN training, over the standard graph partitioning model which does not accurately encode the communication costs. Experiments performed on real-world graph datasets demonstrate that the proposed algorithms achieve considerable speedups over alternative solutions. The optimizations achieved on communication costs become even more pronounced at high scalability with many processors. The performance benefits are preserved in deeper GCNs having more layers as well as on billion-scale graphs.

Modified transfer matrix method for vibration analysis of beam structures including branches and rigid bodies

Beam frames can be found in many application scenarios, ranging from big-scale structure engineering of buildings, bridges to small systems of micromachines, sensors, robotics and metamaterials. However, several challenging issues, e.g. serial-parallel configurations, beam and rigid body coupling along with inconsistent dimension of state variables, often lead to a complicated dynamic modeling. In this paper, we show that the free vibration analysis of a class of beam structures with serial-parallel branches and rigid bodies can be significantly streamlined by modifying the transfer matrix method in a graphic manner. Treating general beam structures as a topological graph with standard building blocks enables a modular modeling process and hence offers a new way to divide the complex modeling issue into much easier steps. The transfer matrix of typical building blocks, such as a beam connected to rigid bodies, tree branches and parallel clamped sub-chains, are derived and standardized providing new perspectives and distinct forms in contrast to previous studies. The transfer matrix method exhibits several advantages of easy programming, accurate solution and small degrees-of-freedom for serial structures. The presented approach extends these characteristics to a range of multi-body systems consisting of rigid bodies and beams. Comparative validation with four application case studies confirms these advantages.

Characterizations of the group invertibility of a matrix revisited

Abstract A square complex matrix A A is said to be group invertible if there exists a matrix X X such that A X A = A AXA=A , X A X = X XAX=X , and A X = X A AX=XA hold, and such a matrix X X is called the group inverse of A A . The group invertibility of a matrix is one of the fundamental concepts in the theory of generalized inverses, while group inverses of matrices have many essential applications in matrix theory and other disciplines. The purpose of this article is to reconsider the characterization problem of the group invertibility of a matrix, as well as the constructions of various algebraic equalities in relation to group invertible matrices. The coverage includes collecting and establishing a family of existing and new necessary and sufficient conditions for a matrix to be group invertible and giving many algebraic matrix equalities that involve Moore-Penrose inverses and group inverses of matrices through the skillful use of a series of highly selective formulas and facts about ranks, ranges, and generalized inverses of matrices, as well as block matrix operations.

Testing and signal identification for two-sample high-dimensional covariances via multi-level thresholding

The paper considers testing and signal identification for covariance matrices from two populations of marginally sub-Gaussian distributed. A multi-level thresholding procedure is proposed for testing the equality of two high-dimensional covariance matrices, which is designed to detect sparse and faint differences between the covariances. A novel U-statistic composition is developed to establish the asymptotic distribution of the thresholding statistics in conjunction with the matrix blocking and the coupling techniques. It is shown that the proposed test is more powerful than the existing tests in detecting sparse and weak signals in covariances. Multiple testing procedures are constructed to discover different covariances and the sub-groups of variables with different covariance structures between the two populations. The proposed procedures are based on the multi-level thresholding test, which are able to control the false discovery proportion (FDP) with high power. Simulation experiments and a case study on the returns of the S&P 500 stocks before and after the COVID-19 pandemic are conducted to demonstrate and compare the utilities of the proposed methods.

RESULTS ON BUILDING FRACTIONAL MATRIX DIFFERENTIAL EQUATION SYSTEMS USING A CLASS OF BLOCK MATRICES

In this paper, some important objectives have been achieved, which are as follows: First, we present a method of the inverse for a class of non-singular block matrices and some associated properties. Also, the accuracy of a new method is verified with some illustrated examples by applying the MATLAB lines. Second, applying a class of block matrices, we give the exact solution for fractional matrix differential equation systems using the Laplace fractional transformation method. Finally, illustrative examples and individual cases are also presented and discussed to demonstrate our new approach.

