Matrix Partitioning Method Research Articles

Optimization of Large-Scale Sparse Matrix-Vector Multiplication on Multi-GPU Systems

Sparse matrix-vector multiplication (SpMV) is one of the important kernels of many iterative algorithms for solving sparse linear systems. The limited storage and computational resources of individual GPUs restrict both the scale and speed of SpMV computing in problem-solving. As real-world engineering problems continue to increase in complexity, the imperative for collaborative execution of iterative solving algorithms across multiple GPUs is increasingly apparent. Although the multi-GPU-based SpMV takes less kernel execution time, it also introduces additional data transmission overhead, which diminishes the performance gains derived from parallelization across multi-GPUs. Based on the non-zero elements distribution characteristics of sparse matrices and the trade-off between redundant computations and data transfer overhead, this paper introduces a series of SpMV optimization techniques tailored for multi-GPU environments and effectively enhances the execution efficiency of iterative algorithms on multiple GPUs. Firstly, we propose a two-level non-zero elements-based matrix partitioning method to increase the overlap of kernel execution and data transmission. Then, considering the irregular non-zero elements distribution in sparse matrices, a long-row-aware matrix partitioning method is proposed to hide more data transmissions. Finally, an optimization using redundant and inexpensive short-row execution to exchange costly data transmission is proposed. Our experimental evaluation demonstrates that, compared with the SpMV on a single GPU, the proposed method achieves an average speedup of 2.00x and 1.85x on platforms equipped with two RTX 3090 and two Tesla V100-SXM2, respectively. The average speedup of 2.65x is achieved on a platform equipped with four Tesla V100-SXM2.

Numerical Methods for Calculating Component Modes for Geometric Mistuning Reduced-Order Models

AbstractGeometric mistuning models formulated from component mode synthesis methods often require the calculation of component modes, particularly constraint and fixed interface normal modes (FINMs), during substructuring. For integrally bladed rotors, these calculations are required for each sector. This paper proposes methods that reuse information garnered from solving the constraint modes of a single sector on the remaining sectors to reduce memory requirements and solution times. A mesh metamorphosis tool is used to ensure finite element models match geometry obtained from a three-dimensional optical scanner. This tool also produces a common mesh pattern from sector-to-sector. This is exploited to produce common permutation matrices and symbolic factorizations of sector stiffness matrices that are proposed for reuse in solving subsequent constraint modes. Furthermore, a drop tolerance is introduced to remove small values during constraint mode calculation to reduce memory requirements. It is proposed to reuse this dropping pattern produced from a single sector on the remaining sectors. Approaches are then extended to a parallel processing scheme to propose effective matrix partitioning methods. Finally, information gathered during the constraint mode calculations is reused during the solution of the FINMs to improve solution time. Results show reusing permutation matrices and symbolic factorizations from sector-to-sector improves solution time and introduces no error. Using a drop tolerance is shown to reduce storage requirements of a constraint mode matrix, while reusing the dropping pattern introduces minimal error. Similarly, reusing constraint mode information in calculating normal modes greatly improves the performance.

A Numerical Method for Flexural Vibration Band Gaps in A Phononic Crystal Beam with Locally Resonant Oscillators

The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.

An improved HASM method for dealing with large spatial data sets

Surface modeling with very large data sets is challenging. An efficient method for modeling massive data sets using the high accuracy surface modeling method (HASM) is proposed, and HASM_Big is developed to handle very large data sets. A large data set is defined here as a large spatial domain with high resolution leading to a linear equation with matrix dimensions of hundreds of thousands. An augmented system approach is employed to solve the equality-constrained least squares problem (LSE) produced in HASM_Big, and a block row action method is applied to solve the corresponding very large matrix equations. A matrix partitioning method is used to avoid information redundancy among each block and thereby accelerate the model. Experiments including numerical tests and real-world applications are used to compare the performances of HASM_Big with its previous version, HASM. Results show that the memory storage and computing speed of HASM_Big are better than those of HASM. It is found that the computational cost of HASM_Big is linearly scalable, even with massive data sets. In conclusion, HASM_Big provides a powerful tool for surface modeling, especially when there are millions or more computing grid cells.

