Block Version Research Articles

Блочное исполнение дополнительной защиты электродвигателей погрузчика-зернометателя

Новые технологии, Химическая промышленность, Химическая технология, Научные журналы, Защита от коррозии, Технология металлов, Свойства материалов, Технические журналы, Наука и технологии, Справочная литература, Научные разработки

Block Gram-Schmidt algorithms and their stability properties

Block Gram-Schmidt algorithms serve as essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This work provides a comprehensive categorization of block Gram-Schmidt algorithms, particularly those used in Krylov subspace methods to build orthonormal bases one block vector at a time. Known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. Additionally, new block versions of low-synchronization variants are derived, and their efficacy and stability are demonstrated for a wide range of challenging examples. Numerical examples are computed with a versatile Matlab package hosted at https://github.com/katlund/BlockStab, and scripts for reproducing all results in the paper are provided. Block Gram-Schmidt implementations in popular software packages are discussed, along with a number of open problems. An appendix containing all algorithms type-set in a uniform fashion is provided.

Greedy Motzkin–Kaczmarz methods for solving linear systems

AbstractThe famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy randomized Motzkin–Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. The computational complexities of the proposed methods are also given. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices and some realistic practical problems, and their block versions have almost the same performance for most of cases.

GPU Preconditioning for Block Linear Systems Using Block Incomplete Sparse Approximate Inverses

Solving sparse triangular systems is the building block for incomplete LU- (ILU-) based preconditioning, but parallel algorithms, such as the level-scheduling scheme, are sometimes limited by available parallelism extracted from the sparsity pattern. In this study, the block version of the incomplete sparse approximate inverses (ISAI) algorithm is studied, and the block-ISAI is considered for preconditioning by proposing an efficient algorithm and implementation on graphical processing unit (GPU) accelerators. Performance comparisons are carried out between the proposed algorithm and serial and parallel block triangular solvers from PETSc and cuSPARSE libraries. The experimental results show that GMRES (30) with the proposed block-ISAI preconditioning achieves accelerations 1.4 × –6.9 × speedups over that using the cuSPARSE library on NVIDIA Tesla V100 GPU.

On convergence to eigenvalues and eigenvectors in the block-Jacobi EVD algorithm with dynamic ordering

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A(k) converge to zero. However, this fact alone does not necessarily guarantee that A(k) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q(k). We prove that under certain assumptions A(k) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q(k) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q(k) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example.

An algorithm for embedding information in digital images based on discrete wavelet transform and learning automata

The paper presents a new algorithm for embedding information in the frequency domain of the discrete wavelet-transform (DWT) of digital images. A block version of quantization index modulation (QIM) is used as a basic embedding operation. A distinctive feature of the algorithm consists in the adaptive selection of the data block size depending on the local properties of the cover image. It has been shown experimentally that for image areas containing a larger number of edge pixels, it is necessary to select blocks of greater length in the corresponding frequency domains. In addition, the problem of optimization the distortions in the blocks of DWT coefficients is substantiated and solved in order to improve the quality of embedding. A computing model of learning automata is used to solve this problem. The advantage of the obtained algorithm is that the receiver of the stego-image does not need additional information to extract the embedded message. The algorithm is highly efficient in terms of the main criteria of embedding quality and can be used both for embedding digital watermarks and arbitrary messages.

The Stability of Block Variants of Classical Gram--Schmidt

The block version of the classical Gram--Schmidt (\tt BCGS) method is often employed to efficiently compute orthogonal bases for Krylov subspace methods and eigenvalue solvers, but a rigorous proof...

Parallel and Communication Avoiding Least Angle Regression

We are interested in parallelizing the least angle regression (LARS) algorithm for fitting linear regression models to high-dimensional data. We consider two parallel and communication avoiding versions of the basic LARS algorithm. The two algorithms have different asymptotic costs and practical performance. One offers more speedup and the other produces more accurate output. The first is bLARS, a block version of the LARS algorithm, where we update $b$ columns at each iteration. Assuming that the data are row-partitioned, bLARS reduces the number of arithmetic operations, latency, and bandwidth by a factor of $b$. The second is tournament-bLARS (T-bLARS), a tournament version of LARS where processors compete by running several LARS computations in parallel to choose $b$ new columns to be added in the solution. Assuming that the data are column-partitioned, T-bLARS reduces latency by a factor of $b$. Similarly to LARS, our proposed methods generate a sequence of linear models. We present extensive numerical experiments that illustrate speedups up to 4x compared to LARS without any compromise in solution quality.

A New Algorithm for Solving Large-Scale Generalized Eigenvalue Problem Based on Projection Methods

In this paper, we consider four methods for determining certain eigenvalues and corresponding eigenvectors of large-scale generalized eigenvalue problems which are located in a certain region. In these methods, a small pencil that contains only the desired eigenvalue is derived using moments that have obtained via numerical integration. Our purpose is to improve the numerical stability of the moment-based method and compare its stability with three other methods. Numerical examples show that the block version of the moment-based (SS) method with the Rayleigh–Ritz procedure has higher numerical stability than respect to other methods.

