arXiv subjects

Zhimin Zhang

Publications and source records attributed to Zhimin Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2023

Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa--Holm system with small energy

A rotation-two-component Camassa-Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are $O(τ^2 + h^2)$ for the velocity in the $L^\infty$-norm and the surface elevation in the $L^2$-norm, where $τ$ denotes the temporal stepsize and $h$ the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Comparing with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.

math.NA↗

arXiv2023

Nonconforming finite elements for the Brinkman and $-\text{curl}Δ\text{curl}$ problems on cubical meshes

We propose two families of nonconforming elements on cubical meshes: one for the $-\text{curl}Δ\text{curl}$ problem and the other for the Brinkman problem. The element for the $-\text{curl}Δ\text{curl}$ problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter $ν$. The lowest-order elements for the $-\text{curl}Δ\text{curl}$ and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of $H(\text{curl};Ω)$ and $H(\text{div};Ω)$, and they, as nonconforming approximation to $H(\text{gradcurl};Ω)$ and $[H^1(Ω)]^3$, can form a discrete Stokes complex together with the Lagrange element and the $L^2$ element.

math.NA↗

arXiv2022

When Six Degrees of Separation Meets Online Social Networks: How Low Can the Degree Be?

The proposal of Six Degrees of Separation makes people realize that the world is not as big as we imagined. Even if the world's population now exceeds 7 billion, two strangers can still get in touch through a limited intermediary. When online social networks have taken the world by storm, can people connect through shorter distances? This issue is worth thinking about. This paper describes Six Degrees of Separation and the limitations of this theory in practical applications. Combined with online social networks, the paper analyzes the development and change of the degrees of connection between people. Finally, the paper considers the actual coverage of online social networks and other issues, and rationalizes the possible degrees of the future Six Degrees of Separation.

cs.SI↗

arXiv2022

Improving Stack Overflow question title generation with copying enhanced CodeBERT model and bi-modal information

Context: Stack Overflow is very helpful for software developers who are seeking answers to programming problems. Previous studies have shown that a growing number of questions are of low quality and thus obtain less attention from potential answerers. Gao et al. proposed an LSTM-based model (i.e., BiLSTM-CC) to automatically generate question titles from the code snippets to improve the question quality. However, only using the code snippets in the question body cannot provide sufficient information for title generation, and LSTMs cannot capture the long-range dependencies between tokens. Objective: This paper proposes CCBERT, a deep learning based novel model to enhance the performance of question title generation by making full use of the bi-modal information of the entire question body. Method: CCBERT follows the encoder-decoder paradigm and uses CodeBERT to encode the question body into hidden representations, a stacked Transformer decoder to generate predicted tokens, and an additional copy attention layer to refine the output distribution. Both the encoder and decoder perform the multi-head self-attention operation to better capture the long-range dependencies. This paper builds a dataset containing around 200,000 high-quality questions filtered from the data officially published by Stack Overflow to verify the effectiveness of the CCBERT model. Results: CCBERT outperforms all the baseline models on the dataset. Experiments on both code-only and low-resource datasets show the superiority of CCBERT with less performance degradation. The human evaluation also shows the excellent performance of CCBERT concerning both readability and correlation criteria.

cs.CL↗

arXiv2022

A new family of nonconforming elements with $\pmb{H}(\mathrm{curl})$-continuity for the three-dimensional quad-curl problem

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of $\pmb{H}(\mathrm{curl})$, but not of $\pmb{H}(\mathrm{grad}~\mathrm{curl})$, which are different from the existing nonconforming ones. The well-posedness of the discrete problem is proved and optimal error estimates in discrete $\pmb{H}(\mathrm{grad}~\mathrm{curl})$ norm, $\pmb{H}(\mathrm{curl})$ norm and $\pmb{L}^2$ norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.

math.NA↗

arXiv2022

Nonconforming finite element approximations and the analysis of Nitsche's method for a singularly perturbed quad-curl problem in three dimensions

We introduce and analyze a robust nonconforming finite element method for a three dimensional singularly perturbed quad-curl model problem. For the solution of the model problem, we derive proper a priori bounds, based on which, we prove that the proposed finite element method is robust with respect to the singular perturbation parameter $\varepsilon$ and the numerical solution is uniformly convergent with order $h^{1/2}$. In addition, we investigate the effect of treating the second boundary condition weakly by Nitsche's method. We show that such a treatment leads to sharper error estimates than imposing the boundary condition strongly when the parameter $\varepsilon< h$. Finally, numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.

math.NA↗

arXiv2022

An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/nonconforming meshes and h-version/p-version refinement. The optimal h-version convergence rate and the exponential p-version convergence rate are observed.

math.NA↗

arXiv2022

$\bf H(\mathrm{curl}^2)$ conforming element for Maxwell's transmission eigenvalue problem using fixed-point approach

Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, we solve the Maxwell's transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf H}(\mathrm{curl})$-semi-norm) are established.