Robust multi-view subspace enhanced representation based on collaborative constraints and HSIC induction

The design of effective multi-view subspace clustering (MSC) algorithms has recently garnered significant research attention. Herein, to effectively improve the recognition performance and anti-noise interference ability of an MSC model, we propose a novel MSC algorithm, termed as robust multi-view subspace enhancement representation, based on collaborative constraints and a Hilbert–Schmidt independence criterion (HSIC) induction method. To mine the complementary information between different views, we apply the HSIC as a diversity regularization term. Specifically, to enhance the diagonal block structure of a subspace representation, a new sparse constraint is introduced on the product of itself and the transpose of the subspace representation matrix in a multi-view subspace learning model. Furthermore, hypergraph regularization and a low-rank idea are considered to capture the local geometric structure and clean data. In addition, to optimize our model, we adopt an augmented Lagrangian multiplier method and discuss the convergence of the model. Extensive experiments on six challenging datasets reveal that the proposed method achieves a highly competent objective performance with and without noisy views, as compared with several state-of-the-art multi-view clustering methods.

Pore‐network modelling of combined molecular diffusion and gravity drainage mechanisms in a porous matrix block: The competitive role of driving forces

AbstractA large part of the world's hydrocarbon resources are located in fractured reservoirs, and mass transfer phenomena play a crucial role in enhanced hydrocarbon recovery from these reservoirs. Pore‐network models have been widely used to study kinetic and pore‐scale micro‐mechanisms. Molecular diffusion involves mass transfer and liquid–vapour phase change and can be simulated by a modified invasion percolation model. Despite the existence of separate pore‐scale studies on molecular diffusion and gravity drainage, no articles have been published that evaluate the combined effect of both mechanisms. This study investigates the competitive roles of the two phenomena and the effective factors controlling each mechanism with the aid of pore‐network models. According to the results obtained, gravity drainage and molecular diffusion would have a synergic effect when they are simultaneously active. Although for a single‐component liquid system, there would be a capillary holdup residual saturation in the pure gravity drainage process (between 11% and 14% for the evaluated cases) and a slow and lengthy evaporation in pure molecular diffusion (between 47% and 57% longer for the cases under study), our investigation revealed that when the two mechanisms coexist, a faster process with no residual liquid is expected. Our findings clarify that when the system is strongly gravity dominated, the liquid body remains integrated, gas–liquid contact recedes in a piston‐like manner, and three‐stage liquid desaturation is observed. Furthermore, highly clustered liquid saturation is observed in strongly capillary‐dominated systems, and the liquid desaturation curve in a capillary‐dominated model has two distinguishable stages. The competitive contribution of gravity drainage and molecular diffusion as the main driving forces of liquid extraction from a single‐block model is quantified for the entire period of desaturation. Depending on the dominance of the production mechanisms, the process is either gravity‐assisted molecular diffusion or diffusion‐assisted gravity drainage.

BMPMDA: Prediction of MiRNA-Disease Associations Using a Space Projection Model Based on Block Matrix.

With the high-quality development of bioinformatics technology, miRNA-disease associations (MDAs) are gradually being uncovered. At present, convenient and efficient prediction methods, which solve the problem of resource-consuming in traditional wet experiments, need to be further put forward. In this study, a space projection model based on block matrix is presented for predicting MDAs (BMPMDA). Specifically, two block matrices are first composed of the known association matrix and similarity to increase comprehensiveness. For the integrity of information in the heterogeneous network, matrix completion (MC) is utilized to mine potential MDAs. Considering the neighborhood information of data points, linear neighborhood similarity (LNS) is regarded as a measure of similarity. Next, LNS is projected onto the corresponding completed association matrix to derive the projection score. Finally, the AUC and AUPR values for BMPMDA reach 0.9691 and 0.6231, respectively. Additionally, the majority of novel MDAs in three disease cases are identified in existing databases and literature. It suggests that BMPMDA can serve as a reliable prediction model for biological research.

