STABLE SOLUTION OF TIME DOMAIN INTEGRAL EQUATION METHODS USING QUADRATIC B-SPLINE TEMPORAL BASIS FUNCTIONS

This paper is concerned with stable solutions of time domain integral equation （TDIE） methods for transient scattering problems with 3D conducting objects. We use the quadratic B-spline function as temporal basis functions, which permits both the induced currents and induced charges to be properly approximated in terms of completeness. Because the B-spline function has the least support width among all polynomial basis functions of the same order, the resulting system matrices seem to be the sparsest. The TDIE formula-tions using induced electric polarizations as unknown function are adopted and justified. Numerical results demonstrate that the proposed approach is accurate and efficient, and no late-time instability is observed.     

EQUIVALENCE OF SEMI-LAGRANGIAN AND LAGRANGE-GALERKIN SCHEMES UNDER CONSTANT ADVECTION SPEED

We compare in this paper two major implementations of large time-step schemes for advection equations, i.e., Semi-Lagrangian and Lagrange-Galerkin techniques. We show that SL schemes are equivalent to exact LG schemes via a suitable definition of the basis functions. In this paper, this equivalence will be proved assuming some simplifying hypoteses, mainly constant advection speed, uniform space grid, symmetry and translation invariance of the cardinal basis functions for interpolation. As a byproduct of this equivalence, we obtain a simpler proof of stability for SL schemes in the constant-coefficient case.     

THE STABILITY AND APPROXIMATION PROPERTIES OF RITZ-VOLTERRA PROJECTION AND APPLICATION,I

The purpose of this paper is to study the stability and approximation properties of Ritz-Volterra projection. Through constructing a new type of Green functions and making use of various properties and estimates related with the functions, we prove that the Ritz-Volterra projection defined on the finite-dimensional subspace S_h of H_o~1 possesses the W_p~1-stability and the optimal approxi mation properties in W_p~1 and L_p for 2≤p≤∞. Our results, in this paper, can be applied to the finite element approximations for many evolution equations such as parabolic and hyperbolic integrodifferential equations,Sobolevequations and visco-elasticity, etc.     

Jackknife empirical likelihood test for the equality of degrees of freedom in t-copulas

It has been argued that fitting a t-copula to financial data is superior to a normal copula. To overcome the shortcoming that a t-copula only has one parameter for the degrees of freedom, the t-copula with multiple parameters of degrees of freedom has been proposed in the literature, which generalizes both the t-copulas and the grouped t-copulas. Like the inference for a t-copula, a computationally efficient inference procedure is to first estimate the correlation matrix via Kendall's τ and then to estimate the parameters of degrees of freedom via pseudo maximum likelihood estimation. This paper proposes a jackknife empirical likelihood test for testing the equality of some parameters of degrees of freedom based on this two-step inference procedure, and shows that the Wilks theorem holds.     

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.     

旋转锥面$l_\infty$拟合的截断光滑化牛顿法

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than 12 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large amount of cones repeatedly, moreover, when large amount of measured data are to be fitted to one rotated cone, the number of components in the maximum function is large. So it is necessary to develop efficient solution methods. To solve such optimization problems efficiently, a truncated smoothing Newton method is presented. At first, combining aggregate smoothing technique to the maximum function as well as the absolute value function and a smoothing function to the square root function, a monotonic and uniform smooth approximation to the objective function is constructed. Using the smooth approximation, a smoothing Newton method can be used to solve the problem. Then, to reduce the computation cost, a truncated aggregate smoothing technique is applied to give the truncated smoothing Newton method, such that only a small subset of component functions are aggregated in each iteration point and hence the computation cost is considerably reduced.     