math.NA↗

arXiv2022

Some mathematical aspects of Anderson localization: boundary effect, multimodality, and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.

math-ph↗

arXiv2022

Unsupervised Manga Character Re-identification via Face-body and Spatial-temporal Associated Clustering

In the past few years, there has been a dramatic growth in e-manga (electronic Japanese-style comics). Faced with the booming demand for manga research and the large amount of unlabeled manga data, we raised a new task, called unsupervised manga character re-identification. However, the artistic expression and stylistic limitations of manga pose many challenges to the re-identification problem. Inspired by the idea that some content-related features may help clustering, we propose a Face-body and Spatial-temporal Associated Clustering method (FSAC). In the face-body combination module, a face-body graph is constructed to solve problems such as exaggeration and deformation in artistic creation by using the integrity of the image. In the spatial-temporal relationship correction module, we analyze the appearance features of characters and design a temporal-spatial-related triplet loss to fine-tune the clustering. Extensive experiments on a manga book dataset with 109 volumes validate the superiority of our method in unsupervised manga character re-identification.

cs.CV↗

arXiv2022

Variable step-size BDF3 method for Allen-Cahn equation

In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergence analysis are well established in [Liao and Zhang, \newblock Math. Comp., \textbf{90} (2021) 1207--1226; Chen, Yu, and Zhang, \newblock SIAM J. Numer. Anal., Major Revised]. However, the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial, since the DOC kernels are not always positive. By developing a novel spectral norm inequality, the unconditional stability and convergence are rigorously proved under the updated step ratio restriction $r_k:=τ_k/τ_{k-1}\leq 1.405$ (compared with $r_k\leq 1.199$ in [Calvo and Grigorieff, \newblock BIT. \textbf{42} (2002) 689--701]) for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

math.NA↗

arXiv2022

Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn-Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise $L^2$ norm error estimate is established at the first time under an updated step-ratio restriction $0 <r_k:=τ_k/τ_{k-1}\leq r_{\mathrm{user}}$, where $r_{\mathrm{user}}$ can be chosen by the user such that $r_{\mathrm{user}}<4.864$. The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if $0<r_k\le r_{\mathrm{user}}$, such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state.

math.NA↗

arXiv2021

Spurious solutions for high order curl problems

We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. To explain the numerical results, we clarify the cohomological structures in high order curl problems by relating the partial differential equations to the Hodge-Laplacian boundary problems of the gradcurl-complexes.

math.NA↗

arXiv2021

A family of finite element Stokes complexes in three dimensions

We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition. We demonstrate the validity of the algorithms by numerical experiments.

math.NA↗

arXiv2021

Hierarchy-Aware T5 with Path-Adaptive Mask Mechanism for Hierarchical Text Classification

Hierarchical Text Classification (HTC), which aims to predict text labels organized in hierarchical space, is a significant task lacking in investigation in natural language processing. Existing methods usually encode the entire hierarchical structure and fail to construct a robust label-dependent model, making it hard to make accurate predictions on sparse lower-level labels and achieving low Macro-F1. In this paper, we propose a novel PAMM-HiA-T5 model for HTC: a hierarchy-aware T5 model with path-adaptive mask mechanism that not only builds the knowledge of upper-level labels into low-level ones but also introduces path dependency information in label prediction. Specifically, we generate a multi-level sequential label structure to exploit hierarchical dependency across different levels with Breadth-First Search (BFS) and T5 model. To further improve label dependency prediction within each path, we then propose an original path-adaptive mask mechanism (PAMM) to identify the label's path information, eliminating sources of noises from other paths. Comprehensive experiments on three benchmark datasets show that our novel PAMM-HiA-T5 model greatly outperforms all state-of-the-art HTC approaches especially in Macro-F1. The ablation studies show that the improvements mainly come from our innovative approach instead of T5.

cs.CL↗

arXiv2021

Simple curl-curl-conforming finite elements in two dimensions

We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite elements have only 6 and 8 degrees of freedom, respectively. Numerical experiments for each family demonstrate the convergence and efficiency of the elements for solving the quad-curl problem.

math.NA↗

arXiv2021

FE-Holomorphic Operator Function Method for Nonlinear Plate Vibrations with Elastically Added Masses

Vibrations of structures subjected to concentrated point loads have many applications in mechanical engineering. Experiments are expensive and numerical methods are often used for simulations. In this paper, we consider the plate vibration with nonlinear dependence on the eigen-parameter. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function. The Bogner-Fox-Schmit element is used for the discretization and the spectral indicator method is employed to compute the eigenvalues. The convergence is proved using the abstract approximation theory of Karma. Numerical examples are presented for validations.

math.NA↗

arXiv2021

Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains

It is well known that the usual mixed method for solving the biharmonic eigenvalue problem by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of $\mathcal{O}({h}^{2α})$ ($ 0<α<π/ω$, $ω> π$ is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$ convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.

math.NA↗