Discrete Matrix Research Articles

Fully coupled hydro-mechanical modeling of two-phase flow in deformable fractured porous media with discontinuous and continuous Galerkin method

Discrete fracture and matrix modeling of coupled flow and geomechanics is instrumental for understanding flow in fractured media for various geoengineering applications such as enhanced geothermal energy systems and groundwater remediation. We employ the governing equations for two-phase flow in deformable fractured porous media with a stress-dependent porosity and permeability model of matrix, and variable fracture aperture and the corresponding permeability. A finite element framework is presented, in which fractures are regarded as low-dimensional objects, to discretize the coupled two-phase flow and geomechanics in fractured porous media. A hybrid method, combining discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods (FEM), is utilized to solve for the two-phase flow, while the solid deformation is approximated using a discontinuous Galerkin FEM approach. Several benchmark cases are utilized to examine the accuracy of the proposed hybrid DG-CG FEM method. Further validation is performed using more complex, realistic fracture configurations. As the mechanical characteristics of the fracture and the surrounding matrix differ, the simulation results demonstrate that displacement and stress are discontinuous on both sides of the fracture. While the case with low permeability fractures exhibits pressure jump across the fractures, the pressure changes are reasonably smooth for the conduit fractures.

Highly Efficient Numerical Algorithm for Nonlinear Space Variable-Order Fractional Reaction–Diffusion Models

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational approximation. A splitting technique is used to address the issues related to computational efficiency and the stability of the method. Two linear systems need to be solved using the same real-valued discretization matrix. The stability and convergence of the method are discussed analytically and demonstrated through numerical experiments by solving test problems from the literature. The variable-order diffusion effects on the solution profiles are illustrated through graphs. Finally, numerical experiments demonstrate the superiority of the presented method in terms of computational efficiency, accuracy, and reliability.

Robust gradient‐based neural networks for solving online the discrete periodic Lyapunov matrix equations

AbstractHere, a gradient‐based neural network (GNN) model is constructed for solving the discrete periodic Lyapunov matrix equation (DPLME) associated with discrete‐time linear periodic systems. In practical applications, the recurrent neural network model should not only converge rapidly, but also be able to tolerate noise. However, the influence of noise on GNN models was seldom considered in the past. In order to improve the convergence and robustness of the GNN model, a novel type of non‐linear activation function is applied to the GNN model. Compared with the traditional activation functions, the activation function used here makes the GNN model to achieve fixed‐time convergence. Besides, when disturbed by bounded noise, the unique positive definite solution of the DPLME can still be obtained by using the GNN model. Finally, simulation experiment is performed to verify the effectiveness and superiority of the proposed GNN model.

Drop edges and adapt: A fairness enforcing fine-tuning for graph neural networks

The rise of graph representation learning as the primary solution for many different network science tasks led to a surge of interest in the fairness of this family of methods. Link prediction, in particular, has a substantial social impact. However, link prediction algorithms tend to increase the segregation in social networks by disfavouring the links between individuals in specific demographic groups. This paper proposes a novel way to enforce fairness on graph neural networks with a fine-tuning strategy. We Drop the unfair Edges and, simultaneously, we Adapt the model’s parameters to those modifications, DEA in short. We introduce two covariance-based constraints designed explicitly for the link prediction task. We use these constraints to guide the optimization process responsible for learning the new ‘fair’ adjacency matrix. One novelty of DEA is that we can use a discrete yet learnable adjacency matrix in our fine-tuning. We demonstrate the effectiveness of our approach on five real-world datasets and show that we can improve both the accuracy and the fairness of the link prediction tasks. In addition, we present an in-depth ablation study demonstrating that our training algorithm for the adjacency matrix can be used to improve link prediction performances during training. Finally, we compute the relevance of each component of our framework to show that the combination of both the constraints and the training of the adjacency matrix leads to optimal performances.

Error analysis of finite difference scheme for American option pricing under regime-switching with jumps

This paper mainly focuses on evaluating American options under regime-switching jump-diffusion models (Merton’s and Kou’s models). An efficient numerical method is designed for the concerned problems. The problem of American option pricing under regime-switching jump-diffusion models can be described as a free-boundary problem or a complementarity problem with integral and differential terms on an unbounded domain. By analyzing the relation of optimal exercise boundaries among several options, we truncate the solving domain of regime-switching jump-diffusion options, and present reasonable boundary conditions. For the integral terms of the truncated model, a composite trapezoidal formula is applied, which guarantees that the integral discretized matrix is a Toeplitz matrix. Meanwhile, a finite difference scheme is proposed for the resulting system, which leads to a linear complementary problem (LCP) with a unique solution. Moreover, we also prove the stability, monotonicity, and consistency of the discretization scheme and estimate the convergence order. In consideration of the characteristics of the discrete matrix, a projection and contraction method is suggested to solve the discretized LCP. Numerical experiments are carried out to verify the efficiency of the proposed scheme.

