Waveforms for Gravitational Radiation from Cosmic String Loops

We obtain general formulae for the plus- and cross- polarized waveforms of gravitational radiation emitted by a cosmic string loop in transverse, traceless (synchronous, harmonic) gauge. These equations are then specialized to the case of piecewise linear loops, and it is shown that the general waveform for such a loop is a piecewise linear function. We give several simple examples of the waveforms from such loops. We also discuss the relation between the gravitational radiation by a smooth loop and by a piecewise linear approximation to it.Comment: 16 pages, 6 figures, Revte

Improved ZZ A Posteriori Error Estimators for Diffusion Problems: Conforming Linear Elements

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;\Omega)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.Comment: arXiv admin note: substantial text overlap with arXiv:1407.437

Nonconforming tetrahedral mixed finite elements for elasticity

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.Comment: 13 pages, 2 figure

Modeling the AgInSbTe Memristor

The AgInSbTe memristor shows gradual resistance tuning characteristics, which makes it a potential candidate to emulate biological plastic synapses. The working mechanism of the device is complex, and both intrinsic charge-trapping mechanism and extrinsic electrochemical metallization effect are confirmed in the AgInSbTe memristor. Mathematical model of the AgInSbTe memristor has not been given before. We propose the flux-voltage controlled memristor model. With piecewise linear approximation technique, we deliver the flux-voltage controlled memristor model of the AgInSbTe memristor based on the experiment data. Our model fits the data well. The flux-voltage controlled memristor model and the piecewise linear approximation method are also suitable for modeling other kinds of memristor devices based on experiment data

Understanding deep neural networks from the perspective of piecewise linear property

In recent years, deep learning models have been widely used and are behind major breakthroughs across many fields. Deep learning models are usually considered to be black boxes due to their large model structures and complicated hierarchical nonlinear transformations. As deep learning technology continues to develop, the understanding of deep learning models is raising concerns, such as the understanding of the training and prediction behaviors and the internal mechanism of models. In this thesis, we study the model understanding problem of deep neural networks from the perspective of piecewise linear property. First, we introduce the piecewise linear property. Next, we review the role and progress of deep learning understanding from the perspective of the piecewise linear property. The piecewise linear property reveals that deep neural networks with piecewise linear activation functions can generally divide the input space into a number of small disjointed regions that correspond to a local linear function within each region. Next, we investigate two typical understanding problems, namely model interpretation, and model complexity. In particular, we provide a series of derivations and analyses of the piecewise linear property of deep neural networks with piecewise linear activation functions. We propose an approach for interpreting the predictions given by such models based on the piecewise linear property. Next, we propose a method to provide local interpretation to a black box deep model by mimicking a piecewise linear approximation from the deep model. Then, we study deep neural networks with curve activation functions with the aim of providing piecewise linear approximations for these networks that would let them benefit from the piecewise linear property. After proposing a piecewise linear approximation framework, we investigate model complexity and model interpretation using the approximation. The thesis concludes by discussing future directions for understanding deep neural networks from the perspective of the piecewise linear property