arXiv subjects

Wenjie Liu

Publications and source records attributed to Wenjie Liu.

117 records · Page 7Linked to original sources

arXiv2019

A Generalized-Jacobi-Function Spectral Method for Space-Time Fractional Reaction-Diffusion Equations with Viscosity Terms

In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.

math.NA↗

arXiv2019

Bloch oscillations of multi-magnon excitations in a Heisenberg XXZ chain

The studies of multi-magnon excitations will extend our understandings of quantum magnetism and strongly correlated matters. Here, by using the time-evolving block decimation algorithm, we investigate the Bloch oscillations of two-magnon excitations under a gradient magnetic field. Through analyzing the symmetry of the Hamiltonian, we derive a rigorous and universal relation between ferromagnetic and anti-ferromagnetic systems. Under strong interactions, in addition to free-magnon Bloch oscillations, there appear fractional bounded-magnon Bloch oscillations which can be understood by an effective single-particle model. To extract the frequencies of Bloch oscillations and determine the gradient of magnetic field, we respectively calculate the fidelity in the time domain and the sub-standard deviation in the frequency domain. Our study not only explore the interaction-induced Bloch oscillations of multi-magnon excitations, but also provides an alternative approach to determine the gradient of magnetic field via ultracold atoms in optical lattices.

quant-ph↗

arXiv2018

Uniform bounds and asymptotics of Generalized Gegenbauer functions of fractional degree

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(λ)}_ν(x)$ (right GGF-Fs) and ${}^{l}G^{(λ)}_ν(x)$ (left GGF-Fs) with $x\in (-1,1),$ $λ>-1/2$ and real $ν\ge 0,$ are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for $λ,ν>0$ and $x=\cosθ$ with $θ\in (0,π),$ \begin{equation*}\label{IntRep-0N} (\sin φ)^λ\,{}^{r\!}G_ν^{(λ)}(\cos φ)= \frac{2^λΓ(λ+1/2)}{\sqrtπ {(ν+λ)^λ}} \, {\cos ((ν+λ)φ- λπ/2)} +{\mathcal R}_ν^{(λ)} (φ), \end{equation*} and derive the precise expression of the "residual" term ${\mathcal R}_ν^{(λ)} (φ).$ With this at our disposal, we obtain the bounds of GGF-Fs uniform in $ν.$ Under an appropriate weight function, the bounds are uniform for $θ\in [0,π]$ as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree $ν.$ The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.

math.NA↗

arXiv2018

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations.

math.NA↗

arXiv2018

Feature Fusion through Multitask CNN for Large-scale Remote Sensing Image Segmentation

In recent years, Fully Convolutional Networks (FCN) has been widely used in various semantic segmentation tasks, including multi-modal remote sensing imagery. How to fuse multi-modal data to improve the segmentation performance has always been a research hotspot. In this paper, a novel end-toend fully convolutional neural network is proposed for semantic segmentation of natural color, infrared imagery and Digital Surface Models (DSM). It is based on a modified DeepUNet and perform the segmentation in a multi-task way. The channels are clustered into groups and processed on different task pipelines. After a series of segmentation and fusion, their shared features and private features are successfully merged together. Experiment results show that the feature fusion network is efficient. And our approach achieves good performance in ISPRS Semantic Labeling Contest (2D).

cs.CV↗

arXiv2018

3D super-resolved multi-angle TIRF via polarization modulation

Measuring the three dimension nanoscale organization of protein or cellular structures is challenging, especially when the structure is dynamic. Owing to the informative total internal reflection fluorescence (TIRF) imaging under varied illumination angles, multi-angle (MA) TIRF has been examined to offer a nanoscale axial and a sub-second temporal resolution. However, conventional MA-TIRF still performs badly in lateral resolution and fail to characterize the depth-image in densely-distributed regions, leaving a huge contrast between the nanoscale axial resolution and the diffraction limited lateral resolution. Moreover, the previous reconstructions are highly restricted by the efficiency with the increased amount of illumination angles. Here, we for the first time, emphasize the lateral super-resolution in the MA-TIRF and exampled by simply introducing polarization modulation into the illumination procedure. Equipped with a sparsity and accelerated proximal algorithm, we examine a more precise 3D sample structure compared with previous methods, enabling live cell imaging with temporal-resolution of 2 seconds, recovering high-resolution mitochondria fission and fusion process. Since the introduced vortex half wave retarder is an add-on component to the existing MA-TRIF system and the algorithm is the first open sourced and the fastest, we anticipate that the method and algorithm introduced here would be adopted rapidly by the biological community.

physics.optics↗

arXiv2017

DeepUNet: A Deep Fully Convolutional Network for Pixel-level Sea-Land Segmentation

Semantic segmentation is a fundamental research in remote sensing image processing. Because of the complex maritime environment, the sea-land segmentation is a challenging task. Although the neural network has achieved excellent performance in semantic segmentation in the last years, there are a few of works using CNN for sea-land segmentation and the results could be further improved. This paper proposes a novel deep convolution neural network named DeepUNet. Like the U-Net, its structure has a contracting path and an expansive path to get high resolution output. But differently, the DeepUNet uses DownBlocks instead of convolution layers in the contracting path and uses UpBlock in the expansive path. The two novel blocks bring two new connections that are U-connection and Plus connection. They are promoted to get more precise segmentation results. To verify our network architecture, we made a new challenging sea-land dataset and compare the DeepUNet on it with the SegNet and the U-Net. Experimental results show that DeepUNet achieved good performance compared with other architectures, especially in high-resolution remote sensing imagery.

cs.CV↗

arXiv2017

High-order implicit Galerkin-Legendre spectral method for the two-dimensional Schrodinger equation

In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives, and then employ the implicit Runge-Kutta method for the time integration of the resulting linear first-order system of ordinary differential equations in complex domain. We derive the spectral rate of convergence for the proposed method in the L^2-norm for the semidiscrete formulation. Numerical experiments show our formulation have high-order accurate, and have the exponential rates of convergence in space.

math.NA↗

arXiv2013

Quantum Private Comparison: A Review

As an important branch of quantum secure multiparty computation, quantum private comparison (QPC) has attracted more and more attention recently. In this paper, according to the quantum implementation mechanism that these protocols used, we divide these protocols into three categories: The quantum cryptography QPC, the superdense coding QPC, and the entanglement swapping QPC. And then, a more in-depth analysis on the research progress, design idea, and substantive characteristics of corresponding QPC categories is carried out, respectively. Finally, the applications of QPC and quantum secure multi-party computation issues are discussed and, in addition, three possible research mainstream directions are pointed out.

cs.CR↗