Efficiency of Some Predictor–Corrector Methods with Fourth-Order Compact Scheme for a System of Free Boundary Options

The trade-off between numerical accuracy and computational cost is always an important factor to consider when pricing options numerically, due to the inherent irregularity and existence of non-linearity in many models. In this work, we first present fast and accurate (1,2) and (2,2) predictor–corrector methods with a fourth-order compact finite difference scheme for pricing coupled system of the non-linear free boundary option pricing problem consisting of the option value and delta sensitivity. To predict the optimal exercise boundary, we set up a high-order boundary scheme, which is strategically derived using a combination of the fourth-order Robin boundary scheme and the fourth-order compact finite difference scheme near boundary. Furthermore, we implement a three-step high-order correction scheme for computing interior values of the option value and delta sensitivity. The discrete matrix system of this correction scheme has a tri-diagonal structure and strictly diagonal dominance. This nice feature allows for the implementation of the Thomas algorithm, thereby enabling fast computation. The optimal exercise boundary value is also corrected in each of the three correction steps with the derived Robin boundary scheme. Our implementations are fast on both coarse and very refined grids and provide highly accurate numerical approximations. Moreover, we recover a reasonable convergence rate. Further extensions to high-order predictor two-step corrector schemes are elaborated.

Optimizing Strategy of Computer Teaching Mode in Universities Based on Continuous Discrete Method

Abstract To improve the teaching effect of computer courses, make students master computer knowledge, and improve computer operation skills, this paper proposes a computer teaching model of continuous, discrete methods in colleges and universities. This paper investigates the teaching mode of traditional computer teaching methods in colleges and universities and proposes a continuous, discrete computer teaching model for the drawbacks of traditional computer teaching methods. The discrete learning teaching matrix and teaching process are constructed using the information entropy-based rough set discretization algorithm. Moreover, the teaching effects are compared with traditional methods regarding students’ learning interests, homework completion, and test scores. In terms of homework completion rate: the correct homework rate of the class using the discrete learning method was 20% higher than that of the class using the traditional teaching method. Regarding skill and overall test scores, the number of students who excelled in the classes using discrete learning reached about 50%, 20% more than the conventional teaching method. 20% fewer students scored below 60 than in the classes using the conventional teaching method. Moreover, after implementing discrete learning concept teaching, students’ interest in learning increased by more than 40%. Therefore, it can be seen that the teaching mode of the discrete learning concept is beneficial to the teaching of computer courses.

Rank-r Discrete Matrix Factorization for Anchor Graph Clustering

Considering many graph clustering methods are with quadratic or cubic time complexity and need post-processing to obtain the discrete solution. Combining with the anchor graph, we study a novel graph clustering model called Rank-<inline-formula><tex-math notation="LaTeX">$r$</tex-math></inline-formula> Discrete Matrix Factorization (DMF-RR), which is linear time complexity, and motivated by nonnegative matrix factorization (NMF). Instead of constraining the factor matrices of NMF to be nonnegative as many existed methods, we constrain them to indicator matrices. Thus, DMF-RR can obtain the discrete solution by directly solving the original problem without post-processing. Furthermore, considering the greater similarity between samples of the same category, an anchor graph is constructed as an input to capture essential clustering structure by utilizing the duality information between samples and anchors. Subsequently, an efficient and simple algorithm is proposed due to the nature of indicator matrices. Extensive experiments performed on synthetic and real-world datasets demonstrate the superiority of DMF-RR.

Efficient Top-K Feature Selection Using Coordinate Descent Method

Sparse learning based feature selection has been widely investigated in recent years. In this study, we focus on the l2,0-norm based feature selection, which is effective for exact top-k feature selection but challenging to optimize. To solve the general l2,0-norm constrained problems, we novelly develop a parameter-free optimization framework based on the coordinate descend (CD) method, termed CD-LSR. Specifically, we devise a skillful conversion from the original problem to solving one continuous matrix and one discrete selection matrix. Then the nontrivial l2,0-norm constraint can be solved efficiently by solving the selection matrix with CD method. We impose the l2,0-norm on a vanilla least square regression (LSR) model for feature selection and optimize it with CD-LSR. Extensive experiments exhibit the efficiency of CD-LSR, as well as the discrimination ability of l2,0-norm to identify informative features. More importantly, the versatility of CD-LSR facilitates the applications of the l2,0-norm in more sophisticated models. Based on the competitive performance of l2,0-norm on the baseline LSR model, the satisfactory performance of its applications is reasonably expected. The source MATLAB code are available at: https://github.com/solerxl/Code_For_AAAI_2023.

