scholarly journals Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1174 ◽  
Author(s):  
Philipp Bader ◽  
Sergio Blanes ◽  
Fernando Casas
Keyword(s):  
Numerical Experiments ◽  
State Of The Art ◽  
Matrix Exponential ◽  
Superior Performance ◽  
Taylor Polynomial ◽  
Polynomial Evaluation ◽  
The Matrix ◽  
Similar Accuracy ◽  
Matrix Norms ◽  
Range Of Values

A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Padé approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.

scholarly journals Enhancing Matrix Completion Using a Modified Second-Order Total Variation

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Wendong Wang ◽  
Jianjun Wang
Keyword(s):  
Total Variation ◽  
Optimization Problem ◽  
State Of The Art ◽  
Matrix Completion ◽  
Second Order ◽  
Low Rank ◽  
Superior Performance ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
The Matrix

In this paper, we propose a new method to deal with the matrix completion problem. Different from most existing matrix completion methods that only pursue the low rank of underlying matrices, the proposed method simultaneously optimizes their low rank and smoothness such that they mutually help each other and hence yield a better performance. In particular, the proposed method becomes very competitive with the introduction of a modified second-order total variation, even when it is compared with some recently emerged matrix completion methods that also combine the low rank and smoothness priors of matrices together. An efficient algorithm is developed to solve the induced optimization problem. The extensive experiments further confirm the superior performance of the proposed method over many state-of-the-art methods.

scholarly journals The smallest singular value of certain Toeplitz-related parametric triangular matrices

2021 ◽  
Vol 9 (1) ◽  
pp. 103-111
Author(s):  
Maryam Shams Solary ◽  
Alexander Kovačec ◽  
Stefano Serra Capizzano
Keyword(s):  
Numerical Experiments ◽  
Singular Values ◽  
Singular Value ◽  
Triangular Matrices ◽  
Matrix Size ◽  
The Matrix ◽  
Lower Triangular ◽  
Triangular Toeplitz Matrix ◽  
Range Of Values ◽  
Smallest Singular Value

Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a 1, a 2, ..., ap , a 1, ..., ap , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a 1, a 2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a 1, ..., ap . It depends on the asymptotics in µ of the l 2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

scholarly journals Advances in the Approximation of the Matrix Hyperbolic Tangent

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1219
Author(s):  
Javier Ibáñez ◽  
José M. Alonso ◽  
Jorge Sastre ◽  
Emilio Defez ◽  
Pedro Alonso-Jordá
Keyword(s):  
Taylor Series ◽  
State Of The Art ◽  
Matrix Exponential ◽  
The Other ◽  
Hyperbolic Tangent ◽  
Matrix Polynomials ◽  
Matrix Products ◽  
The Matrix ◽  
Computationally Expensive ◽  
Exponential Matrix

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

scholarly journals Fast Taylor polynomial evaluation for the computation of the matrix cosine

2019 ◽  
Vol 354 ◽  
pp. 641-650 ◽  
Author(s):  
Jorge Sastre ◽  
Javier Ibáñez ◽  
Pedro Alonso-Jordá ◽  
Jesús Peinado ◽  
Emilio Defez
Keyword(s):  
Taylor Polynomial ◽  
Polynomial Evaluation ◽  
The Matrix

State-of-the-Art of Modeling Surface Water Impoundments

1982 ◽  
Vol 14 (1-2) ◽  
pp. 241-261 ◽  
Author(s):  
P A Krenkel ◽  
R H French
Keyword(s):  
Water Quality ◽  
Surface Water ◽  
State Of The Art ◽  
Rate Coefficients ◽  
Two Dimensional ◽  
One Dimensional ◽  
Integral Energy ◽  
Range Of Values ◽  
Dimensional Integral ◽  
Water Impoundment

The state-of-the-art of surface water impoundment modeling is examined from the viewpoints of both hydrodynamics and water quality. In the area of hydrodynamics current one dimensional integral energy and two dimensional models are discussed. In the area of water quality, the formulations used for various parameters are presented with a range of values for the associated rate coefficients.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

scholarly journals RLC-GNN: An Improved Deep Architecture for Spatial-Based Graph Neural Network with Application to Fraud Detection

2021 ◽  
Vol 11 (12) ◽  
pp. 5656
Author(s):  
Yufan Zeng ◽  
Jiashan Tang
Keyword(s):  
Numerical Experiments ◽  
State Of The Art ◽  
Single Layer ◽  
Fraud Detection ◽  
Layer By Layer ◽  
Residual Structure ◽  
Detection Algorithms ◽  
Deep Architecture ◽  
Graph Neural Networks ◽  
Node Embeddings

