分类: 数学 >> 计算数学 提交时间: 2024-01-04
摘要: This paper presents error analysis of stabilizer free weak Galerkin finite element method (SFWG-FEM) for a second order elliptic equation with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However, if the solutions are in $H^{1+s}$ with $0< s < 1$, numerical experiments show that the SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard $H^{2}$ finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The ($P_{k}(T),P_{k-1}(e), P_{k+1}(T) ^d$) elements with dimensions of space $d = 2,3$ are employed and the numerical examples are tested to confirm the theory.
分类: 数学 >> 计算数学 提交时间: 2023-12-25
摘要: This paper introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a-priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.
分类: 物理学 >> 核物理学 分类: 数学 >> 计算数学 提交时间: 2023-09-29
摘要: Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easyto solve using traditional numerical methods albeit still challenging using neural networks for a wide rangeof practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Net#2;work (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neu#2;tron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowesteigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface condi#2;tions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue byminimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN,PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains fromthe field of nuclear reactor physics. The methods were compared with the standard finite element method. Theapplicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNNand DRM.
分类: 数学 >> 计算数学 提交时间: 2023-04-20
摘要: We present the Parareal-CG algorithm for time-dependent differential equations in this work. The algorithm is a parallel in time iteration algorithm utilizes Chebyshev-Gauss spectral collocation method for fine propagator F and backward Euler method for coarse propagator G. As far as we know, this is the first time that the spectral method used as the F propagator of the parareal algorithm. By constructing the stable function of the Chebyshev-Gauss spectral collocation method for the symmetric positive definite (SPD) problem, we find out that the Parareal-CG algorithm and the Parareal-TR algorithm, whose F propagator is chosen to be a trapezoidal ruler, converge similarly, i.e., the Parareal-CG algorithm converge as fast as Parareal-Euler algorithm with sufficient Chebyhsev-Gauss points in every coarse grid. Numerical examples including ordinary differential equations and time-dependent partial differential equations are given to illustrate the high efficiency and accuracy of the proposed algorithm.
分类: 数学 >> 计算数学 提交时间: 2023-02-15 合作期刊: 《桂林电子科技大学学报》
