Search

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin (WG) mixed-FEM based on Banach spaces for the stationary Navier--Stokes equation in pseudostress-velocity formulation. More precisely, a modified pseudostress tensor, called $ \boldsymbol{\sigma} $, depending on the pressure, and the diffusive and convective terms has been introduced in the proposed technique, and a dual-mixed variational formulation has been derived where the aforementioned pseudostress tensor and the velocity, are the main unknowns of the system, whereas the pressure is computed via a post-processing formula. Thus, it is sufficient to provide a WG space for the tensor variable and a space of piecewise polynomial vectors of total degree at most 'k' for the velocity. Moreover, in order to define the weak discrete bilinear form, whose continuous version involves the classical divergence operator, the weak divergence operator as a well-known alternative for the classical divergence operator in a suitable discrete subspace is proposed. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babu\v{s}ka-Brezzi theory and the Banach-Ne\v{c}as-Babu\v{s}ka theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method's good performance and confirming the theoretical rates of convergence are presented.

As we are aware, various types of methods have been proposed to approximate the Caputo fractional derivative numerically. A common challenge of the methods is the non-local property of the Caputo fractional derivative which leads to the slow and memory consuming methods. Diffusive representation of fractional derivative is an efficient tool to overcome the mentioned challenge. This paper presents two new diffusive representations to approximate the Caputo fractional derivative of order $0<\alpha<1$. Error analysis of the newly presented methods together with some numerical examples are provided at the end.

In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson-Nernst-Planck (PNP) and Navier--Stokes (NS) equations modeling microfluidic and electrochemical systems (diffuse transport of charged species within incompressible fluids coupled through electrostatic forces). A mixed VEM is employed to discretize the NS equations whereas classical VEM in primal form is used to discretize the PNP equations. The stability, existence and uniqueness of solution of the associated VEM are proved by fixed point theory. Global mass conservation and electric energy decay of the scheme are also proved. Also, we obtain unconditionally optimal error estimates for both the electrostatic potential and ionic concentrations of PNP equations in the $H^{1}$-norm, as well as for the velocity and pressure of NS equations in the $\mathbf{H}^{1}$- and $L^{2}$-norms, respectively. Finally, several numerical experiments are presented to support the theoretical analysis of convergence and to illustrate the satisfactory performance of the method in simulating the onset of electrokinetic instabilities in ionic fluids, and studying how they are influenced by different values of ion concentration and applied voltage. These tests relate to applications in the desalination of water.

Users' reviews contain valuable information which are not taken into account in most recommender systems. According to the latest studies in this field, using review texts could not only improve the performance of recommendation, but it can also alleviate the impact of data sparsity and help to tackle the cold start problem. In this paper, we present a neural recommender model which recommends items by leveraging user reviews. In order to predict user rating for each item, our proposed model, named MatchPyramid Recommender System (MPRS), represents each user and item with their corresponding review texts. Thus, the problem of recommendation is viewed as a text matching problem such that the matching score obtained from matching user and item texts could be considered as a good representative of their joint extent of similarity. To solve the text matching problem, inspired by MatchPyramid (Pang, 2016), we employed an interaction-based approach according to which a matching matrix is constructed given a pair of input texts. The matching matrix, which has the property of hierarchical matching patterns, is then fed into a Convolutional Neural Network (CNN) to compute the matching score for the given user-item pair. Our experiments on the small data categories of Amazon review dataset show that our proposed model gains from 1.76% to 21.72% relative improvement compared to DeepCoNN model, and from 0.83% to 3.15% relative improvement compared to TransNets model. Also, on two large categories, namely AZ-CSJ and AZ-Mov, our model achieves relative improvements of 8.08% and 7.56% compared to the DeepCoNN model, and relative improvements of 1.74% and 0.86% compared to the TransNets model, respectively., Comment: 19 pages, 3 figures

In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection-reaction-diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank--Nicolson finite difference formulation. In the stochastic direction, we also employ a random variable $W$ based on the $Q-$Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation. Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.

In this work, we apply two meshless methods for the numerical solution of the time-dependent transport equation defined on the sphere in spherical coordinates. The first technique, which was introduced by Mirzaei (BIT Numerical Mathematics, 54 (4) 1041-1063, 2017) in Cartesian coordinates is a generalized moving least squares approximation, and the second one, which is developed here, is moving kriging least squares interpolation on the sphere. These methods do not depend on the background mesh or triangulation, and they can be implemented on the transport equation in spherical coordinates easily using different distribution points. Furthermore, the time variable is approximated by a second-order backward differential formula. The obtained fully discrete scheme is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner at each time step. Three well-known test problems namely solid body rotation, vortex roll-up, and deformational flow are solved to demonstrate our developments.

One of the most important tasks in network management is identifying different types of traffic flows. As a result, a type of management service, called Network Traffic Classifier (NTC), has been introduced. One type of NTCs that has gained huge attention in recent years applies deep learning on packets in order to classify flows. Internet is an imbalanced environment i.e., some classes of applications are a lot more populated than others e.g., HTTP. Additionally, one of the challenges in deep learning methods is that they do not perform well in imbalanced environments in terms of evaluation metrics such as precision, recall, and $\mathrm{F_1}$ measure. In order to solve this problem, we recommend the use of augmentation methods to balance the dataset. In this paper, we propose a novel data augmentation approach based on the use of Long Short Term Memory (LSTM) networks for generating traffic flow patterns and Kernel Density Estimation (KDE) for replicating the numerical features of each class. First, we use the LSTM network in order to learn and generate the sequence of packets in a flow for classes with less population. Then, we complete the features of the sequence with generating random values based on the distribution of a certain feature, which will be estimated using KDE. Finally, we compare the training of a Convolutional Recurrent Neural Network (CRNN) in large-scale imbalanced, sampled, and augmented datasets. The contribution of our augmentation scheme is then evaluated on all of the datasets through measurements of precision, recall, and F1 measure for every class of application. The results demonstrate that our scheme is well suited for network traffic flow datasets and improves the performance of deep learning algorithms when it comes to above-mentioned metrics., Comment: Submitted to IFIP Networking 2019