A gradient set matching pursuit algorithm with anti-base mismatch ability for sparse reconstruction under low SNR

This paper is concerned with designing an effective algorithm for recovering mismatched sparse signals from noise underdetermined measurements. The sparse recovery problem is formulated as the gradient optimization model of the residual cost function in a multidimensional continuous parameter space. The mismatched atoms of the discrete dictionary matrix are recovered as the accurate target sparse coefficient by solving a gradient descent algorithm with a hyperparameters joint adaptive adjustment function. To estimate the target parameters of frequency agile radar, we propose a gradient set matching pursuit algorithm. The algorithm uses the prior knowledge of resolution of the two-dimensional ambiguity function to build a discrete dictionary and obtains the support set according to the orthogonal matching pursuit principle. We divide the gradient set space in the neighborhood of the support set and calculate the mean gradient as the judgment direction and iteration step of the initial optimization process to overcome the residual function distortion and gradient error caused by random noise. The iterative convergence process of the proposed algorithm in the support set vector space is more robust, and the problem of falling into local optimum is avoided. Compared with the conventional sparse recovery algorithm and gradient descent algorithm, the proposed algorithm boosts its reconstruction performance of target parameters in the scenario of low SNR and mismatch of multiple scattering points. Simulation results show the effectiveness of the proposed algorithm.

Unified One-Step Multi-View Spectral Clustering

Multi-view spectral clustering, which exploits the complementary information among graphs of diverse views to obtain superior clustering results, has attracted intensive attention recently. However, most existing multi-view spectral clustering methods obtain the clustering partitions in a two-step scheme, i.e., spectral embedding and subsequent k-means. This two-step scheme inevitably seeks sub-optimal clustering results due to the information loss during the two-steps processes. Besides, existing multi-view spectral clustering methods do not jointly utilize the information of graphs and embedding matrices, which also degrades final clustering results. To solve these issues, we propose a unified one-step multi-view spectral clustering method, which integrates the spectral embedding and <formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex>$k$</tex></formula> -means into a unified framework to obtain discrete clustering labels with a one-step strategy. Under the observation that the inner product of the embedding matrix is a low-rank approximation of the graph, we combine graphs and embedding matrices of different views to obtain a unified graph. Then, we directly capture the discrete clustering indicator matrix from the unified graph. Furthermore, we design an effective optimization algorithm to solve the resultant problem. Finally, a set of experiments on various datasets are conducted to verify the effectiveness of the proposed method. The demo code of this work is publicly available at https://github.com/guanyuezhen/UOMvSC.

Multitasking neural network to jointly map discrete fracture structures and matrix transmissivity by inverting hydraulic data acquired in 2D fractured aquifers. XNET-fracture

In this paper, we propose a new approach to simultaneously map the fracture/conduit network and the equivalent transmissivity of the rock matrix from hydraulic head measurements acquired during pumping/injection tests in fractured aquifers. The algorithm relies on the use of a multitask neural network to directly approximate the joint inversion operator that links hydraulic head data to aquifer hydraulic properties. In which, the hydraulic head responses are used as input data while the output consists of the fracture structure and matrix transmissivity field. The multitasking is formed by fusing two single-task neural networks, both based on convolutional encoder-decoder architectures, one of which processes the fracture map inversion and the other the transmissivity map. Training neural network then relies on a large database, which consists of thousands of synthetic aquifers characterized by the presence of the fracture/conduit network and a heterogeneous transmissivity field attributed to the rock matrix, both randomly generated. In these aquifer models, the groundwater flow equation is solved in a discrete-continuum concept to numerically simulate the hydraulic head responses associated with sequential pumping/injection tests.Multitasking succeeds in reconstructing the fracture architecture and matrix transmissivity field with higher accuracy in comparison to conventional single-task networks. The difference is due to the transfer mechanism involved in the multi-task network, where the information shared between the tasks effectively enhances the accuracy of both reconstructions. However, as with other inversion techniques, the result accuracy depends on the quality and quantity of the hydraulic head data used in the inversion, as well as the prior information involved in featuring the training models.

Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.

A new ZNN model for finding discrete time-variant matrix square root: From model design to parameter analysis

In recent years, artificial neural network technology has developed very rapidly. More and more experts and scholars use neural network technology to solve the related problems of applied mathematics. As a representative of the Hopfield neural network, zeroing neural network (ZNN) which can deal with the time-variant problem that may not be solved by traditional methods effectively. In this paper, to solve the problem of time-variant matrix square root, the continuous-time ZNN (CT-ZNN) model is presented based on the ZNN design formula. Next, to meet the requirements of computer simulation, the general square-pattern discretization (SPD) formula is applied to obtain the general discrete-time ZNN (G-DT-ZNN) model. Moreover, by changing the value of the selected parameter in the G-DT-ZNN model, it is presented that the effect of different values of the selected parameter on the precision of the G-DT-ZNN model. Finally, in the numerical experiments, three discrete time-variant matrix square root problems are presented as examples for verifying such theoretical results of the proposed G-DT-ZNN model.

