On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.     

Mesh Spacing Estimates and Efficiency Considerations for Moving Mesh Systems

Adaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.     

Harmonic interpolation of Hermite type based on Radon projections with constant distances

We study harmonic interpolation of Hermite type of harmonic functions based on Radon projections with constant distances of chords. We show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords coalesce to some points on the unit circle, we prove that the interpolation polynomials tend to a Hermite interpolation polynomial at the coalescing points.     

一类分形插值函数的分数阶微积分

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘维尔分数阶积分、微分的概念和相关定理.由于分形插值函数满足应用分数阶微积分处理问题的条件,所以利用这些概念及分步积分的方法讨论了折线段分形插值函数的分数阶积分的连续性,可微性及哪些点是不可微的,进一步说明了该插值函数分数阶微分的连续性并指出其不连续点,用黎曼-刘维尔分数阶微积分与分形插值函数结合起来研究,目的是想设法跟经典微积分一样,能找出函数上在该点的微积分的具体的实际应用意义.这些理论为研究分形插值函数的分数阶微积分的实际应用意义提供了一些理论基础.     

The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.     

线性常微分方程的全局误差估计和优化求解方法

线性常微分方程初值问题求解在许多应用中起着重要作用.目前,已存在很多的数值方法和求解器用于计算离散网格点上的近似解,但很少有对全局误差(global error)进行估计和优化的方法.本文首先通过将离散数值解插值成为可微函数用来定义方程的残差;再给出残差与近似解的关系定理并推导出全局误差的上界;然后以最小化残差的二范数为目标将方程求解问题转化为优化求解问题;最后通过分析导出矩阵的结构,提出利用共轭梯度法对其进行求解.之后将该方法应用于滤波电路和汽车悬架系统等实际问题.实验分析表明,本文估计方法对线性常微分方程的初值问题的全局误差具有比较好的估计效果,优化求解方法能够在不增加网格点的情形下求解出线性常微分方程在插值解空间中的全局最优解.     

Multiple point interpolation in Nevanlinna classes

A Nevanlinna-Pick type interpolation problem for generalized Nevanlinna functions is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points of the upper half plane. An operator theoretic approach is used to parametrize the solutions of this interpolation problem by means of selfadjoint extensions of a certain symmetry.     

Approximation with interpolatory constraints

Given a triangular array of points on satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a near-optimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.

     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.     

Deficient cubic splines with average slope matching

We obtain a deficient cubic spline function which matches the functions with certain area matching over a greater mesh intervals, and also provides a greater flexibility in replacing area matching as interpolation. We also study their convergence properties to the interpolating functions.     

Smooth interpolation of a mesh of curves

The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional vertex enclosure constraint. The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC ¹ surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.Communicated by Wolfgang Dahmen.     

On p-adic differentiability

A p-adic-valued function on the p-adic integers has a continuous derivative, Mahler showed, whenever its interpolation coefficients decay at a certain rate. It is shown here that Mahler's decay condition is equivalent to the strict differentiability of the function. There is a discussion of the Banach-space structure of the space of strictly differentiable functions. It is shown, moreover, that there is no rate of decay common to all functions with continuous derivative. Specifically, given any decay condition, there exists a function with derivative identically zero, whose interpolation coefficients decay more slowly.     

Abschätzung der Normen von Operatoren analytischer Funktionen

This article investigates the norms of certain interpolation operators of analytic functions on the unit disc. In particular, it is shown that the norms of interpolation operators being the identical operator for all n -degree polynomials have a lower bound of order ln n . This result is compared with a recent result regarding trigonometric interpolation of continuous functions on the unit circle. It is shown that opposed to the operators of analytic functions on the unit disc, the method of oversampling can be applied in order to uniformly bound the interpolation operators. Moreover, some practical implications with regard to communication engineering are discussed. It is concluded that in practice the results lead to non-linear interpolation operators.     

Natural bicubic spline fractal interpolation

Fractal Interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied.     

Semi inherited bivariate interpolation

The bivariate interpolation in two dimensional space R2 is more complicated than that in one dimensional space R, because there is no Haar space of continuous functions in R2. Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.     

Interpolation by Polynomials and Radial Basis Functions on Spheres

The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest. March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999.     

Near-optimal data-independent point locations for radial basis function interpolation

The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples. AMS subject classification 41A05, 41063, 41065, 65D05, 65D15This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen.     

SUBSTRUCTURE PRECONDITIONERS FOR NONCONFORMING PLATE ELEMENTS

1.IntroductionInthispaper,wegeneralizetheBPSalgorithm[1]tononconformingelementfproximationsofthebiharmonicequation.WeconstructapreconditionerforMor:elementbysubstructuringonthebasisofafunctiondecompositionfordiscretebibmonicfunctions.Thefunctiondecomposit…     

The optimal recovery of smooth functions

Summary It is shown that there is a positive lower bound,c, to the uniform error in any scheme designed to recover all functions of a certain smoothness from their values at a fixed finite set of points. This lower bound is essentially attained by interpolation at the points by splines with canonical knots. Estimates ofc are also given.     

Generation of Analytic Semigroups by Elliptic Operators with Dirichlet Boundary Conditions in a Cylindrical Domain

We consider an elliptic operator of second order with Dirichlet boundary conditions in a cylindrical domain. We show that a suitable interpretation of this operator in a certain space of continuous functions vanishing on the boundary is the infinitesimal generator of an analytic semigroup in the space of continuous functions. We prove several inclusions of the domain of the infinitesimal generator and of the real interpolation spaces between this domain and the basic space of continuous functions.