arXiv subjects

Yanzhi Zhang

Publications and source records attributed to Yanzhi Zhang.

32 records · Page 2Linked to original sources

arXiv2022

The Development and Prospect of Code Clone

The application of code clone technology accelerates code search, improves code reuse efficiency, and assists in software quality assessment and code vulnerability detection. However, the application of code clones also introduces software quality issues and increases the cost of software maintenance. As an important research field in software engineering, code clone has been extensively explored and studied by researchers, and related studies on various sub-research fields have emerged, including code clone detection, code clone evolution, code clone analysis, etc. However, there lacks a comprehensive exploration of the entire field of code clone, as well as an analysis of the trend of each sub-research field. This paper collects related work of code clones in the past ten years. In summary, the contributions of this paper mainly include: (1) summarize and classify the sub-research fields of code clone, and explore the relative popularity and relation of these sub-research fields; (2) analyze the overall research trend of code clone and each sub-research field; (3) compare and analyze the difference between academy and industry regarding code clone research; (4) construct a network of researchers, and excavate the major contributors in code clone research field; (5) The list of popular conferences and journals was statistically analyzed. The popular research directions in the future include clone visualization, clone management, etc. For the clone detection technique, researchers can optimize the scalability and execution efficiency of the method, targeting particular clone detection tasks and contextual environments, or apply the technology to other related research fields continuously.

cs.SE↗

arXiv2021

Pattern selection in the Schnakenberg equations: From normal to anomalous diffusion

Pattern formation in the classical and fractional Schnakenberg equations is studied to understand the nonlocal effects of anomalous diffusion. Starting with linear stability analysis, we find that if the activator and inhibitor have the same diffusion power, the Turing instability space depends only on the ratio of diffusion coefficients $κ_1/κ_2$. However, the smaller diffusive powers might introduce larger unstable wave numbers with wider band, implying that the patterns may be more chaotic in the fractional cases. We then apply a weakly nonlinear analysis to predict the parameter regimes for spot, stripe, and mixed patterns in the Turing space. Our numerical simulations confirm the analytical results and demonstrate the differences of normal and anomalous diffusion on pattern formation. We find that in the presence of superdiffusion the patterns exhibit multiscale structures. The smaller the diffusion powers, the larger the unstable wave numbers and the smaller the pattern scales.

nlin.PS↗

arXiv2021

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-Δ)^\fracα{2}$ for $α\in (0, 2)$. The main advantage of our method is to easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain the scheme structure and computer implementation unchanged. Moreover, our discretization of the fractional Laplacian results in a symmetric (multilevel) Toeplitz differentiation matrix, which not only saves memory cost in simulations but enables efficient computations via the fast Fourier transforms. The performance of our method in both approximating the fractional Laplacian and solving the fractional Poisson problems was detailedly examined. It shows that our method has an optimal accuracy of ${\mathcal O}(h^2)$ for constant or linear basis functions, while ${\mathcal O}(h^4)$ if quadratic basis functions are used, with $h$ a small mesh size. Note that this accuracy holds for any $α\in (0, 2)$ and can be further increased if higher-degree basis functions are used. If the solution of fractional Poisson problem satisfies $u \in C^{m, l}(\barΩ)$ for $m \in {\mathbb N}$ and $0 < l < 1$, then our method has an accuracy of ${\mathcal O}\big(h^{\min\{m+l,\, 2\}}\big)$ for constant and linear basis functions, while ${\mathcal O}\big(h^{\min\{m+l,\, 4\}}\big)$ for quadratic basis functions. Additionally, our method can be readily applied to study generalized fractional Laplacians with a symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

math.NA↗

arXiv2021

A universal solution scheme for fractional and classical PDEs

We propose a unified meshless method to solve classical and fractional PDE problems with $(-Δ)^{\fracα{2}}$ for $α\in (0, 2]$. The classical ($α= 2$) and fractional ($α< 2$) Laplacians, one local and the other nonlocal, have distinct properties. Therefore, their numerical methods and computer implementations are usually incompatible. We notice that for any $α\ge 0$, the Laplacian $(-Δ)^{\fracα{2}}$ of generalized inverse multiquadric (GIMQ) functions can be analytically written by the Gauss hypergeometric function, and thus propose a GIMQ-based method. Our method unifies the discretization of classical and fractional Laplacians and also bypasses numerical approximation to the hypersingular integral of fractional Laplacian. These two merits distinguish our method from other existing methods for the fractional Laplacian. Extensive numerical experiments are carried out to test the performance of our method. Compared to other methods, our method can achieve high accuracy with fewer number of unknowns, which effectively reduces the storage and computational requirements in simulations of fractional PDEs. Moreover, the meshfree nature makes it free of geometric constraints and enables simple implementation for any dimension $d \ge 1$. Additionally, two approaches of selecting shape parameters, including condition number-indicated method and random-perturbed method, are studied to avoid the ill-conditioning issues when large number of points.

math.NA↗

arXiv2020

A unified meshfree pseudospectral method for solving both classical and fractional PDEs