Evaluating the soft error sensitivity of a GPU-based SoC for matrix multiplication

System-on-Chip (SoC) devices can be composed of low-power multicore processors combined with a small graphics accelerator (or GPU) which offers a trade-off between computational capacity and low-power consumption. In this work we use the LLFI-GPU fault injection tool on one of these devices to compare the sensitivity to soft errors of two different CUDA versions of matrix multiplication benchmark. Specifically, we perform fault injection campaigns on a Jetson TK1 development kit, a board equipped with a SoC including an NVIDIA “Kepler” Graphics Processing Unit (GPU). We evaluate the effect of modifying the size of the problem and also the thread-block size on the behaviour of the algorithms. Our results show that the block version of the matrix multiplication benchmark that leverages the shared memory of the GPU is not only faster than the element-wise version, but it is also much more resilient to soft errors. We also use the cuda-gdb debugger to analyze the main causes of the crashes in the code due to soft errors. Our experiments show that most of the errors are due to accesses to invalid positions of the different memories of the GPU, which causes that the block version suffers a higher percentage of this kind of errors.

Vecchia Approximations of Gaussian-Process Predictions

Gaussian processes are popular and flexible models for spatial, temporal, and functional data, but they are computationally infeasible for large datasets. We discuss Gaussian-process approximations that use basis functions at multiple resolutions to achieve fast inference and that can (approximately) represent any spatial covariance structure. We consider two special cases of this multi-resolution-approximation framework, a taper version and a domain-partitioning (block) version. We describe theoretical properties and inference procedures, and study the computational complexity of the methods. Numerical comparisons and an application to satellite data are also provided.

A Block Version of Orthogonal Rotations for Improving the Accuracy of Hybrid State Estimators

This paper addresses the problem of hybrid state estimation, which relies on data provided by both conventional SCADA systems and phasor measurement units. A previously proposed cascaded architecture is employed, whose first estimation module processes the asynchronous measurements through a conventional, nonlinear state estimator. The focus, however, is on the second estimation module, where the SCADA-based estimates are combined with synchrophasor data in order to enhance the quality of the state estimates. For the benefit of computational performance, that task must be performed by a non-iterative and numerically robust algorithm. It is equally important to avoid awkward uncorrelatedness assumptions while defining measurement weighting factors, since they may severely compromise the statistical characterization of the final solution. To accomplish both objectives, a generalized $2\times 2$ block version of the 3-multiplier Givens rotations has been developed and is introduced in this paper. This leads to numerically stable solutions, whose statistical properties are preserved. The features and advantages of the proposed two-stage hybrid state estimator are illustrated through case studies conducted on the IEEE 14-bus, 30-bus, 57-bus, and 300-bus test systems.

Truncated model reduction methods for linear time-invariant systems via eigenvalue computation

This paper provides three model reduction methods for linear time-invariant systems in the view of the Riemannian Newton method and the Jacobi-Davidson method. First, the computation of Hankel singular values is converted into the linear eigenproblem by the similarity transformation. The Riemannian Newton method is used to establish the model reduction method. Besides, we introduce the Jacobi-Davidson method with the block version for the linear eigenproblem and present the corresponding model reduction method, which can be seen as an acceleration of the former method. Both the resulting reduced systems can be equivalent to the reduced system originating from a balancing transformation. Then, the computation of Hankel singular values is transformed into the generalized eigenproblem. The Jacobi-Davidson method is employed to establish the model reduction method, which can also lead to the reduced system equivalent to that resulting from a balancing transformation. This method can also be regarded as an acceleration of a Riemannian Newton method. Moreover, the application for model reduction of nonlinear systems with inhomogeneous conditions is also investigated.

The Block Rational Arnoldi Method

The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions asso...

On-The-Fly Prediction of Macroscopic Cross-Section Spatial Response to TH Perturbations Through Transfer Functions: Theory and First Results

Monte Carlo (MC) codes are widely used for the accurate modeling of nuclear reactors. However, efficient inclusion of thermal-hydraulic (TH) feedback within the MC calculation sequence is still an open problem. The issue is emphasized when coupled MC-TH calculations are needed to model the burnup evolution using multiple depletion steps. Among the techniques proposed to solve this problem is the utilization of stabilized Picard iteration in conjunction with a low-order prediction step. The latter is composed of a prediction block for cross sections and a fast deterministic solver that uses the cross sections to obtain a prediction of the power profile. The predicted power is then used as an improved guess for the next MC calculation, therefore leading to faster convergence for the overall algorithm. In this paper, we propose a new prediction block in which one-group cross sections are calculated through convolution of the TH scalar fields with MC-generated generalized transfer functions (GTFs). First-order perturbation theory is then utilized to calculate the power profile from the updated cross sections. A version of this prediction block using a simple fast Fourier transform–based approximation of the GTF is tested against a boiling water reactor unit-cell with realistic density profile and axial reflectors. The analysis was limited to the feedback between neutronics and coolant density variation. Good agreement was observed for both the spatial power and the one-group macroscopic cross-section profiles, which were compared to the reference MC results. This agreement was also preserved near the boundary, where the spatial flux gradients are maximum due to proximity to the axial reflectors.

