arXiv subjects

Yanzun Meng

Publications and source records attributed to Yanzun Meng.

5 recordsLinked to original sources

arXiv2025

Wavelet-based Global Orientation and Surface Reconstruction for Point Clouds

Unoriented surface reconstruction is an important task in computer graphics and has extensive applications. Based on the compact support of wavelet and orthogonality properties, classic wavelet surface reconstruction achieves good and fast reconstruction. However, this method can only handle oriented points. Despite some improved attempts for unoriented points, such as iWSR, these methods perform poorly on sparse point clouds. To address these shortcomings, we propose a wavelet-based method to represent the mollified indicator function and complete both the orientation and surface reconstruction tasks. We use the modifying kernel function to smoothen out discontinuities on the surface, aligning with the continuity of the wavelet basis function. During the calculation of coefficient, we fully utilize the properties of the convolutional kernel function to shift the modifying computation onto wavelet basis to accelerate. In addition, we propose a novel method for constructing the divergence-free function field and using them to construct the additional homogeneous constraints to improve the effectiveness and stability. Extensive experiments demonstrate that our method achieves state-of-the-art performance in both orientation and reconstruction for sparse models. We align the matrix construction with the compact support property of wavelet basis functions to further accelerate our method, resulting in efficient performance on CPU. Our source codes will be released on GitHub.

cs.CG↗

arXiv2024

A seamless local-nonlocal coupling diffusion model with $H^1$ vanishing nonlocality convergence

Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in $H^1$ norm can be derived.

math.AP↗

arXiv2024

Asymptotically Compatible Error Bound of Finite Element Method for Nonlocal Diffusion Model with An Efficient Implementation

This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes, the error is bounded by $O(h^k + \delta)$, where $h$ is the mesh size, $\delta$ is the nonlocal horizon, and $k$ is the order of the FEM basis. Without shape regularity, the bound becomes $O(h^{k+1}/\delta + \delta)$. In addition, we present an efficient implementation of the finite element method of nonlocal model. The direct implementation of the finite element method of nonlocal model requires computation of $2n$-dimensional integrals which are very expensive. For the nonlocal model with Gaussian kernel function, we can decouple the $2n$-dimensional integral to 2-dimensional integrals which reduce the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.

math.NA↗

arXiv2024

Maximum principle preserving nonlocal diffusion model with Dirichlet boundary condition

In this paper, we propose nonlocal diffusion models with Dirichlet boundary. These nonlocal diffusion models preserve the maximum principle and also have corresponding variational form. With these good properties, we can prove the well-posedness and the vanishing nonlocality convergence. Furthermore, by specifically designed weight function, we can get a nonlocal diffusion model with second order convergence which is optimal for nonlocal diffusion models.

math.AP↗

arXiv2024

A Second-Order Nonlocal Approximation to Manifold Poisson Models with Neumann Boundary

In this paper, we propose a class of nonlocal models to approximate the Poisson model on manifolds with homogeneous Neumann boundary condition, where the manifolds are assumed to be embedded in high dimensional Euclid spaces. In comparison to the existing nonlocal approximation of Poisson models with Neumann boundary, we optimize the truncation error of model by adding an augmented function involving the second order normal derivative along the $2\delta$ layer of boundary, with $2\delta$ be the nonlocal interaction horizon. The 2nd normal derivative is expressed as the difference between the interior Laplacian and the boundary Laplacian. The concentration of our paper is on the construction of nonlocal model, the well-posedness of model, and its second-order convergence rate to its local counterpart. The localization rate of our nonlocal model is currently optimal among all related works even for the case of high dimensional Euclid spaces.

math.NA↗