Hierarchy paths in the Hatcher-Thurston complex

Inspired by works of Masur-Minsky and Mahan Mj, we observe that the Hatcher-Thurston complex of a surface F is an interpolating complex between the pants complex and the curve complex, then we give a hierarchical structure for the Hatcher-Thurston complex. By this hierarchical structure, we show a distance formula in the Hatcher-Thurston complex related to subsurfaces of positive genera. As a corollary, we show that the Hatcher-Thurston complex has one end for surface with genus at least three, the proof runs the same line as a result of Masur-Schleimer, the key tool is the distance formula.     

DIRECTIONAL DO-NOTHING CONDITION FOR THE NAVIER-STOKES EQUATIONS

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations diseretized by finite elements is the “do-nothing” condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a “directional do-nothing” condition. In contrast to the usual “do-nothing” condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.     

基于离散点和法矢的曲线重构算法

This paper presents a curve reconstruction algorithm based on discrete data points and normal vectors using B-splines.The proposed algorithm has been improved in three steps:parameterization of the discrete data points with tangent vectors,the B-spline knot vector determination by the selected dominant points based on normal vectors,and the determination of the weight to balancing the two errors of the data points and normal vectors in fitting model.Therefore,we transform the B-spline fitting problem into three sub-problems,and can obtain the B-spline curve adaptively.Compared with the usual fitting method which is based on dominant points selected only by data points,the B-spline curves reconstructed by our approach can retain better geometric shape of the original curves when the given data set contains high strength noises.     

An orthogonal basis for non-uniform algebraic-trigonometric spline space

Non-uniform algebraic-trigonometric B-splines shares most of the properties as those of the usual polynomial B-splines. But they are not orthogonal. We construct an orthogonal basis for the n-order(n ≥ 3) algebraic-trigonometric spline space in order to resolve the theoretical problem that there is not an explicit orthogonal basis in the space by now. Motivated by the Legendre polynomials, we present a novel approach to define a set of auxiliary functions,which have simple and explicit expressions. Then the proposed orthogonal splines are given as the derivatives of these auxiliary functions.     

基于层次T网格的分片孔斯曲面重构

本文提出一种基于任意层次T网格的多项式(PHT)样条空间$S(3,3,1,1,T)$的一个新的曲面重构算法.该算法由分片插值于层次T网格上每个小矩形单元对应4个顶点的16个参数的孔斯曲面形式给出.对于一个给定的T网格和相应基点处的几何信息(函数值,两个一阶偏导数和混合导数值),可得到与$S(3,3,1,1,T)$的PHT样条曲面相同的结果,且曲面表达形式更简单,同时,在离散数据点的曲面拟合中,我们给出了自适应的曲面加细算法.数值算例显示,该自适应算法能够有效的拟合离散数据点.     

A hierarchical basis preconditioner for the biharmonic equation on the sphere

^** Email: jan.maes{at}cs.kuleuven.be In this paper, we propose a natural way to extend a bivariatePowell–Sabin (PS) B-spline basis on a planar polygonaldomain to a PS B-spline basis defined on a subset of the unitsphere in [graphic: see PDF] . The spherical basis inherits many properties of the bivariatebasis such as local support, the partition of unity propertyand stability. This allows us to construct a C¹ continuous hierarchicalbasis on the sphere that is suitable for preconditioning fourth-orderelliptic problems on the sphere. We show that the stiffnessmatrix relative to this hierarchical basis has a logarithmicallygrowing condition number, which is a suboptimal result comparedto standard multigrid methods. Nevertheless, this is a hugeimprovement over solving the discretized system without preconditioning,and its extreme simplicity contributes to its attractiveness.Furthermore, we briefly describe a way to stabilize the hierarchicalbasis with the aid of the lifting scheme. This yields a waveletbasis on the sphere for which we find a uniformly well-conditionedand (quasi-) sparse stiffness matrix.     

EXPLICIT BOUNDS OF EIGENVALUES FOR STIFFNESS MATRICES BY QUADRATIC HIERARCHICAL BASIS METHOD)

The bounds for the eigenvalues of the stiffness matrices in the finite element discretiza-tion corresponding to Lu:=-u″with zero boundary conditions by quadratic hierarchical basic are shown explicitly.The condition numberof the resulting system behaves like O(1/h)where h is the mesh size.We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to O(1).     

Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.     

联图的圈基

MacLane于1937年给出了圈基方面的重要定理: 图G是平面图, 当且仅当图G有2-重基. 连通图G_1和G_2的联图G_1\vee G_2指的是在它们的不交并G_1\bigcup G_2上添加边集(u,v)|u\in V(G_1), v\in V(G_2). 对G_1和G_2的联图G_1\vee G_2的圈基重数进行了研究, 得到了一个上界, 改进了Zare的结果. 并在此基础之上, 进一步得到特殊联图C_m\vee C_n的圈基重数的一个上界.     

HIERARCHICAL BASIS METHOD FOR COVOLUME METHOD FOR NONSYMMETRIC AND INDEFINITE ELLIPTIC PROBLEM

In this paper, hierarchical basis method for second order nonsymmetric and indefinite elliptic problem on a polygonal domain (possibly nonconvex) discreted by a vertex-centered covolume method is constructed.     

Geometry + Simulation Modules: Implementing Isogeometric Analysis

Isogeometric analysis (IGA) is a recently developed simulation method that allows integration of finite element analysis (FEA) with conventional computer-aided design (CAD) software [1,3]. This goal requires new software design strategies, in order to enable the use of CAD data in the analysis pipeline. To this end, we have initiated G + SMO (Geometry+Simulation Modules), an open-source, C++ library for IGA. G + SMO is an object-oriented, template library, that implements a generic concept for IGA, based on abstract classes for discretization basis, geometry map, assembler, solver and so on. It makes use of object polymorphism and inheritance techniques to provide a common framework for IGA, for a variety of different basis-types which are available. A highlight of our design is the dimension independent code, realized by means of template meta-programming. Some of the features already available include computing with B-spline, Bernstein, NURBS bases, as well as hierarchical and truncated hierarchical bases of arbitrary polynomial order. These basis functions are used in continuous and discontinuous Galerkin approximation of PDEs over (non-)conforming multi-patch computational (physical) domains. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

-error estimates for ``shifted' surface spline interpolation on Sobolev space

The accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant is reasonably smooth. However, for functions which have smoothness below a certain order associated with the basis function , no approximation power has yet been established. Hence, the purpose of this study is to discuss the -approximation order ( ) of interpolation to functions in the Sobolev space with \max(0,d/2-d/p)$">. We are particularly interested in using the ``shifted' surface spline, which actually includes the cases of the multiquadric and the surface spline. Moreover, we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met.

     

Hierarchical spline spaces: quasi-interpolants and local approximation estimates

A local approximation study is presented for hierarchical spline spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement in any dimensionality. We provide approximation estimates for general hierarchical quasi-interpolants expressed in terms of the truncated hierarchical basis. Under some mild assumptions, we prove that such hierarchical quasi-interpolants and their derivatives possess optimal local approximation power in the general q-norm with \(1\leq q\leq \infty \). In addition, we detail a specific family of hierarchical quasi-interpolants defined on uniform hierarchical meshes in any dimensionality. The construction is based on cardinal B-splines of degree p and central factorial numbers of the first kind. It guarantees polynomial reproduction of degree p and it requires only function evaluations at grid points (odd p) or half-grid points (even p). This results in good approximation properties at a very low cost, and is illustrated with some numerical experiments.     

Changeable degree spline basis functions

A B-spline basis function is a piecewise function of polynomials of equal degree on its support interval. This paper extends B-spline basis functions to changeable degree spline (CD-spline for short) basis functions, each of which may consist of polynomials of different degrees on its support interval. The CD-spline basis functions possess many B-spline-like properties and include the B-spline basis functions as subcases. Their corresponding parametric curves, called CD-spline curves, are like B-spline curves and also have many good properties. If we use the CD-spline basis functions to design a curve made up of polynomial segments of different degrees, the number of control points may be decreased.