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| 1 | A Parallel Algorithm for Adaptive Local Refinement of Tetrahedral Meshes Using Bisection显示文摘Local mesh refinement is one of the key steps in the implementations of adaptive finite element methods.This paper presents a parallel algorithm for distributed memory parallel computers for adaptive local refinement of tetrahedral meshes using bisection.This algorithm is used in PHG,Parallel Hierarchical Grid (http://gffzz8065c2988e934515hbbfcb9qnfxx96k6w.ffgz.tsg.suse.edu.cn/phg/),a toolbox under active development for parallel adaptive finite element solutions of partial differential equations.The algorithm proposed is characterized by allowing simultaneous refinement of submeshes to arbitrary levels before synchronization between submeshes and without the need of a central coordinator process for managing new vertices.Using the concept of canonical refinement, a simple proof of the independence of the resulting mesh on the mesh partitioning is given,which is useful in better understanding the behaviour of the bisectioning refinement procedure. | Lin-Bo Zhang | 2009 | Numerical Mathematics(Theory,Methods and Applications)2009，2，1： | 27 |
| 2 | Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations显示文摘The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations(FDEs)with Caputo derivatives,is an efficient and stable scheme.We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order indexα>0. | Jianfei Huang Yifa Tang Luis Vázquez | 2012 | Numerical Mathematics(Theory,Methods and Applications)2012，5，2： | 11 |
| 3 | Fast Texture Segmentation Based on Semi-Local Region Descriptor and Active Contour显示文摘In this paper,we present an efficient approach for unsupervised segmentation of natural and textural images based on the extraction of image features and a fast active contour segmentation model.We address the problem of textures where neither the gray-level information nor the boundary information is adequate for object extraction.This is often the case of natural images composed of both homogeneous and textured regions.Because these images cannot be in general directly processed by the gray-level information,we propose a new texture descriptor which intrinsically defines the geometry of textures using semi-local image information and tools from differential geometry.Then,we use the popular Kullback-Leibler distance to design an active contour model which distinguishes the background and textures of interest.The existence of a minimizing solution to the proposed segmentation model is proven.Finally, a texture segmentation algorithm based on the Split-Bregman method is introduced to extract meaningful objects in a fast way.Promising synthetic and real-world results for gray-scale and color images are presented. | Nawal Houhou Jean-Philippe Thiran Xavier Bresson | 2009 | Numerical Mathematics(Theory,Methods and Applications)2009，2，4： | 10 |
| 4 | Advances in Studies and Applications of Centroidal Voronoi Tessellations显示文摘Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating. | Qiang Du Max Gunzburger Lili Ju | 2010 | Numerical Mathematics(Theory,Methods and Applications)2010，3，2： | 6 |
| 5 | On the Approximation of the Derivatives of Spline Quasi-Interpolation in Cubic Spline Space S_(3)^(1,2)(∆_(mn)^((2)))显示文摘In this paper,based on the basis composed of two sets of splines with distinct local supports,cubic spline quasi-interpolating operators are reviewed on nonuniform type-2 triangulation.The variation diminishing operator is defined by discrete linear functionals based on a fixed number of triangular mesh-points,which can reproduce any polynomial of nearly best degrees.And by means of the modulus of continuity,the estimation of the operator approximating a real sufficiently smooth function is reviewed as well.Moreover,the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation.And then the convergence results are worked out. | Jiang Qian Fan Wang | 2014 | Numerical Mathematics(Theory,Methods and Applications)2014，7，1： | 6 |
| 6 | A Coordinate Gradient Descent Method for Nonsmooth Nonseparable Minimization显示文摘This paper presents a coordinate gradient descent approach for minimizing the sum of a smooth function and a nonseparable convex function.We find a search direction by solving a subproblem obtained by a second-order approximation of the smooth function and adding a separable convex function.Under a local Lipschitzian error bound assumption,we show that the algorithm possesses global and local linear convergence properties.We also give some numerical tests(including image recovery examples) to illustrate the efficiency of the proposed method. | Zheng-Jian Bai Michael K. Ng Liqun Qi | 2009 | Numerical Mathematics(Theory,Methods and Applications)2009，2，4： | 6 |
| 7 | A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation显示文摘In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results. | Ningning Yan Zhaojie Zhou | 2008 | Numerical Mathematics(Theory,Methods and Applications)2008，1，3： | 5 |
| 8 | A Compound Algorithm of Denoising Using Second-Order and Fourth-Order Partial Differential Equations显示文摘In this paper,we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LET model with a parameter functionθ.The numerical experiments demonstrate that our compound algorithm is efficient and preserves the main advantages of the two models.In particular, the errors of the compound algorithm in L2 norm between the exact images and corresponding restored images are the smallest among the three models.For images with strong noises,the restored images of the compound algorithm are the best in the corresponding restored images.The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration.It is found that the combination of these methods is efficient and robust in the image restoration. | Qianshun Chang Xuecheng Tai Lily Xing | 2009 | Numerical Mathematics(Theory,Methods and Applications)2009，2，4： | 5 |
| 9 | The Simultaneous Approximation Average Errors for Bernstein Operators on the r-Fold Integrated Wiener Space显示文摘For weighted approximation in Lp-norm,we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the r-fold integrated Wiener space. | Guiqiao Xu | 2012 | Numerical Mathematics(Theory,Methods and Applications)2012，5，3： | 5 |
