QR Algorithm Research Articles

QZ algorithm with two‐sided generalized Rayleigh quotient shifts

AbstractWe generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift.

Spatiotemporal optimization of groundwater monitoring networks using data-driven sparse sensing methods

Abstract. Groundwater monitoring and specific collection of data on the spatiotemporal dynamics of the aquifer are prerequisites for effective groundwater management and determine nearly all downstream management decisions. An optimally designed groundwater monitoring network (GMN) will provide the maximum information content at the minimum cost (Pareto optimum). In this study, PySensors, a Python package containing scalable, data-driven algorithms for sparse sensor selection and signal reconstruction with dimensionality reduction is applied to an existing GMN in 1D (hydrographs) and 2D (gridded groundwater contour maps). The algorithm first fits a basis object to the training data and then applies a computationally efficient QR algorithm that ranks existing monitoring wells (for 1D) or suitable sites for additional monitoring (for 2D) in order of importance, based on the state reconstruction of this tailored basis. This procedure enables a network to be reduced or extended along the Pareto front. Moreover, we investigate the effect of basis choice on reconstruction performance by comparing three types typically used for sparse sensor selection (i.e., identity, random projection, and SVD, respectively, PCA). We define a gridded cost function for the extension case that penalizes unsuitable locations. Our results show that the proposed approach performs better than the best randomly selected wells. The optimized reduction makes it possible to adequately reconstruct the removed hydrographs with a highly reduced subset with low loss. With a GMN reduced by 94 %, an average absolute reconstruction accuracy of 0.1 m is achieved, in addition to 0.05 m with a reduction by 69 % and 0.01 m with 18 %.

Modelling multiple quantiles together with the mean based on SA‐ConvLSTM for taxi pick‐up prediction

A more complete taxi pick-ups prediction going beyond the conditional expectation would be more beneficial to allocate taxis effectively. Prior works have predicted the conditional expectation of taxi demand and improved the prediction accuracy. But the ability to convey calibrated uncertainty estimates in prediction was less studied. Here, a multi-output multi-quantile deep neural network architecture is presented for predicting the mean and the multiple quantiles simultaneously, which jointly models multiple quantiles together with the conditional expectation based on self-attention ConvLSTM. Using the NYC taxi dataset, we have analyzed the performance of the developed architecture. For the mean prediction, the proposed method has an error of 8.584 in terms of RMSE criteria and Linear QR, Mcdropout-Gal, ConvLSTM and SA-ConvLSTM methods, which have error values of 8.800, 8.671, 8.599, and 8.597, respectively. According to the MAE criterion, this method has a value of 5.934. The other methods have obtained values of 6.614, 5.990, 5.986, and 5.959, respectively. For multiple quantiles prediction, the proposed method produces narrower intervals while obtaining the target coverage percentage. Overall, the proposed model not only outperforms the other models, but also provides a rich description about the predictive density.

A novel hybrid model for flow image segmentation and bubble pattern extraction

Direct imaging method has been proved to be quite convenient and importance for measuring bubble shape parameters of gas–liquid two-phase flow due to its advantages of non-contact and visualization. The accuracy of flow pattern segmentation and measurement has an importantly impact on flow characteristic. For bubble recognition (pattern segmentation and extraction) with low-quality image, the aim of this paper is to propose a novel hybrid image analysis model (U-net-QR-EMD) combining the U-net algorithm with the quantile regression method (QR) and empirical mode decomposition (EMD). For a direct-contact gas–liquid mixing system, both the synthetic and experimental images of bubble swarm are respectively captured and used to verify the feasibility of the new model. The main findings of our study are as follows. (1) Computing and experimental results show that it is effective in the recognition of bubbles with irregular shape and have a potential in recognizing other bubbles taken under different working conditions. The number of iterations with the proposed model is 90 in the training process, leading to the prediction accuracy with more than 90%. Specifically, it can accurately and reliably identify the individual small and large bubbles in a direct-contact gas–liquid mixing system, but the classical Otsu and K-means methods cannot. (2) Using QR method, it is found that with the increase of gas–liquid mixing time, the number of the identified bubbles decreases but the gas holdup increases. Meanwhile, 90% of the bubble number are between the 5% and 95% quantile regression curves. Both the time series of bubble number and gas holdup tend to be non-stationary and non-linear, but intrinsic mode function (IMF) components and residual obtained by EMD display a higher stability and stronger periodicity. (3) The six parameters (area, perimeter, deformation coefficient, equivalent diameter, aspect ratio and eccentricity) related to bubble patterns can be measured and agree with each other in the evolution process of the mixing state quality with the help of the new model. The proposed model can not only successfully identify the micro-details of small-scale bubbles in low-quality images of gas–liquid two-phase flow, but also improve the ability to obtain low-quality deep processing of other two-phase flow taking advantage of direct imaging method.