PH/PH/1 Bulk Arrival and Bulk Service Queue

This paper studies two stochastic bulk arrivals and bulk services PH/PH/1 queue Models (A) and (B) with k1 and k2 as the number of phases of PH arrival and PH service distributions respectively. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes is greater than the maximum of the service sizes and the infinitesimal generator is partitioned as blocks of k1k2 times the maximum of the arrival sizes for analysis. In Model (B) the maximum of the arrival sizes is less than the maximum of the service sizes. The generator is partitioned using blocks of k1k2 times the maximum of the service sizes. Block circulant matrix structure is noticed in the basic system generator. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using matrix geometric methods. Numerical examples are presented for illustration.

M/M/1 Bulk Arrival and Bulk Service Queue with Randomly Varying Environment

This paper studies two stochastic bulk arrival and bulk service queueing Models (A) and (B) with k varying environments. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes in all environments is greater than the maximum of the service sizes in all environments and the infinitesimal generator is partitioned as blocks of k times the maximum of the arrival sizes for analysis and in Model (B) the maximum of the arrival sizes in all environments is less than the maximum of the service sizes in all environments where the generator is partitioned using blocks of k times the maximum of the service sizes. The basic system generator is seen as a block circulant matrix. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using the rate matrix iterated. Numerical examples are presented for illustration. Bulk arrival and bulk service M/PH/1 and PH/M/1 queues become special cases of the models when the PH generator is identified as the generator of the randomly varying environment.

High Performance Solvers for Implicit Particle in Cell Simulation

A three-dimensional implicit particle-in-cell (iPIC3D) method implemented by S. Markidis et. al. in [“Multiscale simulations of plasma with iPIC3D”, Mathematics and Computers in Simulation, 80(2010), 1509-1519] allows time steps at magnetohy- drodynamics time scale. The code requires the solution of two linear systems: a Poisson system related to divergence cleaning, and a system related to a second order formulation of Maxwell equation. In iPIC3D, the former is the most costly.To reduce the cost of solving the Poisson system, a parallel matrix assembly and partitioning method are implemented, and conjugate gradient and algebraic multigrid (AMG) solvers from the Hypre library are called. The scalability of AMG as a solver is studied for 1D and 3D partitionings and compared to that of CG.

Macro-elementwise preconditioning methods

Extremely large scale problems, modelled by partial differential equations, arise in various applications, and must be solved by properly preconditioned iterative methods. Frequently, the corresponding medium is heterogeneous. Recursively constructed two-by-two block matrix partitioning methods and elementwise constructed preconditioners for the arising pivot block and Schur complement matrices have turned out to be very efficient methods, and are analysed in this paper. Thereby special attention is paid to macroelementwise partitionings, which can be particularly efficient in the modelling of materials with large and narrow variations and can also provide efficient implementations on parallel computers.

Development of a design structure matrix partitioning method towards effective design collaboration

The difficulties in design collaboration stem from the managerial complexity in terms of the interactions and interdependencies among the respective development activities. This paper introduces a simple structural partitioning algorithm of design structure matrix, basing on dependency relationships among tasks via tracing the input tasks chain. The algorithm is composed of four modules, which are used to organise and transform design tasks into input model, cycle model, group model and execution ordering model correspondingly. Cycle matrix is created to partition the task set into individual groups that are with different interdependency levels. Coupled and unordered individual task set could be converted into uncoupled, ordered task clusters. The illustrative case studies on automotive industry has verified the efficiency of the proposed algorithm, which could partition the development processes into different execution clusters to improve the concurrence and collaboration of product development tasks.

A Matrix Partitioning Interface to PaToH in MATLAB

We present the PaToH MATLAB Matrix Partitioning Interface. The interface provides support for hypergraph-based sparse matrix partitioning methods which are used for efficient parallelization of sparse matrix–vector multiplication operations. The interface also offers tools for visualizing and measuring the quality of a given matrix partition. We propose a novel, multilevel, 2D coarsening-based 2D matrix partitioning method and implement it using the interface. We have performed extensive comparison of the proposed method against our implementation of orthogonal recursive bisection and fine-grain methods on a large set of publicly available test matrices. The conclusion of the experiments is that the new method can compete with the fine-grain method while also suggesting new research directions.

MPM: a hierarchical clustering algorithm using matrix partitioning method for non-numeric data

The Publisher regrets that the original article incorrectly listed the authors’ location as the “People’s Republic of China”. This was a typesetting error. Their correct location is the “Republic of China”, which is also known as Taiwan. Their full affiliation is Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan.