| 10 | Superconvergence and L^(∞)-Error Estimates of RT1Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint显示文摘In this paper,we investigate the superconvergence property and the L∞-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint.The state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear functions.We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions.Moreover,we derive L∞-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions.Finally,some numerical examples are given to demonstrate the theoretical results. | Yanping Chen Tianliang Hou | 2012 | Numerical Mathematics(Theory,Methods and Applications)2012，5，3： | 5 |
| 11 | Higher Order Triangular Mixed Finite Element Methods for Semilinear Quadratic Optimal Control Problems显示文摘In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results. | Kang Deng Yanping Chen Zuliang Lu | 2011 | Numerical Mathematics(Theory,Methods and Applications)2011，4，2： | 4 |
| 12 | A Second-Order Method for the Electromagnetic Scattering from a Large Cavity显示文摘In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media. | Yingxi Wang Kui Du Weiwei Sun | 2008 | Numerical Mathematics(Theory,Methods and Applications)2008，1，4： | 4 |
| 13 | A Collocation Method for Initial Value Problems of Second-Order ODEs by Using Laguerre Functions显示文摘We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions.This new process provides global numerical solutions.Numerical results demonstrate the efficiency of the proposed algorithm. | Jian-Ping Yan Ben-Yu Guo 无 | 2011 | Numerical Mathematics(Theory,Methods and Applications)2011，4，2： | 4 |
| 14 | A Source Transfer Domain Decomposition Method For Helmholtz Equations in Unbounded Domain Part II: Extensions显示文摘In this paper we extend the source transfer domain decomposition method(STDDM)introduced by the authors to solve the Helmholtz problems in two-layered media,the Helmholtz scattering problems with bounded scatterer,and Helmholtz problems in 3D unbounded domains.The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers.The details of STDDM is given for each extension.Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper. | Zhiming Chen Xueshuang Xiang | 2013 | Numerical Mathematics(Theory,Methods and Applications)2013，6，3： | 4 |
| 15 | High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations显示文摘As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other areas.This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin(DG)methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space.The Caputo derivative is chosen as the representation of spatial derivative,because it may represent the fractional derivative by an integral operator.Some numerical examples show that the convergence orders of the proposed local Pk–DG methods are O(hk+1)both in one and two dimensions,where Pk denotes the space of the real-valued polynomials with degree at most k. | Xia Ji Huazhong Tang | 2012 | Numerical Mathematics(Theory,Methods and Applications)2012，5，3： | 4 |
| 16 | A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation显示文摘A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation;we construct a finite difference scheme on a priori (se-quentially) adapted meshes and study its convergence.The scheme on a priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution,which allows us to find a priori a subdomain where the computed solution requires a further improvement.This subdomain is defined by the perturbation parameterε,the step-size of a uniform mesh in x,and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for im- proving the solution.To solve the discrete problems aimed at the improvement of the solution,we use uniform meshes on the subdomains.The error of the numerical so- lution depends weakly on the parameterε.The scheme converges almostε-uniformly, precisely,under the condition N^(-1)=o(ε~v),where N denotes the number of nodes in the spatial mesh,and the value v=v(K) can be chosen arbitrarily small for suitable K. | Grigory I.Shishkin | 2008 | Numerical Mathematics(Theory,Methods and Applications)2008，1，2： | 4 |
| 17 | Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations显示文摘A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods. | Yunxia Wei Yanping Chen | 2011 | Numerical Mathematics(Theory,Methods and Applications)2011，4，3： | 3 |
| 18 | Numerical Analysis of a System of Singularly Perturbed Convection-Diffusion Equations Related to Optimal Control显示文摘We consider an optimal control problem with an 1D singularly perturbed differential state equation.For solving such problems one uses the enhanced system of the state equation and its adjoint form.Thus,we obtain a system of two convectiondiffusion equations.Using linear finite elements on adapted grids we treat the effects of two layers arising at different boundaries of the domain.We proof uniform error estimates for this method on meshes of Shishkin type.We present numerical results supporting our analysis. | Hans-Görg Roos Christian Reibiger | 2011 | Numerical Mathematics(Theory,Methods and Applications)2011，4，4： | 3 |
| 19 | Partial Shape Matching Without Point-Wise Correspondence显示文摘Partial similarity of shapes is a challenging problem arising in many important applications in computer vision,shape analysis,and graphics,e.g.when one has to deal with partial information and acquisition artifacts.The problem is especially hard when the underlying shapes are non-rigid and are given up to a deformation.Partial matching is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise non-bijective correspondence between the two,taking into account possibly different parts.In this paper,we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation.We use region-wise local descriptors and optimize over the integration domains on which the integral descriptors of the two parts match.The problem is regularized using the Mumford-Shah functional.We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular point clouds and meshes,and present experiments demonstrating the success of the proposed method. | Jonathan Pokrass Alexander M.Bronstein Michael M.Bronstein | 2013 | Numerical Mathematics(Theory,Methods and Applications)2013，6，1： | 3 |
| 20 | Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions显示文摘In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry. | Xiao-Yong Zhang Ben-Yu Guo Yu-Jian Jiao | 2009 | Numerical Mathematics(Theory,Methods and Applications)2009，2，1： | 3 |
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