Quantile regression analysis of the association between parental rearing and interpersonal sensitivity in Chinese adolescents

BackgroundParental rearing is well documented as an important influencing factor of interpersonal sensitivity (IS). However, little research has focused on the extent by which various aspects of parental rearing in fluence IS. This study aimed to analyze the effects of parental rearing on IS, using quantile regression. We analyzed the extent of the influence of parental rearing on IS by quantile regression to provide definitive evidence on the family education of adolescents with IS problems.MethodsThe multiple cross-sectional studies were conducted among 3345 adolescents from Harbin, China, in 1999, 2006, 2009 and 2016. Furthermore, a multistage sampling method (stratified random cluster) was used to select participants. IS was assessed using a subscale of the Symptom Checklist-90-Revision. Perceived parental rearing was assessed using the Egna Minnen av. Barndoms Uppfostran. The ordinary least squares (OLS) linear regression was used to determine the average effect of parental rearing on IS. The quantile regression was conducted to examine the established associations and to further explain the association.ResultsPaternal emotional warmth was found to be associated with IS across the quantile, especially after the 0.6 quantiles; however, this association was not found for maternal emotional warmth. Paternal punishment was associated with IS at the 0.22–0.27 and 0.60 quantile; however, maternal punishment had no significant effect on IS. QR method found that paternal overinvolvement was associated with IS at the 0.48–0.65 quantiles, but paternal overprotection was associated with IS across the quantile; however, maternal overinvolvement and overprotection was positively correlated with IS at the 0.07–0.95 quantiles. The correlation between paternal rejection and IS was found at the 0.40–0.75 and > 0.90 quantiles; maternal rejection was associated with IS within the 0.05–0.92 quantiles.ConclusionsParental rearing practices predict different magnitudes of IS at varying levels. This study provides suggestions for parents to assess purposefully and systematically, intervene, and ameliorate adolescent IS problems. We also highlight the role of paternal rearing in children’s IS problems, providing new ideas for family education.

BP Neural Network Retrieval for Remote Sensing Atmospheric Profile of Ground-Based Microwave Radiometer

Vertical distributions of temperature and humidity are two essential factors for understanding the atmospheric structure, extreme weather events, and regional and global climate. The ground-based microwave radiometer (MWR), which acts as a passive sensor and operates continuously under all weather conditions, has an irreplaceable role in measuring the vertical information of the temperature and water content in the atmosphere. In this letter, we proposed a four-layer back-propagation neural network (BPNN) method to retrieve temperature and relative humidity (RH) profiles from the bright temperature measured by the MWR. In contrast to the traditional BPNN, this method has greater advantages in dealing with the problems of overfitting, gradient disappearance, and gradient explosion in vertical atmospheric retrieval. By adding dropout layers, it can also help to describe the nonlinear relationships for RH profiles. Results showed that the performance of the four-layer BPNN method was better than the quadratic regression (QR, provided by MWR manufacturer) method under both cloud and cloud-free conditions. Compared with measurements of radiosonde data, root-mean-square error of temperature and RH, BPNN achieves 1.88 K and 19.30% under cloud conditions and 2.03 K and 15.10% under cloud-free conditions, respectively, whereas the corresponding values by using the QR method were only 3.07 K and 24.28% under cloud conditions and 4.14 K and 18.96% under cloud-free conditions, respectively. Temperature and RH profiles retrieval with high precision have increased the efficiency of the MWR observations and provided a data foundation for further atmospheric climate research.

Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

А Gpu-based Orthogonal Matrix Factorization Algorithm that Produces a Two-Diagonal Shape


 
 
 With the development of the Big Data sphere, as well as those fields of study that we can relate to artificial intelligence, the need for fast and efficient computing has become one of the most important tasks nowadays. That is why in the recent decade, graphics processing unit computations have been actively developing to provide an ability for scientists and developers to use thousands of cores GPUs have in order to perform intensive computations. The goal of this research is to implement orthogonal decomposition of a matrix by applying a series of Householder transformations in Java language using JCuda library to conduct a research on its benefits. Several related papers were examined. Malaschonok and Savchenko in their work have introduced an improved version of QR algorithm for this purpose [4] and achieved better results, however Householder algorithm is more promising for GPUs according to another team of researchers – Lahabar and Narayanan [6]. However, they were using Float numbers, while we are using Double, and apart from that we are working on a new BigDecimal type for CUDA. Apart from that, there is still no solution for handling huge matrices where errors in calculations might occur.
 The algorithm of orthogonal matrix decomposition, which is the first part of SVD algorithm, is researched and implemented in this work. The implementation of matrix bidiagonalization and calculation of orthogonal factors by the Hausholder method in the jCUDA environment on a graphics processor is presented, and the algorithm for the central processor for comparisons is also implemented. Research of the received results where we experimentally measured acceleration of calculations with the use of the graphic processor in comparison with the implementation on the central processor are carried out. We show a speedup up to 53 times compared to CPU implementation on a big matrix size, specifically 2048, and even better results when using more advanced GPUs. At the same time, we still experience bigger errors in calculations while using graphic processing units due to synchronization problems. We compared execution on different platforms (Windows 10 and Arch Linux) and discovered that they are almost the same, taking the computation speed into account. The results have shown that on GPU we can achieve better performance, however there are more implementation difficulties with this approach.
 
 

Node Generation for RBF-FD Methods by QR Factorization

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

Reducing Approximation Error with Rapid Convergence Rate for Non-Negative Matrix Factorization (NMF)

Non-Negative Matrix Factorization (NMF) is utilized in many important applications. This paper presents development of an efficient low rank approximate NMF algorithm for feature extraction related to text mining and spectral data analysis. NMF can be used for clustering. NMF factorizes a positive matrix A to two positive matrices W and H matrices where A=WH. The proposal uses k-means clustering algorithm to determine the centroid of each cluster and assigns the centroid coordinates of each cluster as one column for W matrix. The initial choice of W matrix is positive. The H matrix is determined with gradient descent algorithm based on thin QR optimization. The performance comparison of the proposed NMF algorithm is illustrated with results. The accurate choice of initial positive W matrix reduces approximation error and the use of thin QR algorithm in combination with gradient descent approach provides rapid convergence rate for NMF. The proposed algorithm is implemented with the randomly generated matrix in MATLAB environment. The number of significant singular values of the generated matrix is selected as the number of clusters. The error and convergence rate comparison of the proposed algorithm with the current algorithms are demonstrated in this research. The accurate measurement of execution time for individual program is not possible in MATLAB. The average time execution over 200 iterations is therefore calculated with an increasing iteration count of the proposed algorithm and the comparative results are presented.

A Study on Radar Target Detection using Space Time Adaptive Processing Algorithm and LCMV Algorithm

In this paper, we study the directionof arrival estimation of the desired target in adaptive array MV algorithm to update the weight, and the optimized weight removes the interference signal. The target signal is estimated using the optimized weight vector and the high resolution the direction of arrival estimation MUSIC algorithm. We calculate the inverse of the correlation matrix using the QR method to reduce the processing power consumption of the optimized weight. The optimal weight vector is applied to the proposed algorithm to estimate the desired target direction from the output power spectrum. The performance of the proposed method is compared with the existing method by simulation. The experimental method estimates three targets from the antenna received signal. The existing method did not estimate the three desired targets at [-30o,-20, -10o]. The proposed method accurately estimates the desired three targets at [-30o,-20, -10o]. In the [-10o, 0, 10o] target estimation, the existing method reduces the estimated resolution of the target, but the proposed method accurately estimates the target. We proved that the proposed method in the simulation was superior to the existing method.

