Numerical methods for nonlinear Dirac equation

This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.Comment: 39 pages, 13 figure

Multi-Label Learning with Label Enhancement

The task of multi-label learning is to predict a set of relevant labels for the unseen instance. Traditional multi-label learning algorithms treat each class label as a logical indicator of whether the corresponding label is relevant or irrelevant to the instance, i.e., +1 represents relevant to the instance and -1 represents irrelevant to the instance. Such label represented by -1 or +1 is called logical label. Logical label cannot reflect different label importance. However, for real-world multi-label learning problems, the importance of each possible label is generally different. For the real applications, it is difficult to obtain the label importance information directly. Thus we need a method to reconstruct the essential label importance from the logical multilabel data. To solve this problem, we assume that each multi-label instance is described by a vector of latent real-valued labels, which can reflect the importance of the corresponding labels. Such label is called numerical label. The process of reconstructing the numerical labels from the logical multi-label data via utilizing the logical label information and the topological structure in the feature space is called Label Enhancement. In this paper, we propose a novel multi-label learning framework called LEMLL, i.e., Label Enhanced Multi-Label Learning, which incorporates regression of the numerical labels and label enhancement into a unified framework. Extensive comparative studies validate that the performance of multi-label learning can be improved significantly with label enhancement and LEMLL can effectively reconstruct latent label importance information from logical multi-label data.Comment: ICDM 201

Non-coherent Massive SIMO Systems in ISI Channels: Constellation Design and Performance Analysis

A massive single-input multiple-output (SIMO) system with a single transmit antenna and a large number of receive antennas in intersymbol interference (ISI) channels is considered. Contrast to existing energy detection (ED)-based non-coherent receiver where conventional pulse amplitude modulation (PAM) is employed, we propose a constellation design which minimizes the symbol-error rate (SER) with the knowledge of channel statistics. To make a comparison, we derive the SERs of the ED-based receiver with both the proposed constellation and PAM, namely $P_{e\_opt}$ and $P_{e\_pam}$. Specifically, asymptotic behaviors of the SER in regimes of a large number of receive antennas and high signal-to-noise ratio (SNR) are investigated. Analytical results demonstrate that the logarithms of both $P_{e\_opt}$ and $P_{e\_pam}$ decrease approximately linearly with the number of receive antennas, while $P_{e\_opt}$ degrades faster. It is also shown that the proposed design is of less cost, because compared with PAM, less antennas are required to achieve the same error rate