Block Versions of R functions in 'generalCorr' for Generalized Correlations and Causal Paths

Karl Pearson developed the correlation coefficient r(X,Y) in 1890s vastly underestimates dependence between two series. Vinod(2014} develops new generalized correlation coefficients so that when r*(Y|X) > r*(X|Y) then X is the cause'' of Y. Vinod (2015) reports simulations favoring kernel causality. An R software package called 'generalCorr' (at r-project.org) computes generalized correlations, partial correlations, and plausible causal paths. This short paper describes the block versions of various R functions newly added to the 'generalCorr' package in October 2019. Newly published Vinod (2019) has the latest rendering of the theory behind causal paths including theorems with proofs. The function 'causeSummBlk(.)' is recommended for practitioners.

Mutual Information of Wireless Channels and Block-Jacobi Ergodic Operators

Shannon's mutual information of a random multiple antenna and multipath channel is studied in the general case where the channel impulse response is an ergodic and stationary process which is assumed to be available at the receiver. From this viewpoint, the channel is represented by an ergodic self-adjoint block-Jacobi operator, which is close in many aspects to a block version of a random Schr\"odinger operator. The mutual information is then related to the so-called density of states of this operator. In this paper, it is shown that under the weakest assumptions on the channel, the mutual information can be expressed in terms of a matrix-valued stochastic process coupled with the channel process. This allows numerical approximations of the mutual information in this general setting. Moreover, assuming further that the channel impulse response is a Markov process, a representation for the mutual information offset in the large Signal to Noise Ratio regime is obtained in terms of another related Markov process. This generalizes previous results from Levy~\emph{et.al.}. It is also illustrated how the mutual information expressions that are closely related to those predicted by the random matrix theory can be recovered in the large dimensional regime.

Video summarization via block sparse dictionary selection

The explosive growth of video data has raised new challenges for many video processing tasks such as video browsing and retrieval, hence, effective and efficient video summarization (VS) is urgently demanded to automatically summarize a video into a succinct version. Recent years have witnessed the advancements of sparse representation based approaches for VS. However, video frames are analyzed individually for keyframe selection in existing methods, which could lead to redundancy among selected keyframes and poor robustness to outlier frames. Due to that adjacent frames are visually similar, candidate keyframes often occur in temporal blocks, in addition to sparse presence. Therefore, in this paper, the block-sparsity of candidate keyframes is taken into consideration, by which the VS problem is formulated as a block sparse dictionary selection model. Moreover, a simultaneous block version of Orthogonal Matching Pursuit (SBOMP) algorithm is designed for model optimization. Two keyframe selection strategies are also explored for each block. Experimental results on two benchmark datasets, namely VSumm and TVSum datasets, demonstrate that the proposed SBOMP based VS method clearly outperforms several state-of-the-art sparse representation based methods in terms of F-score, redundancy among keyframes and robustness to outlier frames.

A class of multi-resolution approximations for large spatial datasets

Gaussian processes are popular and flexible models for spatial, temporal, and functional data, but they are computationally infeasible for large datasets. We discuss Gaussian-process approximations that use basis functions at multiple resolutions to achieve fast inference and that can (approximately) represent any spatial covariance structure. We consider two special cases of this multi-resolution-approximation framework, a taper version and a domain-partitioning (block) version. We describe theoretical properties and inference procedures, and study the computational complexity of the methods. Numerical comparisons and an application to satellite data are also provided.

Projection Methods in Krylov Subspaces

The paper considers preconditioned iterative methods in Krylov subspaces for solving large systems of linear algebraic equations with sparse coefficient matrices arising in solving multidimensional boundary-value problems by finite volume or finite element methods of different orders on unstructured grids. Block versions of the weighted Cimmino methods, based on various orthogonal and/or variational approaches and realizing preconditioning functions for two-level multi-preconditioned semi-conjugate residual algorithms with periodic restarts, are proposed. At the inner iterations between restarts, additional acceleration is achieved by applying deflation methods, providing low-rank approximations of the original matrix and playing the part of an additional preconditioner. At the outer level of the Krylov process, in order to compensate the convergence deceleration caused by restricting the number of the orthogonalized direction vectors, restarted approximations are corrected by using the least squares method. Scalable parallelization of the methods considered, based on domain decomposition, where the commonly used block Jacobi–Schwarz iterative processes is replaced by the block Cimmino–Schwarz algorithm, is discussed. Hybrid programming technologies for implementing different stages of the computational process on heterogeneous multi-processor systems with distributed and hierarchical shared memory are described.