Subjects: Mathematics >> Computational Mathematics. submitted time 2024-01-04
Abstract： 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.
Peer Review Status:
Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2023-12-25
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Physics >> Nuclear Physics Subjects: Mathematics >> Computational Mathematics. submitted time 2023-09-29
Abstract： 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 easy
to solve using traditional numerical methods albeit still challenging using neural networks for a wide range
of 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 lowest
eigenvalue 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 by
minimizing 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 from
the field of nuclear reactor physics. The methods were compared with the standard finite element method. The
applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN
and DRM.
Subjects: Mathematics >> Computational Mathematics. submitted time 2023-04-20
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2023-02-15 Cooperative journals: 《桂林电子科技大学学报》
Abstract： The aim of this paper is to estimate the risk of Zika and dengue fever (DEN)infection that imports from Asia and
causes local outbreaks. Based on the oversea epidemic data and the floating population between 2015 and 2017, an input
model was built to estimate the number of imported cases, and the local epidemic transmission probability and basic regeneration
number were calculated via the branching process under different temperatures and herd immunity levels. The imported
cases of Zika in China mainly came from Singapore, Thailand and Vietnam, with the predicted number of cases as 7.0
(95% CI:6.5-7.5), 2.0 (95% CI:1.8-2.2) and 1.0 (95% CI:0.9-1.1). The imported DEN cases were mainly from Thailand,
Malaysia, Singapore, Vietnam, Philippines, Indonesia, India and Korea, the predicted number of cases as 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:268.6-285.3),
241.20 (95% CI:233.6-248.8), 67.03 (95% CI:59.6-74.5), 9.1 (95% CI:6.7-11.3) and 3.0 (95% CI:1.9-4.1). The
optimum temperature for Zika and DEN transmission is around 28.9 ℃, in which the risk probability of local transmission is
24.4% and 99.9%, respectively. When the human herd immunity level is 0, 0.2 and 0.6, the basic reproduction numbers
of Zika and DEN are 8.1, 6.7, 3.2 and 3.2, 2.7, 1.3, respectively. The imported cases mainly come from South Asia.
South-central and south-eastern China are top-risk areas for local transmission, especially in June-August. The infection in
Singapore is more likely to cause Zika outbreak in China, while the infection in Thailand, Vietnam, Malaysia and Singapore
are the biggest cause of local transmission of DEN in China.
Subjects: Mathematics >> Computational Mathematics. submitted time 2023-02-15 Cooperative journals: 《桂林电子科技大学学报》
Abstract： In order to further improve the numerical accuracy of solving Volterra integro-differential, two kinds of Legendre
spectral Galerkin numerical integration methods are investigated for the Volterra-type integro-differential equation with variable
coefficients. Firstly, the Galerkin Legendre numerical integration is applied to deal with the integral term of the Volterra-
type integro-differential equations. Secondly, the Legendre tau scheme is developed for the Volterra-type integral-differential
equations with variable coefficient, and the Chebyshev-Gauss-Lobatto collocation point is used to the calculation of the
variable coefficient and integral term. Finally, by decomposing the definition interval of the function, the multi-interval Legendre
spectral Galerkin numerical integration method is also designed. Its scheme of the proposed method has symmetric
structure for odd-order model. In addition, by introducing the least squares function of the Volterra type integro-differential
equation, the Legendre spectral Galerkin least-squares numerical integration method of is constructed. The corresponding
coefficient matrix of the algebraic equation is symmetric positive. Some numerical examples are given to test the high-order
accuracy and the effectiveness of our methods.
Subjects: Mathematics >> Computational Mathematics. submitted time 2022-08-25
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2022-03-22
Abstract：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。
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2020-10-19
Abstract： Before entering into a recommender system, an entity name must be embedded into a vector. Some popular models, such as word2vec, are based on the principle “words which are in the same syntactic position should embedded into similar vectors”. However, sequence of entity names has no syntactic structure, which led to the low quality of name vectors. Based on the principle “neighbouring names should embedded into similar vectors”, this paper proposes a novel algorithm named name2vec. Name2vec has new features: vector length equals 1, relative weight which has solved the low frequency problem, optimization objective function is mean square error rather than cross entropy. The quality of embedding is measured by the similarity of entity names. On there datasets from WEIBO.COM, name2vec has a better performance than word2veec.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science submitted time 2020-03-16
Abstract： "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2019-11-26
Abstract： This paper proposes a novel clustring algorithm named Sliding Means, aiming to take the place of k-means algorithm which is widely used in internet applications. Sliding means has the ability to handle with very large datasets, and to automatically determine the number of clusters. With the help of shuffling samples, bad initial centroids have little chance to be selected. Sliding means is also able to drop some bad centroids on the fly. On the iris dataset and optdigits dataset, sliding means achieves better performance(Adjusted Rand Index) than k-means by 9.93% and 5.17% respectively.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2019-08-27
Abstract： A generalized conjugate gradient method is proposed to solve eigenvalue problems. This method is designed by combining the dumping block inverse power scheme, subspace projection method. Furthermore, based on the properties of the proposed method, a series of optimization techniques is developed to improve the stability, computing efficiency and scalability. We also introduce a computing package GCGE (Generalized Conjugate Gradient Eigensolver) which is developed based on the proposed method here. Some numerical examples are provided to validate the stability, computing efficiency and scalability of the method in this paper. The corresponding computing package can be downloaded from the web site: https://github.com/pase2017/GCGE-1.0.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2019-04-10
Abstract： This paper proposes a novel method named Polyhedron Regression(PR) for Click-Through-Rate prediction, aiming to take the place of Factorization Machines(FM). PR constructs a convex polyhedra with hyperplanes to separate positive samples from negative samples. PR has intuitionistic geometrical interpretations and a Lipschitz continuous surface, converges to global optimum point from arbitrary initial values. Compared with FM, PR has better classification accuracy, interpretability and surface smoothness on the three artificial datasets. With comparable parameters and computation, PR achieves better AUC than FM on Avazu and Criteo datasets.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2018-04-03
Abstract： In this paper, I found the two reasons of overfitting of logistic regression: boundary samples occupy a larger and larger share as the length of normal vector becomes longer and longer, boundary samples do not fit their probability density function well. With the help of insight in overfitting, I propose a acceleration method for logistic regression and got a training speedup of 38.25 on MNIST dataset, a training speedup of 5.61 on CIFAR10 dataset.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. submitted time 2018-03-22
Abstract： In this paper, I found the two reasons of overfitting of logistic regression: boundary samples occupy a larger and larger share as the length of normal vector becomes longer and longer, boundary samples do not fit their probability density function well. With the help of insight in overfitting, I propose a acceleration method for logistic regression and got a training speedup of 38.25 on MNIST dataset, a training speedup of 5.61 on CIFAR10 dataset.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Computational Mathematics. Subjects: Mathematics >> Applied Mathematics submitted time 2017-08-22
Abstract：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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. Subjects: Mathematics >> Computational Mathematics. submitted time 2016-07-11
Abstract：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.
Peer Review Status:Awaiting Review
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