Graph neural networks (GNNs) have been very successful at solving fraud detection tasks. The GNN-based detection algorithms learn node embeddings by aggregating neighboring information. Recently, CAmouflage-REsistant GNN (CARE-GNN) is proposed, and this algorithm achieves state-of-the-art results on fraud detection tasks by dealing with relation camouflages and feature camouflages. However, stacking multiple layers in a traditional way defined by hop leads to a rapid performance drop. As the single-layer CARE-GNN cannot extract more information to fix the potential mistakes, the performance heavily relies on the only one layer. In order to avoid the case of single-layer learning, in this paper, we consider a multi-layer architecture which can form a complementary relationship with residual structure. We propose an improved algorithm named Residual Layered CARE-GNN (RLC-GNN). The new algorithm learns layer by layer progressively and corrects mistakes continuously. We choose three metrics—recall, AUC, and F1-score—to evaluate proposed algorithm. Numerical experiments are conducted. We obtain up to 5.66%, 7.72%, and 9.09% improvements in recall, AUC, and F1-score, respectively, on Yelp dataset. Moreover, we also obtain up to 3.66%, 4.27%, and 3.25% improvements in the same three metrics on the Amazon dataset.

scholarly journals Utilization of Eco-Friendly Waste Generated Nanomaterials in Water-Based Drilling Fluids; State of the Art Review

Materials ◽  
2021 ◽  
Vol 14 (15) ◽  
pp. 4171
Author(s):  
Rabia Ikram ◽  
Badrul Mohamed Jan ◽  
Akhmal Sidek ◽  
George Kenanakis
Keyword(s):  
State Of The Art ◽  
Superior Performance ◽  
Future Research ◽  
Drilling Fluids ◽  
Drill Cuttings ◽  
Environmental Friendly ◽  
Water Based ◽  
Drilling Operations ◽  
Filtration Properties

An important aspect of hydrocarbon drilling is the usage of drilling fluids, which remove drill cuttings and stabilize the wellbore to provide better filtration. To stabilize these properties, several additives are used in drilling fluids that provide satisfactory rheological and filtration properties. However, commonly used additives are environmentally hazardous; when drilling fluids are disposed after drilling operations, they are discarded with the drill cuttings and additives into water sources and causes unwanted pollution. Therefore, these additives should be substituted with additives that are environmental friendly and provide superior performance. In this regard, biodegradable additives are required for future research. This review investigates the role of various bio-wastes as potential additives to be used in water-based drilling fluids. Furthermore, utilization of these waste-derived nanomaterials is summarized for rheology and lubricity tests. Finally, sufficient rheological and filtration examinations were carried out on water-based drilling fluids to evaluate the effect of wastes as additives on the performance of drilling fluids.

scholarly journals 3D reconstruction of genomic regions from sparse interaction data

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Julen Mendieta-Esteban ◽  
Marco Di Stefano ◽  
David Castillo ◽  
Irene Farabella ◽  
Marc A Marti-Renom
Keyword(s):  
3D Models ◽  
Genomic Region ◽  
Gene Promoters ◽  
Chromosome Conformation ◽  
The Matrix ◽  
Chromatin Structural ◽  
A Cell ◽  
Similar Accuracy ◽  
Genomic Regions ◽  
Interaction Matrices

Abstract Chromosome conformation capture (3C) technologies measure the interaction frequency between pairs of chromatin regions within the nucleus in a cell or a population of cells. Some of these 3C technologies retrieve interactions involving non-contiguous sets of loci, resulting in sparse interaction matrices. One of such 3C technologies is Promoter Capture Hi-C (pcHi-C) that is tailored to probe only interactions involving gene promoters. As such, pcHi-C provides sparse interaction matrices that are suitable to characterize short- and long-range enhancer–promoter interactions. Here, we introduce a new method to reconstruct the chromatin structural (3D) organization from sparse 3C-based datasets such as pcHi-C. Our method allows for data normalization, detection of significant interactions and reconstruction of the full 3D organization of the genomic region despite of the data sparseness. Specifically, it builds, with as low as the 2–3% of the data from the matrix, reliable 3D models of similar accuracy of those based on dense interaction matrices. Furthermore, the method is sensitive enough to detect cell-type-specific 3D organizational features such as the formation of different networks of active gene communities.

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