Learning ability analysis for linear discrete delay systems with iteration-varying trial length

This study investigates finite-time tracking issues of linear discrete delay systems with iteration-varying trial length (IVTL). With the assistance of explicit solutions expressed by the discrete matrix delayed exponential emA⋅, the obstacle in explicitly expressing the tracking error is overcome. Iterative learning control (ILC) update laws using the current state feedback control are proposed to reduce greatly acute error jitter caused by IVTL. The convergence conditions are presented for realizable systems and learning ability of the delay systems. The relation between ‖emA⋅‖1 and probability distribution of the trial length is given to imply how to guarantee convergence of tracking error. Finally, a numerical simulation is presented to illustrate that the ILC scheme can improve transient tracking performance. Moreover, convergence speed of the delay systems can be expedited.

Solving discrete first-order matrix linear control problems with general parametric uncertainties: A probability-density-based approach

This paper deals with the probabilistic analysis of discrete first-order linear control models with uncertainties. For the sake of generality in our stochastic analysis, we assume that all model parameters (the initial and final states, the matrix containing the free dynamics part, and the control’s coefficient) are random variables with an arbitrary joint probability density function. We then combine some results from classical Control Theory with Probability Theory to obtain, under very general hypotheses, the first probability density function of the control and the solution, which are parametric stochastic processes. To illustrate our theoretical findings, we also show two numerical examples and a classical discrete macroeconomic model whose parameters are treated as random variables.

Joint discrete and continuous matrix distribution modeling

In this paper, we introduce a bivariate distribution on arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.

Analysis of a nonuniform guiding structure by the adaptive finite‐difference and singular value decomposition methods

AbstractThis paper presents a flexible finite‐difference technique for analyzing the nonuniform guiding structures. Because the voltage and current variations along the nonuniform structure differ for each segment, this work considers the adaptable discretization steps. This technique increases the accuracy of the final response. Moreover, by applying the singular value decomposition and discarding the nonprincipal singular values, an optimal lower rank approximation of the discretization matrix is obtained. The computational cost of the introduced method is significantly reduced using the optimal discretization matrix. Also, the proposed method can be extended to the nonuniform waveguides. The technique is verified by analyzing several practical transmission lines and waveguides with nonuniform profiles.

Static and Dynamic Compliance Analyses of Curved-Axis Flexure Hinges: A Discrete Beam Transfer Matrix

Abstract A discrete beam transfer matrix method is introduced to enhance the existing approaches for the static and dynamic compliance solutions of curved-axis flexure hinges with variable curvatures and nonuniform profiles. An idea of discretizing curved-axis flexure hinges as a series of constant beam segments parallel to the centroidal axis is developed. As a result, only a concise beam transfer matrix with decoupled longitudinal and transverse components is needed to establish the compliance model. A step-by-step modeling procedure with simple formulas is provided as well qualifying for curved-axis and folded hinges. With this modeling idea, the small-deflection compliance matrix in the common sense of statics and particularly in a viewpoint of frequency-dependent dynamics can be simultaneously obtained. A typical curved-axis flexure hinge available in the literature is analyzed and compared as a study case. In addition, the static and dynamic design for a compliant guiding mechanism composed of folded flexure hinges is efficiently implemented with the presented method.

EDMH: Efficient discrete matrix factorization hashing for multi-modal similarity retrieval

Hashing has been an emerging topic and has recently attracted widespread attention in multi-modal similarity search applications. However, most existing approaches rely on relaxation schemes to generate binary codes, leading to large quantization errors. In addition, amounts of existing approaches embed labels into the pairwise similarity matrix, leading to expensive time and space costs and losing category information. To address these issues, we propose an Efficient Discrete Matrix factorization Hashing (EDMH). Specifically, EDMH first learns the latent subspaces for individual modality through matrix factorization strategy, which preserves the semantic structure representation information of each modality. In particular, we develop a semantic label offset embedding learning strategy, improving the stability of label embedding regression. Furthermore, we design an efficient discrete optimization scheme to generate compact binary codes discretely. Eventually, we present two efficient learning strategies EDMH-L and EDMH-S to pursue high-quality hash functions. Extensive experiments on various widely-used databases verify that the proposed algorithms produce significant performance and outperform some state-of-the-art approaches, with an average improvement of 2.50% (for Wiki), 2.66% (for MIRFlickr) and 2.25% (for NUS-WIDE) over the best available results, respectively.