Numerical Methods for Partial Differential Equations最新文献

Compactness results for a Dirichlet energy of nonlocal gradient with applications 非局部梯度迪里夏特能量的紧凑性结果及其应用

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-09-14 DOI: 10.1002/num.23149

Zhaolong Han, Tadele Mengesha, Xiaochuan Tian

引用次数: 0

Layer‐parallel training of residual networks with auxiliary variable networks 残差网络与辅助变量网络的层并行训练

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-09-06 DOI: 10.1002/num.23147

Qi Sun, Hexin Dong, Zewei Chen, Jiacheng Sun, Zhenguo Li, Bin Dong

{"title":"Layer‐parallel training of residual networks with auxiliary variable networks","authors":"Qi Sun, Hexin Dong, Zewei Chen, Jiacheng Sun, Zhenguo Li, Bin Dong","doi":"10.1002/num.23147","DOIUrl":"https://doi.org/10.1002/num.23147","url":null,"abstract":"Gradient‐based methods for training residual networks (ResNets) typically require a forward pass of input data, followed by back‐propagating the error gradient to update model parameters, which becomes time‐consuming as the network structure goes deeper. To break the algorithmic locking and exploit synchronous module parallelism in both forward and backward modes, auxiliary‐variable methods have emerged but suffer from communication overhead and a lack of data augmentation. By trading off the recomputation and storage of auxiliary variables, a joint learning framework is proposed in this work for training realistic ResNets across multiple compute devices. Specifically, the input data of each processor is generated from its low‐capacity auxiliary network (AuxNet), which permits the use of data augmentation and realizes forward unlocking. Backward passes are then executed in parallel, each with a local loss function derived from the penalty or augmented Lagrangian (AL) method. Finally, the AuxNet is adjusted to reproduce updated auxiliary variables through an end‐to‐end training process. We demonstrate the effectiveness of our method on ResNets and WideResNets across CIFAR‐10, CIFAR‐100, and ImageNet datasets, achieving speedup over the traditional layer‐serial training approach while maintaining comparable testing accuracy.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Error bound of the multilevel fast multipole method for 3‐D scattering problems 三维散射问题多级快速多极子方法的误差约束

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-09-04 DOI: 10.1002/num.23148

Wenhui Meng

引用次数: 0

An explicit fourth‐order hybrid‐variable method for Euler equations with a residual‐consistent viscosity 具有残差一致粘度的欧拉方程的显式四阶混合变量法

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-26 DOI: 10.1002/num.23146

Xianyi Zeng

引用次数: 0

Exponential time difference methods with spatial exponential approximations for solving boundary layer problems 用于解决边界层问题的空间指数近似指数时差法

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-26 DOI: 10.1002/num.23145

Liyong Zhu, Xinwei Wang, Tianzheng Lu

引用次数: 0

A nonlocal convection–diffusion model with Gaussian‐type kernels and meshfree discretization 采用高斯型核和无网格离散化的非局部对流扩散模型

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-15 DOI: 10.1002/num.23141

Hao Tian, Xiaojuan Liu, Chenguang Liu, Lili Ju

{"title":"A nonlocal convection–diffusion model with Gaussian‐type kernels and meshfree discretization","authors":"Hao Tian, Xiaojuan Liu, Chenguang Liu, Lili Ju","doi":"10.1002/num.23141","DOIUrl":"https://doi.org/10.1002/num.23141","url":null,"abstract":"Nonlocal models have demonstrated their indispensability in numerical simulations across a spectrum of critical domains, ranging from analyzing crack and fracture behavior in structural engineering to modeling anomalous diffusion phenomena in materials science and simulating convection processes in heterogeneous environments. In this study, we present a novel framework for constructing nonlocal convection–diffusion models using Gaussian‐type kernels. Our framework uniquely formulates the diffusion term by correlating the constant diffusion coefficient with the variance of the Gaussian kernel. Simultaneously, the convection term is defined by integrating the variable velocity field into the kernel as the expectation of a multivariate Gaussian distribution, facilitating a comprehensive representation of convective transport phenomena. We rigorously establish the well‐posedness of the proposed nonlocal model and derive a maximum principle to ensure its stability and reliability in numerical simulations. Furthermore, we develop a meshfree discretization scheme tailored for numerically simulating our model, designed to uphold both the discrete maximum principle and asymptotic compatibility. Through extensive numerical experiments, we validate the efficacy and versatility of our framework, demonstrating its superior performance compared to existing approaches.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model 非局部 Ohta-Kawasaki 模型的渐近兼容方案

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-15 DOI: 10.1002/num.23143

Wangbo Luo, Yanxiang Zhao

{"title":"Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model","authors":"Wangbo Luo, Yanxiang Zhao","doi":"10.1002/num.23143","DOIUrl":"https://doi.org/10.1002/num.23143","url":null,"abstract":"We study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A variable‐step high‐order scheme for time‐fractional advection‐diffusion equation with mixed derivatives 具有混合导数的时间分数平流-扩散方程的可变步长高阶方案

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-14 DOI: 10.1002/num.23140

Junhong Feng, Pin Lyu, Seakweng Vong

引用次数: 0

Improved error bounds on a time splitting method for the nonlinear Schrödinger equation with wave operator 带波算子的非线性薛定谔方程时间分割法的改进误差范围

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-13 DOI: 10.1002/num.23139

Jiyong Li

引用次数: 0

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme 用列维函数方案数值求解椭圆方程的边界值问题

IF 3.9 3区数学

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-13 DOI: 10.1002/num.23142

Jinchao Pan, Jijun Liu

{"title":"Numerical solution of the boundary value problem of elliptic equation by Levi function scheme","authors":"Jinchao Pan, Jijun Liu","doi":"10.1002/num.23142","DOIUrl":"https://doi.org/10.1002/num.23142","url":null,"abstract":"For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0