In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish it from other numerical methods for fractional PDEs. Moreover, our method is simple and easy to handle complex geometry and local refinement, and its computer program implementation remains the same for any dimension $d \ge 1$. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving PDE problems. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the literature. Our studies suggest that to obtain good accuracy the shape parameter cannot be too small or too big, and the optimal shape parameter might depend on the RBF center points and the solution properties.

math.NA↗

arXiv2020

Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian

In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$ formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the $M$-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from $\mathcal{O}(MN^3 + M^2N)$ by a direct solver to $\mathcal{O}(MN(\log N + N_{exp}))$ per preconditioned Krylov subspace iteration and a memory requirement from $O(MN^2)$ to $O(NN_{exp})$, where $N$ and $(N_{exp} \ll)~M$ are the number of spatial and temporal grid nodes. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

math.NA↗

arXiv2020

Numerical simulations for the energy-supercritical nonlinear wave equation

We carry out numerical simulations of the defocusing energy-supercritical nonlinear wave equation for a range of spherically-symmetric initial conditions. We demonstrate numerically that the critical Sobolev norm of solutions remains bounded in time. This lends support to conditional scattering results that have been recently established for nonlinear wave equations.

math.NA↗

arXiv2019

Numerical approximations for the tempered fractional Laplacian: Error analysis and applications

In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of $O(h^ε)$, for $u \in C^{0, α+ε} (\barΩ)$ if $α< 1$ (or $u \in C^{1, α-1+ε} (\barΩ)$ if $α\ge 1$) with $ε> 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in C^{2, α+ε} (\barΩ)$ if $α< 1$ (or $u \in C^{3, α- 1 + ε} (\barΩ)$ if $α\ge 1$). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solution $u \in C^{1,1}(\barΩ)$ for any $0<α<2$. Moreover, if the solution of tempered fractional Poisson problems satisfies $u \in C^{p, s}(\barΩ)$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy of $O(h^{p+s})$. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen-Cahn equation and Gray-Scott equations.

math.NA↗

arXiv2019

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-Δ)^{\fracα{2}}$ ($0 < α< 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $α\to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-Δ$. We prove that our methods are consistent if $u \in C^{\lfloorα\rfloor, α-\lfloorα\rfloor+ε}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^ε)$, with $ε> 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloorα\rfloor, α-\lfloorα\rfloor+ε}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $α\in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.

math.NA↗

arXiv2017

A comparative study on nonlocal diffusion operators related to the fractional Laplacian

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as α\to 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k \in N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any α\in (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size δis sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(δ^{-α}). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as α\to 2, it generally provides inconsistent result from that of the fractional Laplacian if α\ll 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

math.NA↗

arXiv2014

Computing the ground and first excited states of the fractional Schrodinger equation in an infinite potential well

In this paper, we numerically study the ground and first excited states of the fractional Schrodinger equation in an infinite potential well. Due to the non-locality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrodinger equation either analytically or numerically. We first introduce a fractional gradient flow with discrete normalization and then discretize it by using the trapezoidal type quadrature rule in space and the semi-implicit Euler method in time. Our method can be used to compute the ground and first excited states not only in the linear cases but also in the nonlinear cases. Moreover, it can be generalized to solve the fractional partial differential equations (PDEs) with Riesz fractional derivatives in space. Our numerical results suggest that the eigenfunctions of the fractional Schrodinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrodinger equation. In addition, we find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrodinger equation.

math-ph↗

arXiv2013

An efficient spectral method for computing dynamics of rotating two-component Bose--Einstein condensates via coordinate transformation

In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose--Einstein condensates (BECs) which is described by coupled Gross--Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in literature. Numerical results show that to achieve the same accuracy, our method needs much shorter computing time. We also applied our method to study the dynamic properties of rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross--Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.

math.NA↗

arXiv2013

A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via a rotating Lagrangian coordinate

We propose a simple, efficient and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with/without a long-range dipole-dipole interaction. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range dipole-dipole interaction, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential and review some dynamical laws related to the 2D and 3D GPE. By introducing a rotating Lagrangian coordinate system, the original GPEs are re-formulated to GPEs without the angular momentum rotation which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable and very efficient in computation. It is spectral order accurate in space and second-order accurate in time, and conserves the mass in the discrete level. Extensive numerical results are reported to demonstrate the efficiency and accuracy of the new numerical method. Finally, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation and center-of-mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without/with the long-range dipole-dipole interaction.

cond-mat.quant-gas↗

arXiv2007

Dynamics of vortices in weakly interacting Bose-Einstein condensates

We study the dynamics of vortices in ideal and weakly interacting Bose-Einstein condensates using a Ritz minimization method to solve the two-dimensional Gross-Pitaevskii equation. For different initial vortex configurations we calculate the trajectories of the vortices. We find conditions under which a vortex-antivortex pair annihilates and is created again. For the case of three vortices we show that at certain times two additional vortices may be created, which move through the condensate and annihilate each other again. For a noninteracting condensate this process is periodic, whereas for small interactions the essential features persist, but the periodicity is lost. The results are compared to exact numerical solutions of the Gross-Pitaevskii equation confirming our analytical findings.

quant-ph↗