MPM: a hierarchical clustering algorithm using matrix partitioning method for non-numeric data

Clustering has been widely adopted in numerous applications, including pattern recognition, data analysis, image processing, and market research. When performing data mining, traditional clustering algorithms which use distance-based measurements to calculate the difference between data are unsuitable for non-numeric attributes such as nominal, Boolean, and categorical data. Applying an unsuitable similarity measurement in clustering may cause some valuable information embedded in the data attributes to be lost, and hence low quality clusters will be created. This paper proposes a novel hierarchical clustering algorithm, referred to as MPM, for the clustering of non-numeric data. The goals of MPM are to retain the data features of interest while effectively grouping data objects into clusters with high intra-similarity and low inter-similarity. MPM achieves these goals through two principal methods: (1) the adoption of a novel similarity measurement which has the ability to capture the "characterized properties" of information, and (2) the application of matrix permutation and matrix participation partitioning to the results of the similarity measurement (constructed in the form of a similarity matrix) in order to assign data to appropriate clusters. This study also proposes a heuristic-based algorithm, the Heuristic_MPM, to reduce the processing times required for matrix permutation and matrix partitioning, which together constitute the bulk of the total MPM execution time.

Numerical Study of Approximate Inverse Preconditioner for Two-Dimensional Engine Inlet Problems

The method of moments has been widely used in the analysis of electromagnetic scattering problems. Recently, significant advances in the development of fast and efficient techniques for solving large problems have been reported. In such methods, iterative matrix solvers are preferred by virtue of their speed and low memory requirements. But for near resonant and strong multiple scattering problems, e.g., involving an aircraft engine inlet, a large number of iterations is required for convergence. In this paper, an efficient approximate inverse based preconditioner is used to reducing this number of iterations. By using the matrix partitioning method, the computational cost for obtaining the approximate inverse is reduced to O(N). We apply this preconditioner to an O(N log N) algorithm, the multilevel fast multipole algorithm, for the aircraft engine inlet problem. The numerical results show the efficiency of this preconditioner.

Matrix partitioning methods for interior point algorithms

We consider here a linear programming problem whose rows of the constraint matrix can be partitioned into two parts. Such natural partitions exist in several special linear programs, including the assignment problem, the transportation problem, the generalized upper-bounded variable problem, the block diagonal linear program; and can also be used to induce sparsity patterns in Cholesky factorizations. In this paper, we propose a matrix partitioning method for interior point algorithms. The proposed method independently generates Cholesky factorizations of each part, and reduces the complexity to that of solving generally, a dense linear system involving several rank one updates of the identity matrix. Here, we propose solving this linear system by an inductive use of the Sherman-Morrison-Woodbury formula. The proposed method is easily implemented on a vector, parallel machine as well as on a distributed system. Such partitioning methods have been popular in the context of the simplex method, where the special structure of the basis matrix is exploited.

Compound matrix partitioning and modification for the solution of branched autotransformer traction feeds

The increasing use of autotransformer feed schemes in AC electrified railways, as a result of their obvious advantages, motivates a detailed analysis of the behaviour of these systems. A novel compound matrix partitioning and modification method for the solution of a branched autotransformer feed system is presented. The introduction of branch lines in the autotransformer feed model has stimulated the development of a new solution technique which is very well matched to the problem formation. Results are cited which indicate that the proposed method can achieve a very efficient and reliable solution.

Projection methods in flexible multibody dynamics. Part II: Dynamics and recursive projection methods

AbstractIn Part I of this paper the kinematic relationships between the absolute, elastic and joint accelerations are developed. In this paper, these kinematic equations are used with the generalized Newton‐Euler equations and the relationship between the actual and generalized reaction forces to develop a recursive projection algorithm for the dynamic analysis of open‐loop mechanical systems consisting of a set of interconnected rigid and deformable bodies. Optimal matrix permutation, partitioning and projection methods are used to eliminate the elastic accelerations while maintaining the inertia coupling between the rigid body motion and the elastic deformation. Recursive projection methods are then applied in order to project the inertia of the leaf bodies onto their parent bodies. This leads to an optimal symbolic factorization which recursively yields the absolute and joint accelerations, and the joint reaction forces. The method presented in this paper avoids the use of Newton‐Raphson algorithms in the numerical solution of the constrained dynamic equations of open‐loop kinematic chains since the joint accelerations are readily available from the solution of the resulting reduced system of equations. Furthermore, the method requires only the inversion or decomposition of relatively small matrices and the numerical integration of a minimum number of co‐ordinates. Open‐loop multibody robotic manipulator systems are used to compare the results and efficiency of the recursive methods with that of the augmented formulations that employ Newton‐Raphson algorithms.