Desynchronization of stochastically synchronized neural populations through phase distribution control: a numerical simulation approach

Stochastic synchrony, also known as noise-induced synchronization that leads to phase coherence, arises when a set of uncoupled neurons synchronizes to a common white noise input or other types of non-Gaussian noise. Evidence of abnormally high noise-induced synchronization, or impairment in synchronous activity, has been found in several pathologies such as epilepsy. Therefore, controlling the stochastic synchronization of neurons can have a significant effect on preventing seizures in epilepsy. The main aim of this study is to develop a fast and reliable numerical method to simulate controlling synchronization in a population of noisy and uncoupled neural oscillators. The control algorithm is based on phase reduction and uses the probability phase distribution partial differential equation to change the distribution of oscillators to the desired one. The accuracy and power consumption are two main issues that should be considered in the simulations. In this paper, a new numerical method called the generalized Lagrange Jacobi Gauss–Lobatto collocation method in space and backward-Euler scheme in time is applied to overcome the difficulties of the problem effectively. The resulting full-discrete scheme of the partial differential equation is a linear system of algebraic equations per time step which is solved via QR algorithm. Finally, the proposed algorithm is applied to various neural dynamical models with different phase response curves and investigates them in different factors such as computing time, energy consumption, and accuracy to confirm the applicability of the developed numerical method.

Scattering From Quasi-Planar and Moderate Rough Surfaces: Efficient Method to Fill the EFIE-Galerkin MoM Impedance Matrix and to Solve the Linear System

First, an acceleration to compute the impedance matrix obtained from the electric field integral equation (EFIE) discretized by the Galerkin method of moments (MoM) with Rao–Wilton–Glisson basis functions is addressed. It is based on a far-field approximation and makes it possible to avoid looping on the source and observation triangles. Next, the impedance matrix is split into strong and weak interactions; the latter is compressed by expressing it from Toeplitz submatrices. Then, the linear system is efficiently solved by a bi-iterative scheme. For a given order, the least-squares QR (LSQR) algorithm is applied to solve the sparse linear system related to the strong interactions, while the matrix–vector products, related to the weak interactions, are accelerated by using FFTs. Numerical results of the field scattered by perfectly conducting paraboloid-shape object and Gaussian rough surface are shown.

Structured backward errors in linearizations

A standard approach to calculate the roots of a univariate polynomial is to compute the eigenvalues of an associated confederate matrix instead, such as, for instance, the companion or comrade matrix. The eigenvalues of the confederate matrix can be computed by Francis's QR algorithm. Unfortunately, even though the QR algorithm is provably backward stable, mapping the errors back to the original polynomial coefficients can still lead to huge errors. However, the latter statement assumes the use of a non-structure-exploiting QR algorithm. In [J. L. Aurentz et al., Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942–973] it was shown that a structure-exploiting QR algorithm for companion matrices leads to a structured backward error in the companion matrix. The proof relied on decomposing the error into two parts: a part related to the recurrence coefficients of the basis (a monomial basis in that case) and a part linked to the coefficients of the original polynomial. In this article we prove that the analysis can be extended to other classes of comrade matrices. We first provide an alternative backward stability proof in the monomial basis using structured QR algorithms; our new point of view shows more explicitly how a structured, decoupled error in the confederate matrix gets mapped to the associated polynomial coefficients. This insight reveals which properties have to be preserved by a structure-exploiting QR algorithm to end up with a backward stable algorithm. We will show that the previously formulated companion analysis fits into this framework, and we analyze in more detail Jacobi polynomials (comrade matrices) and Chebyshev polynomials (colleague matrices).

Using cloud storage to support reminiscence

Abstract We consider the problem of people having trouble retrieving their important reminiscence, it’s may get buries in the countless folders. In this study, we present the future of sorting out important reminiscence data using cloud services. Based on our respondents we find out a lot of people having trouble with cloud storage for keeping their reminiscence. To solve this problem we are trying to find the most suitable cloud computing model to support reminiscence and create a cloud services application with the chosen cloud model and the QR algorithm. The result shows that this application is feasible to be made and tried by customers and this app supported by the SaaS model which has a low price and it is easily used for reminiscence purposes because users can upload and easily lookup for the data again.

