Showing papers on "Matrix (mathematics) published in 2022"

An Online and Scalable Model for Generalized Sparse Nonnegative Matrix Factorization in Industrial Applications on Multi-GPU

Abstract: Generalized sparse nonnegative matrix factorization (SNMF) has been proven useful in extracting information and representing sparse data with various types of probabilistic distributions from industrial applications, e.g., recommender systems and social networks.However, current solution approaches for generalized SNMF are based on the manipulation of whole sparse matrices and factor matrices, which will result in large-scale intermediate data.Thus, these approaches cannot describe the high-dimensional and sparse matrices in mainstream industrial and big data platforms, e.g., graphics processing unit (GPU) and multi-GPU, in an online and scalable manner. To overcome these issues, an online, scalable, and single-thread-based SNMF for CUDA parallelization on GPU (CUSNMF) and multi-GPU (MCUSNMF) is proposed in this article. First, theoretical derivation is conducted, which demonstrates that the CUSNMF depends only on the products and sums of the involved feature tuples. Next, the compactness, which can follow the sparsity pattern of sparse matrices, endows the CUSNMF with online learning capability and the fine granularity gives it high parallelization potential on GPU and multi-GPU. Finally, the performance results on several real industrial datasets demonstrate the linear scalability of the time overhead and the space requirement and the validity of the extension to online learning. Moreover, CUSNMF obtains speedup of 7X on a P100 GPU compared to that of the state-of-the-art parallel approaches on a shared memory platform.

Risk Assessment of Water Supply Network Operation Based on ANP-Fuzzy Comprehensive Evaluation Method

Abstract: The water supply network is of great importance for a city, and it faces various risks such as leaks and breaks, so its risk level needs to be evaluated regularly. To propose a new and effi...

Inconsistency thresholds for incomplete pairwise comparison matrices

Abstract: ‘‘Mathematics is the part of physics where experiments are cheap.” 1 (Vladimir Igorevich Arnold: On teaching mathematics) Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.

A novel link prediction algorithm based on inductive matrix completion

Abstract: Link prediction refers to predicting the connection probability between two nodes in terms of existing observable network information, such as network structural topology and node properties. Although traditional similarity-based methods are simple and efficient, their generalization performance varies widely in different networks. In this paper, we propose a novel link prediction approach ICP based on inductive matrix completion, which recoveries node connection probability matrix by applying node features to a low-rank matrix. The approach first explores a comprehensive node feature representation by combining different structural topology information with node importance properties via feature construction and selection. The selected node features are then used as the input of a supervised learning task for solving the low-rank matrix. The node connection probability matrix is finally recovered by a bi-linear function, which predicts the connection probability between two nodes with their features and the low-rank matrix. In order to demonstrate the ICP superiority, we took eleven related efforts including two recent methods proposed in 2020 as baseline methods, and it is shown that ICP has stable performance and good universality in twelve different real networks. Compared with the baseline methods, the improvements of ICP in terms of the average AUC results are ranging from 3.81% ∼ 12.77% and its AUC performance is improved by 0.08% ∼ 3.54% compared with the best baseline method. The limitation of ICP lies in its high computational complexity due to the feature construction, but the complexity can be reduced by replacing complex features with node semantic attributes if there are additional data available. Moreover, it provides a potential link prediction solution for large-scale networks, since inductive matrix completion is a supervised learning task, in which the underlying low-rank matrix can be solved by representative nodes instead of all their nodes.

Sensitivity analysis to geometrical imperfections in shell buckling via a mixed generalized path-following method

Abstract: An efficient continuation method is proposed for a direct sensitivity analysis of the critical point to geometrical imperfections, e.g. expressed as combination of given shapes, for thin-walled structures prone to buckling. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. Remarkable novelties are achieved. Firstly, the Jacobian of the extended system is entirely computed analytically by means of a solid-shell model and a strain-based modeling of the geometrical deviation. Moreover, a mixed scheme with independent element-wise stress variables is devised for a more efficient and robust iterative solution compared to the standard null vector method. The mixed algorithm speeds up the sensitivity analysis allowing larger imperfection steps and much fewer factorizations of the condensed stiffness matrix, only one per imperfection in its modified version with constant matrix over the step. Finally, the critical point derivatives with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection.

A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations

Abstract: In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

Grid Topology Identification With Hidden Nodes via Structured Norm Minimization

Abstract: This letter studies a topology identification problem for an electric distribution grid using sign patterns of the inverse covariance matrix of bus voltage magnitudes and angles, while accounting for hidden buses. Assuming the grid topology is sparse and the number of hidden buses are fewer than those of the observed buses, we express the observed voltages inverse covariance matrix as the sum of three structured matrices: sparse matrix, low-rank matrix with sparse factors , and low-rank matrix . Using the sign patterns of the first two of these matrices, we develop an algorithm to identify the topology of a distribution grid with a minimum cycle length greater than three. To estimate the structured matrices from the empirical inverse covariance matrix, we formulate a novel convex optimization problem with appropriate sparsity and structured norm constraints and solve it using an alternating minimization method. We validate the proposed algorithm’s performance on a modified IEEE 33 bus system.

Reducing conservatism in robust stability analysis of fractional-order-polytopic systems.

Abstract: This paper studies the robust stability of the fractional-order (FO) LTI systems with polytopic uncertainty. Generally, the characteristic polynomial of the system dynamic matrix is not an affine function of the uncertain parameters. Consequently, the robust stability of the uncertain system cannot be evaluated by well-known approaches including LMIs or exposed edges theorem. Here, an over-parameterization technique is developed to convert the main characteristic polynomial into a set of local over-parameterized characteristic polynomials (LOPCPs). It is proved that the robust stability of LOPCPs implies the robust stability of the uncertain system. Then, an algorithm is proposed to explore system's robust stability through investigating the robust stability of these LOPCPs based on the exposed edges idea. For the sake of feasibility comparison, extensive examples are elaborated that reveal the superiority of the proposed algorithm.

Computational Approach via Half-Sweep and Preconditioned AOR for Fractional Diffusion

Abstract: Solving time-fractional diffusion equation using a numerical method has become a research trend nowadays since analytical approaches are quite limited. There is increasing usage of the finite difference method, but the efficiency of the scheme still needs to be explored. A half-sweep finite difference scheme is well-known as a computational complexity reduction approach. Therefore, the present paper applied an unconditionally stable half-sweep finite difference scheme to solve the time-fractional diffusion equation in a one-dimensional model. Throughout this paper, a Caputo fractional operator is used to substitute the time-fractional derivative term approximately. Then, the stability of the difference scheme combining the half-sweep finite difference for spatial discretization and Caputo time-fractional derivative is analyzed for its compatibility. From the formulated half-sweep Caputo approximation to the time-fractional diffusion equation, a linear system corresponds to the equation contains a large and sparse coefficient matrix that needs to be solved efficiently. We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated over relaxation (PAOR) algorithm to achieve a high convergence solution. The convergence of the developed method is analyzed. Finally, some numerical experiments from our research are given to illustrate the efficiency of our computational approach to solve the proposed problems of time-fractional diffusion. The combination of a half-sweep finite difference scheme and PAOR algorithm can be a good alternative computational approach to solve the time-fractional diffusion equation-based mathematical physics model.