Interpolation theorems for self-adjoint operators

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrdinger operator with electro-magnetic potential.     

Interpolation from spaces spanned by monomials

This is an extension and emendation of recent results on the use of Gauss elimination in multivariate polynomial interpolation and, in particular, ideal interpolation. Dedicated to Mariano Gasca on the occasion of his sixtieth birthday     

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.     

Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

球面上的曲面插值

<正>1引言随着现代工业生产的飞速发展,航空、气象、环境监测等领域需要研究解决限制在曲面上的四维数据插值问题,即由有限个位置处的信息推测其它若干位置点的信息.例如,地球上某个地区的温度分布、降雨量分布、大气层的温室效应等;飞行器(飞机、火箭、导弹等)表面压力分布规律、肿瘤的生长规律等.这些在数学上都可归结为限制在曲面上的曲面插值与逼近问题.这个问题自Barnhill提出以后,人们针对限制在球面上     

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm-Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm-Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm-Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm-Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.     

基于时间序列与灰色拓扑的春节火灾损失预测

火灾每年给中国带来了巨大的损失,春节期间的火灾损失更是严重.根据1999-2010年春节期间火灾统计资料,火灾四项指标数据具有时序性以及随机波动性、模糊性.运用时间序列与灰色拓扑预测方法相结合预测春节期间火灾发生规律,且预测出未来3年内的火灾发生情况.结果表明,时间序列预测模型的平均绝对误差较小,且所建立的灰色拓扑预测模型的拟合精度都达到"好"的标准.因此,采用时间序列与灰色拓扑预测模型相结合对春节火灾发生情况进行预测,其结果合理可靠,可供理论研究和消防部门做出相应的预防措施参考,以达到有效控制和预防春节火灾的目的.     

An analytical technique for 3-dimensional interpolation

Given the values of a 3-dimensional function at unevenly spaced grid points on the grid structure of a channel, we describe a new analytical algorithm to determine the function value at an arbitrary point inside the channel. This algorithm has been implemented on a Univac 1100/81 computer in PL/I.     

An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.