Multipath number estimation based on QR decomposition

Accurate multipath number estimation is the prerequisite for MUSIC algorithm to achieve excellent superresolution performance. To solve the problem that the existing multipath number estimation algorithms based on eigenvalue distribution usually require a large amount of computation, this paper proposed a fast multipath number estimation algorithm based on QR decomposition under the OFDM signal model. This algorithm only needs few times QR iteration decomposition to obtain the pseudo-eigenvalues of the matrix, and uses them to estimate the multipath number through two-step comparison. Due to the avoidance of accurate eigenvalues solution, compared with the traditional algorithms based on eigenvalue distribution, proposed algorithm’s computation is smaller. Simulation results showed that the proposed algorithm has better detection performance in low SNR and few snapshots than existing algorithms.

Rank-Structured QR for Chebyshev Rootfinding

We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [Y. Eidelman, L. Gemignani, and I. Gohberg, Numer. Algorithms, 47 (2008...

Rounding Error Analysis of Mixed Precision Block Householder QR Algorithms

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 30 October 2019Accepted: 19 January 2021Published online: 17 May 2021Keywordsmixed precision, fp16, QR factorization, rounding error, block QR, fp32AMS Subject Headings65G50, 65F25, 65F08, 65F10Publication DataISSN (print): 1064-8275ISSN (online): 1095-7197Publisher: Society for Industrial and Applied MathematicsCODEN: sjoce3

Determinant-Based Fast Greedy Sensor Selection Algorithm

In this paper, the sparse sensor placement problem for least-squares estimation is considered, and the previous novel approach of the sparse sensor selection algorithm is extended. The maximization of the determinant of the matrix which appears in pseudo-inverse matrix operations is employed as an objective function of the problem in the present extended approach. The procedure for the maximization of the determinant of the corresponding matrix is proved to be mathematically the same as that of the previously proposed QR method when the number of sensors is less than that of state variables (undersampling). On the other hand, the authors have developed a new algorithm for when the number of sensors is greater than that of state variables (oversampling). Then, a unified formulation of the two algorithms is derived, and the lower bound of the objective function given by this algorithm is shown using the monotone submodularity of the objective function. The effectiveness of the proposed algorithm on the problem using real datasets is demonstrated by comparing with the results of other algorithms. The numerical results show that the proposed algorithm improves the estimation error by approximately 10% compared with the conventional methods in the oversampling case, where the estimation error is defined as the ratio of the difference between the reconstructed data and the full observation data to the full observation. For the NOAA-SST sensor problem, which has more than ten thousand sensor candidate points, the proposed algorithm selects the sensor positions in few seconds, which required several hours with the other algorithms in the oversampling case on a 3.40 GHz computer.

Computed tomography medical image reconstruction on affordable equipment by using Out-Of-Core techniques

Background and objective: As Computed Tomography scans are an essential medical test, many techniques have been proposed to reconstruct high-quality images using a smaller amount of radiation. One approach is to employ algebraic factorization methods to reconstruct the images, using fewer views than the traditional analytical methods. However, their main drawback is the high computational cost and hence the time needed to obtain the images, which is critical in the daily clinical practice. For this reason, faster methods for solving this problem are required.Methods: In this paper, we propose a new reconstruction method based on the QR factorization that is very efficient on affordable equipment (standard multicore processors and standard Solid-State Drives) by using Out-Of-Core techniques.Results: Combining both affordable hardware and the new software proposed in our work, the images can be reconstructed very quickly and with high quality. We analyze the reconstructions using real Computed Tomography images selected from a dataset, comparing the QR method to the LSQR and FBP. We measure the quality of the images using the metrics Peak Signal-To-Noise Ratio and Structural Similarity Index, obtaining very high values. We also compare the efficiency of using spinning disks versus Solid-State Drives, showing how the latter performs the Input/Output operations in a significantly lower amount of time. Conclusions: The results indicate that our proposed me thod and software are valid to efficiently solve large-scale systems and can be applied to the Computed Tomography reconstruction problem to obtain high-quality images.