Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

A non‐conforming piecewise quadratic finite element on triangles

Abstract: It is shown in this paper that non-conforming finite elements on the triangle using second-degree polynomials can be easily built and used. Indeed they appear as an ‘enriched’ version of the standard piecewise quadratic six-node element. This work is divided into two parts. In the first we present the basic properties of the element, namely how it can be built and basic error estimates. We also show that this element exhibits a very peculiar regularity property. In the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials.

Adaptive data analysis via sparse time-frequency representation

Abstract: We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t )c os(θ(t))} ,w herea ≥ 0i s assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L 1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method.

The Influence of the Activation Function in a Convolution Neural Network Model of Facial Expression Recognition

Abstract: The convolutional neural network (CNN) has been widely used in image recognition field due to its good performance. This paper proposes a facial expression recognition method based on the CNN model. Regarding the complexity of the hierarchic structure of the CNN model, the activation function is its core, because the nonlinear ability of the activation function really makes the deep neural network have authentic artificial intelligence. Among common activation functions, the ReLu function is one of the best of them, but it also has some shortcomings. Since the derivative of the ReLu function is always zero when the input value is negative, it is likely to appear as the phenomenon of neuronal necrosis. In order to solve the above problem, the influence of the activation function in the CNN model is studied in this paper. According to the design principle of the activation function in CNN model, a new piecewise activation function is proposed. Five common activation functions (i.e., sigmoid, tanh, ReLu, leaky ReLus and softplus–ReLu, plus the new activation function) have been analysed and compared in facial expression recognition tasks based on the Keras framework. The Experimental results on two public facial expression databases (i.e., JAFFE and FER2013) show that the convolutional neural network based on the improved activation function has a better performance than most-of-the-art activation functions.