A matrix partitioning approach to the calculation of intermolecular potentials. General theory and some examples

A general expression for the interaction energy of two molecules is obtained by using a matrix partitioning method. The wavefunction of the whole system is expanded in terms of antisymmetrized products of free-molecule functions and using a matrix perturbation scheme it is possible to describe the interaction energy in terms of free-molecule quantities (like frequency-dependent polarizabilities and density or spin density matrices) that, in principle, can be evaluated at any level of approximation. By way of example, the interaction potentials are calculated for (N 2) 2 and Ne 2 using wavefunctions of HF and TDHF form. The results are in substantial accord with those available in the literature. Application of these potentials to the calculation of macroscopic properties, however, leads to considerable errors. From the analysis of our results it appears that the dispersion energy is underestimated, probably on account of the neglect of intrasystem correlation energy in the TDHF approximation. The use of more sophisticated methods of evaluation of dynamic polarizabilities will not involve any extension of the approach presented in this work.

Efficient matrix partitioning for optical computing

Because optical computers will be designed to operate on matrices of a fixed size, we must ask how to use them on larger matrices. The matrix partitioning method shown here is simple (no bookkeeping), algorithm-independent, and efficient (making maximum use of all calculation units available).

Explicit Hilbert-space representations of atomic and molecular photoabsorption spectra: Computational studies of Stieltjes-Tchebycheff functions

Explicit Hilbert-space techniques are reported for construction of the discrete and continuum Schr\"odinger states required in atomic and molecular photoexcitation and/or photoionization studies. These developments extend and clarify previously described moment-theory methods for determinations of photoabsorption cross sections from discrete basis-set calculations to include explicit construction of underlying wave functions. The appropriate Stieltjes-Tchebycheff excitation and ionization functions of $n\mathrm{th}$ order are defined as Radau-type eigenstates of an appropriate operator in an $n$-term Cauchy-Lanczos basis. The energies of these states are the Radau quadrature points of the photoabsorption cross section, and their (reciprocal) norms provide the corresponding quadrature weights. Although finite-order Stieltjes-Tchebycheff functions are ${L}^{2}$ integrable, and do not have asymptotic spatial tails in the continuous spectrum, the Radau quadrature weights nevertheless provide information for normalization in the conventional Dirac $\ensuremath{\delta}$-function sense. Since one Radau point can be placed anywhere in the spectrum, appropriately normalized convergent approximations to any of the discrete or continuum Schr\"odinger states are obtained from the development. Connections with matrix partitioning methods are established, demonstrating that $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are optical-potential solutions of the matrix Schr\"odinger equation in the full Cauchy-Lanczos basis. The energies at which the $n\mathrm{th}$-order optical potential vanishes identically correspond to generalized Gaussian quadrature points, in which case the associated Stieltjes eigenfunctions provide an optimal ${L}^{2}$ representation of the Schr\"odinger spectrum of discrete and continuum wave functions. The spectral contributions of individual Schr\"odinger states to $n\mathrm{th}$-order Stieltjes-Tchebycheff functions are obtained in closed form, indicating the latter are spectrally localized in the neighborhoods of the Radau quadrature energies, and spatially localized in accordance with the extent of the target eigenstate. Illustrative studies of the dipole spectra in atomic and molecular hydrogen clarify the nature and convergence of the method. Finally, procedures are indicated for construction of photoemission anisotropies and for performing coupled-channel calculations employing Stieltjes-Tchebycheff functions, and descriptive intercomparisons are made of the present development with more conventional computational procedures.

Non-empirical molecular orbital theory of the electronic structure of molecular crystals

A non-empirical molecular orbital treatment of molecular crystals, based on SCF perturbation theory and matrix partitioning methods is presented.