摘要: 估计寨卡和登革热(DEN)疫情从亚洲输入中国及引起本地暴发的风险。基于20152017年国外疫情和流动人口 数据,构建输入模型估计输入病例数,并计算基于不同温度和群体免疫水平下分支过程的本地疫情传播概率及基本再生 数。中国的寨卡输入病例主要来自新加坡、泰国和越南,预测的病例数分别为7.0(95% CI:6.5~7.5)、2.0(95% CI:1.8~ 2.2)和1.0(95% CI:0.9~1.1);登革热输入病例主要来自泰国、马来西亚、新加坡、越南、菲律宾、印度尼西亚、印度和韩 国,预测的病例数分别为700.0(95% CI:679.8~720.2)、654.1(95% CI:641.8~666.2)、376.3(95%CI:368.2~384.1)、 277.1(95% CI:268.55~285.33)、241.2(95% CI:233.6~248.8)、67.0(95% CI:59.6~74.5)、9.1(95% CI:6.7~11.3)和 3.0(95% CI:1.9~4.1)。温度在28.9 ℃左右是最适宜寨卡和登革热传播的条件,此时发生本地传播的风险概率分别为 24.4%和99.9%。将人类群体免疫水平从0增加到0.2和0.6,寨卡和登革热的基本再生数分别为8.1、6.7、3.2和3.2、 2.7、1.3。输入病例更多来自于南亚,中国中南和东南是暴发本地传播的高风险地区,特别是在6-8月。新加坡疫情更容易 导致寨卡在中国传播,而泰国、越南、马来西亚和新加坡疫情是中国发生登革热本地传播的最大导火索。
分类: 数学 >> 计算数学 提交时间: 2023-02-15 合作期刊: 《桂林电子科技大学学报》
摘要: 为了进一步提高求解Volterra型积分微分的数值精度,针对一种变系数Volterra型积分微分方程,提出了2种 Legendre 谱Galerkin 数值积分法。采用Galerkin Legendre 数值积分对 Volterra 型积分微分方程的积分项进行预处理,对 其构造Legendre tau 格式,同时用Chebyshev-Gauss-Lobatto 配置点对变系数和积分项部分进行计算,并通过对方程的定义 区间进行分解,提出了一种多区间 Legendre 谱Galerkin 数值积分法。该方法的格式对于奇数阶模型具有对称结构。此 外,通过引入Volterra 型积分微分方程的最小二乘函数,构造了Legendre谱Galerkin最小二乘数值积分法。该方法对应的 代数方程系数矩阵是对称正定的。数值算例验证了这2种Legendre 谱Galerkin 数值积分方法的高阶精度和有效性。
分类: 数学 >> 计算数学 提交时间: 2022-08-25
摘要: In this paper, we study the linear complementarity problems on the monotone ex#2;tended second order cones. We demonstrate that the linear complementarity problem on the monotone extended second order cone can be converted into a mixed comple#2;mentarity problem on the non-negative orthant. We prove that any point satisfying the FB equation is a solution of the converted problem. We also show that the semi#2;smooth Newton method could be used to solve the converted problem, and we also provide a numerical example. Finally, we derive the explicit solution of a portfolio optimisation problem based on the monotone extended second order cone.
分类: 数学 >> 计算数学 提交时间: 2022-03-22
摘要: 0.618法是一维线搜索中针对一维单峰函数,应用最为广泛的一种方法。具有良好的收敛性,但其收敛性太慢,因此,本文基于函数在搜索区间端点和区间内任一点函数值的基础上,给出了一种普适性的线搜索加速策略,每步迭代都可以在较大程度上缩小函数值的不确定性区间。数值试验结果表明,其收敛速度较0.618法有所提高,尤其是当初始区间两端函数值相差较大或很大的情况下,本文改进算法可以很大程度上减小区间范围。
分类: 数学 >> 计算数学 提交时间: 2020-10-19
摘要: 在进入推荐系统之前,商品名、人名等实体名字需要嵌入低维向量。word2vec这样的流行嵌入算法的出发点是“相同语法位置上的词具有相似的向量”,而名字序列没有语法结构,导致名字向量的质量不高。 本文从“相邻的名字具有相似的向量”出发,提出一个称为名字嵌入的新方法。名字嵌入使用了一些新技巧:公式比word2vec更简单,向量模长固定为1、用相对权重处理低频名字、优化目标使用简单的均方差。 以名字相似度作为衡量标准,在NBA球队名人造集、球队名微博集和微博点赞集上,名字嵌入均显著优于word2vec。
分类: 数学 >> 计算数学 分类: 信息科学与系统科学 >> 信息科学与系统科学基础学科 提交时间: 2020-03-16
摘要: 针对目前大多数的低秩张量填充(LRTC)模型存在过度稀疏而导致数据的细微特征被忽略的现象, 本文借助框架变换和低秩矩阵分解, 提出了一个基于近似稀疏的低秩张量填充(AS-LRTC) 模型, 进一步设计了块逐次上界极小化(BSUM) 算法求解该模型. 在一定条件下可以证明该算法的收敛性, 大量的实验结果表明本文提出的算法比现有一些经典算法有明显的优势.
分类: 数学 >> 计算数学 提交时间: 2019-11-26
摘要: 本文提出一个名为滑动均值的聚类算法,尝试替代常用的k均值算法。滑动均值能处理大量的样本,自行决定类别数量,用混洗样本来避免出现很差的中心点,能够中途裁减类别数量,聚类效果显著好于k均值。在鸢尾花数据和手写数字数据上,滑动均值的聚类效果比k均值分别高9.93%和5.17%。
分类: 数学 >> 计算数学 提交时间: 2019-08-27
摘要: 本文基于阻尼块反幂法与子空间投影算法设计了一种求解特征值问题的广义共轭梯度算法, 同时也实现了相应的计算软件包. 然后对算法和计算过程进行了一系列的优化来提高算法的稳定性、计算效率和并行可扩展性, 使得本文的算法适合在并行计算环境下求解大规模稀疏矩阵的特征值. 所形成的软件包是基于Matrix-Free和Vector-Free设计的, 可以应用于任意的矩阵向量结构. 针对几种典型矩阵的测试结果表明本文的算法和软件包不但具有良好的数值稳定性, 同时相比于SLEPc软件包中的LOBPCG以及Jacobi-Davidson解法器有2-6倍的效率提升. 软件包的网址: https://github.com/pase2017/GCGE-1.0.