Sample-level weights learning for multi-view clustering on spectral rotation

Multi-view clustering usually yields better results than single-view clustering since it utilizes more information from multi-view data. However, in the original multi-view data, some samples may be corrupted in partial views. Under this situation, the locations of the corrupted data are often unknown. But there is limited literature regarding this problem. Moreover, most existing multi-view spectral clustering methods need a post-processed algorithm k-means after obtaining the partition matrix, which leads to deviations in the clustering results. To resolve these problems, we propose a multi-view spectral clustering method named sample-level weights learning for Multi-view Clustering on Spectral Rotation (SR-MC) in this paper. By learning the weights in the sample level, SR-MC can make full use of the helpful complementary information among different views while reducing the effects of low-quality data for each sample. Therefore, it can deal with various multi-view clustering scenarios such as data are complete or corrupted in partial views. To reduce the deviations of the clustering results, a joint framework is designed for combining the learning of the consensus Laplacian matrix, the real-valued partition matrix and the binary indicator matrix together. The objective function of SR-MC can be efficiently optimized by an alternative optimization algorithm. Compared with other ten baselines, experiments on seven datasets show the superiority of our method.

A Jordan-Schur Algorithm for Solving Sylvester and Lyapunov Matrix Equations

This paper presents a version of the Bartels-Stewart algorithm for solving the Sylvester and Lyapunov equations that utilizes the Jordan-Schur form of the equation matrices. The Jordan-Schur form is a type of Schur form which contains additional information about the Jordan structure of the corresponding matrix. This information can be used to solve more efficiently the Sylvester and Lyapunov equations in some cases. A two-level algorithm is implemented which allows us to find directly non-scalar blocks of the solution matrix. These blocks have sizes that are determined by the Weyr characteristics associated with the eigenvalues of the equation matrices. In the case of large elements of the Weyr characteristics associated with multiple eigenvalues, the determination of the solution blocks can be done more efficiency. Also, the blocks equations can be more appropriate in solving the Sylvester and Lyapunov equations in the case of parallel computations. Results obtained from numerical experiments confirm that the accuracy of the new algorithm is comparable with the accuracy of the Bartels-Stewart algorithm.

Evaluation of the effectiveness of educational activities of a class teacher, taking into account his professional and personal well-being

Introduction. The increasing attention of society and the state to the role of a class teacher actualizes the problem of assessing the effectiveness of his activities, taking into account the professional and personal well-being that determines the success, psychological comfort, satisfaction with professional activities and health in general. The purpose of the study is to develop, justify and test the methodology of assessing the effectiveness of the educational activities of the class teacher, taking into account its professional and personal well-being. Materials and methods. The expert assessment of the methodology involved 122 respondents, including 85 class teachers of Nizhny Novgorod region (the Russian Federation) and 37 undergraduates of Minin University, studying in the field of “Pedagogical education”. The sampling was carried out taking into account the age characteristics and experience of the respondents in this position. The age of the participants ranged from 21 to 71. The younger age group – up to 35 years old amounted to 33%, the middle age group – from 36 to 55 years (46%) and the senior group – over 55 years (21%) of the total number of respondents. Study Results. On the basis of the selected theoretical and methodological foundations, a methodology was developed to assess the effectiveness of the educational activities of the class teacher, taking into account his professional and personal well-being. Its content component is represented by an integrative criteria matrix including a number of criteria and indicators, including professional and personal well-being. As a result of the empirical study, it was found that 76% of the respondents had a positive attitude to the developed methodology, emphasizing its variability. 67.2% of the experts pointed out the importance of the personal block of the integrative criteria matrix, where “professional and personal well-being of the class teacher” is one of the criteria for evaluating effectiveness. 75% of those participating in the examination noted the significance and objectivity of the highlighted indicators of professional and personal well-being. 61.6% emphasized that professional and personal well-being depends on the psychological climate of the educational organization and the attitude of the participants in the educational process towards the activities of the class teacher. Conclusions. The results obtained indicate that the assessment of the effectiveness of the educational activity of a class teacher, taking into account his/her professional and personal well-being, provides a developing, stimulating character, makes it possible to identify educational deficits, thereby creating prerequisites for further professional self-improvement of a class teacher, increasing the quality of the educational process, strengthening and “retention” of novice class teachers in the profession.