Preconditioned Iterative Methods for Algebraic Systems from Multiplicative Half-Quadratic Regularization Image Restorations

<正>Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term.A regularized convex term can usually preserve the image edges well in the restored image.In this paper,we consider a class of convex and edge-preserving regularization functions,i.e.,multiplicative half-quadratic regularizations,and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations.At each Newton iterate,the preconditioned conjugate gradient method,incorporated with a constraint preconditioner,is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived,which can be used to estimate the convergence speed of the preconditioned conjugate gradient method.We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably well.     

Gas-Kinetic BGK Scheme for Three Dimensional Magnetohydrodynamics

<正>The gas-kinetic theory based flux splitting method has been successfully proposed for solving one-and two-dimensional ideal magnetohydrodynamics by Xu et al. [J.Comput.Phys.,1999;2000],respectively.This paper extends the kinetic method to solve three-dimensional ideal magnetohydrodynamics equations,where an adaptive parameter 17 is used to control the numerical dissipation in the flux splitting method. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy and resolve strong discontinuous waves in three dimensional ideal MHD problems.     

MULTIPLE SOLUTIONS TO SINGULAR BVPS WITH VARIABLE COEFFICIENT ON THE HALF-LINE

This paper is concerned with a singular second-order nonlinear boundary value problem with a time depending on derivative operator and posed on the positive half-line. The nonlinearity is derivative-dependent, which has singularities at t=0 and/or x=0, and may change sign. The method of the upper and lower solutions on unbounded domains combined with the topological degree theory are employed to prove the existence and multiplicity of solutions.     

Fast Solvers of Fredholm Optimal Control Problems

<正>The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented.Existence and uniqueness of optimal solutions is proved. A collective Gauss-Seidel scheme and a multigrid scheme are discussed.Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments.     

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

<正>In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging,scientific computing,reverse engineering and geometric modelling.The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation(PDE)-based diffusion model derived by a minimal volume-like variational formulation.The evolution is driven both by the distance from the data set and by the curvature analytically computed by it.The distance function is computed by implicit local interpolants defined in terms of radial basis functions.Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale.The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction.Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions.Moreover,we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.     

Optimal orthogonalization processes

Two optimal orthogonalization processes are devised toorthogonalize,possibly approximately,the columns of a very large and possiblysparse matrix A∈C^n×k.Algorithmically the aim is,at each step,to optimallydecrease nonorthogonality of all the columns of A.One process relies on using translated small rank corrections.Another is a polynomial orthogonalization process forperforming the L?wdin orthogonalization.The steps rely on using iterative methods combined,preferably,with preconditioning which can have a dramatic effect on how fast thenonorthogonality decreases.The speed of orthogonalization depends on howbunched the singular values of A are,modulo the number of steps taken.These methods put the steps of the Gram-Schmidt orthogonalizationprocess into perspective regardingtheir(lack of)optimality.The constructions are entirely operatortheoretic and can be extended to infinite dimensional Hilbert spaces.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

傅种孙——中国现代数学教育的先驱

1 引子 很高兴来到澳门,与澳门大学教育学院的学生——未来的中学数学教师们见面.黄博士拟定的题目——北京师范大学与中学数学教育太大,我有些不敢讲.另一方面,我是个数学老师,没读过太多教育学和心理学的理论,也不大敢在诸位数学教育家面前班门弄斧,所以,就决定给同学们讲几件事,介绍一个人.     

Recent advances in statistical methodologies in evaluating program for high-dimensional data

The era of big data brings opportunities and challenges to developing new statistical methods and models to evaluate social programs or economic policies or interventions. This paper provides a comprehensive review on some recent advances in statistical methodologies and models to evaluate programs with high-dimensional data. In particular, four kinds of methods for making valid statistical inferences for treatment effects in high dimensions are addressed. The first one is the so-called doubly robust type estimation, which models the outcome regression and propensity score functions simultaneously. The second one is the covariate balance method to construct the treatment effect estimators. The third one is the sufficient dimension reduction approach for causal inferences. The last one is the machine learning procedure directly or indirectly to make statistical inferences to treatment effect. In such a way, some of these methods and models are closely related to the de-biased Lasso type methods for the regression model with high dimensions in the statistical literature. Finally, some future research topics are also discussed.     

Novel Kernel Function With a Hyperbolic Barrier Term to Primal-dual Interior Point Algorithm for SDP Problems

In this paper, we introduce for the first time a new eligible kernel function with a hyperbolic barrier term for semidefinite programming(SDP). This add a new type of functions to the class of eligible kernel functions. We prove that the interior-point algorithm based on the new kernel function meets O(n^3/4 logε/n)iterations as the worst case complexity bound for the large-update method. This coincides with the complexity bound obtained by the first kernel function with a trigonometri...     

一道高考题的背景、拓广与应用

问题1(2007年重庆卷,文21)倾斜角为α的直线经过抛物线交于A,两点(图略). 　　(Ⅰ)求抛物线的焦点F的坐标及准线ι的方程; 　　(Ⅱ)若α为锐角,作线段AB的垂直平分线m交X轴于点P,证明|FP|-|FP|cos2α为定值,并求此定值.……     

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.     

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.     

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.     

关于亲和数的一个注记

本文研究了一类整数序列(2n)2n 1的某些性质,利用费玛数和数论函数的某些性质,获得了验证此类整数是否是亲和数和完全数的方法,既不与其他正整数构成亲和数对也不是完全数.     

相同的性质不同的表述

文[1]《2010年四川卷理科20题的引申》一文给出并证明了这样一个性质(表述1.1),这里以焦点在x轴上的椭圆为例.     

打造中国“西屋奖” 成就数学小天才 首届“丘成桐中学数学奖”揭晓

经过10月22日、23日两天的角逐,10月24日,首届"丘成桐中学数学奖"获奖名单在太庙揭晓.有17个团队从海内外800余支报名团队中脱颖而出参加此次总决赛,共有温州中学等5个队摘得金、银、铜奖.丘成桐教授亲自向他们颁发了象征最高荣誉的"金字塔塔尖",同时,金奖团队还将获得15万元奖金.     

扑克牌魔术中的数学

在魔术中有一类扑克牌的魔术,当我们看到魔术师经过眼花缭乱的洗牌、抽牌、翻牌的动作之后,在他的手上出现了所想要的牌,不少人叹为观止.在惊奇之余,人们会想魔术师是怎么做到的?是魔术师的超常记忆力,还是魔术师作弊,蒙骗了众人的眼睛?特别是有些魔术需要观众上台配合以显示魔术     

数学与文学

龟兔赛跑的故事,是大家很熟悉的.兔子本来跑在前面,但由于骄傲,路上睡了一觉,结果输给了乌龟.作为文学作品来说,当然要大力渲染兔子的骄傲自满,乌龟的坚持不懈.而从数学角度来看,则可抽象为下面的行程图.     

爷爷论学习

清华大学教授李文汉老先生是本刊多年的老编委了,十多年前从编委岗位上退下来以后,仍一直关心本刊的发展.今年4月份他将写给正在读初二的孙女的信转给本刊,我们相信李老在信中所谈各点,对广大初中同学会有很大帮助,特刊登此信(略有删减),谢谢李文汉老先生.