分类: 数学 >> 计算数学 提交时间: 2019-04-10
摘要: 本文提出一个名为接圆回归的点击率预测新方法,尝试替代常用的因子分解机(FM)。接圆回归用超平面拼接出一个封闭凸多面体,圈出正样本,有直观的几何解释, 能从任意初始值一次收敛到全局最优解。 拟合出来的曲面Lipschitz连续,变化平缓。在人工设计的星环集、双堆集、双月集上,接圆回归的分类准确性、解释性、平滑性全面超过FM。在同量级参数量、计算量 的条件下,接圆回归在Avazu集和Criteo集上的AUC超过FM。
分类: 数学 >> 计算数学 提交时间: 2018-04-03
摘要: 本文通过严格数学分析找出了逻辑回归过拟合的成因:边界样本的损失贡献比重大且随法向量增长而加速增大、边界样本分布散乱,顺便理清了正则项的作用机理。 利用过拟合机制,本文提出一种反拉方法,既能缓解过拟合,又能减少训练步数,在MNIST数据集上实现加速38.25倍,在CIFAR10数据集上实现加速5.61倍。
分类: 数学 >> 计算数学 提交时间: 2018-03-22
摘要: 本文通过严格数学分析找出了逻辑回归过拟合的成因:边界样本的损失贡献比重大且随法向量增长而加速增大、边界样本分布散乱,顺便理清了正则项的作用机理。 利用过拟合机制,本文提出一种反拉方法,既能缓解过拟合,又能减少训练步数,在MNIST数据集上实现加速38.25倍,在CIFAR10数据集上实现加速5.61倍。
分类: 数学 >> 计算数学 分类: 数学 >> 应用数学 提交时间: 2017-08-22
摘要: Many numerical methods have been proposed in the last 30 years for inverse problems. While very successful in many cases, progress has lagged in other areas of applications which are forced to rely on {\em limited-aperture} measurements. In this paper, we introduce some techniques to retrieve the other data that can not be measured directly. We consider the inverse acoustic scattering of time harmonic plane waves and take the scattering amplitude to be the measurements. Assume that the scattering amplitude can only be measured with observation directions restricted in $S^{n-1}_0$, which is compactly supported in the unit sphere. Based on the reciprocity relation of the scattering amplitude, we prove a special symmetric structure of the corresponding multi-static response matrix. This will also be verified by numerical examples. Combining this, with the help of the Green's formula for the scattered field, we introduce an iterative scheme to retrieve approximate {\em full-aperture} scattering amplitude. As an application, using a recently proposed direct sampling method [28], we consider the fast and robust sampling methods with {\em limited-aperture} measurements. Some numerical simulations are conducted with noisy data, and the results will further verify the effectiveness and robustness of the proposed data retrieval method and of the sampling method for inverse acoustic scattering problems.
分类: 数学 >> 控制和优化 分类: 数学 >> 计算数学 提交时间: 2016-07-11
摘要: In this paper, we construct and analyze an efficient m-step Levenberg-Marquardt method for nonlinear equations. The main advantage of this method is that the m-step LM method could save more Jacobian calculations with frozen $(J_k^TJ_k+\lambda_kI)^{-1}J_k^T$ at every iteration. Under the local error bound condition which is weaker than nonsingularity, the m-step LM method has been proved to have $(m+1)$th convergence order. The global convergence has also been given by trust region technique. Numerical results show that the m-step LM method is efficient and could save many calculations of the Jacobian especially for large scale problems.