Absorbing Boundary Conditions of the Second Order for the Pseudospectral Chebyshev Methods for Wave Propagation

In this paper we describe the implementation of one-way wave equations of the second order in conjuction with pseudospectral methods for wave propagation in two space dimensions. These equations are first reformulated as hyperbolic systems of the first order and the absorbing boundaries are implemented by an appropriate modification of the matrix of this system. The resulting matrix corresponding to one-way wave equation based on Padé approximation has all eigenvalues in the complex negative half plane which allows stable integration of the underlying system by any ODE solver in the sense of eigenvalue stability. The obtained numerical scheme is much more accurate than the schemes obtained before which utilized absorbing boundary conditions of the first order, and is also capable of integrating the wave propagation problems on much larger time intervals than was previously possible.     

A Spectral Mortar Element Discretization of the Poisson Equation with Mixed Boundary Conditions

In this paper, we study a spectral mortar element discretization of the Poisson equation on a square subject to mixed boundary conditions of Dirichlet and Neumann type. We carry out the numerical analysis of the method and derive error estimates. An efficient algorithm for the solution of the problem is proposed and numerical tests confirming the theoretical results are presented.     

Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã = Sã + g(t)w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W ₁–W ₂ and S = S ₁ + S ₂ and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã = W ^–1 Sã + g(t)W ^–1 w are somewhat faster but less convenient to implement since the matrix W ^–1 S is no longer sparse. The pseudospectral methods lead to the system = D + g(t)w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).     

The Variable Order Fast Multipole Method for Boundary Integral Equations of the Second Kind

We discuss the variable order Fast Multipole Method (FMM) applied to piecewise constant Galerkin discretizations of boundary integral equations. In this version of the FMM low-order expansions are employed in the finest level and orders are increased in the coarser levels. Two versions will be discussed, the first version computes exact moments, the second is based on approximated moments. When applied to integral equations of the second kind, both versions retain the asymptotic error of the direct method. The complexity estimate of the first version contains a logarithmic term while the second version is O(N) where N is the number of panels.This work was supported by the NSF under contract DMS-0074553     

The h‐Difference Approach to Controllability and Observability of Fractional Linear Systems with Caputo–Type Operator

In the paper a class of h‐difference linear control systems with n fractional orders is studied. The Caputo‐type h‐difference operator is used. The formula of the solution to the stated initial value problem of the given system is proved. The problems of controllability and observability in a finite number of steps of h‐difference linear control systems with n fractional orders are studied.     

On Solvability of Operator Inclusions: Application to Elliptic Problems with the Neumann Boundary Condition

In studying the surjectivity of set-valued mappings, a modification of the acute-angle lemma (or the equilibrium theorem) is used. This allows one to weaken the coerciveness condition. Some applications to differential equations (inclusions) with Neumann boundary conditions are considered on Sobolev spaces W _p ¹() in which operators are used that are not coercive in the classical sense.     

The equivalence of the constrained Rayleigh quotient and Newton methods for matrix polynomials expressed in different polynomial bases along with the confluent case

We show that using the constrained Rayleigh quotient method to find the eigenvalues of matrix polynomials in different polynomial bases is equivalent to applying the Newton method to certain functions. We find those functions explicitly for a variety of polynomial bases including monomial, orthogonal, Newton, Lagrange and Bernstein bases. In order to do so, we provide explicit symbolic formulas for the right and left eigenvectors of the generalized companion matrix pencils for matrix polynomials expressed in those bases. Using the properties of the Newton basis, we also find two different formulas for the companion matrix pencil corresponding to the Hermite interpolation. We give pairs of explicit LU factors associated with these pencils. Additionally, we explicitly find the right and left eigenvectors for each of these pencils.     

航海雷达电磁波绕射仿真模型及其应用

模拟电磁波浇射并应用于航海雷达模拟器，突破了雷达模拟技术中多年一成不变的几何关系运算。实践证明，这种模拟方法能够大幅度提高航海雷达仿真质量。     

Application of a Numerical Increased-Accuracy Method to the Solution of a Heat Transfer Equation

Based on the Hermite formula, a numerical method for solution of a heat transfer equation is described. Expressions are given for computation of the coefficients of the Hermite formula from free parameters. The use of the method is illustrated by an example of solution of a model problem.     

基于谐波分析和线性光谱模型的耕地信息提取

耕地是重要的农业资源,如何利用遥感技术快速准确地提取耕地信息是目前研究的热点。利用2000年MODIS/EVI时间序列数据提取关中地区耕地资源信息。以不同地类的EVI时间序列数据年内变化差异为分类依据,采用时间序列谐波分析法对全年时间谱EVI数据进行重构分析,减少噪音对信息提取的影响。经最小噪声分离变换(MNF变换)、纯净像元指数(PPI)计算以及N维可视化工具进行人机交互选取植被、耕地、城镇和水体4种端元,基于线性光谱混合模型,获取该地区耕地资源分布信息。通过与同年1∶10万土地利用数据对比验证,本研究提取的耕地总体精度为83%。研究表明：基于时间序列谐波分析法对EVI数据进行重构,利用不同地类的特征差异,采用混合像元分解的方法,可以精确获取耕地资源定量信息。该方法可为长期、大范围、动态的耕地分布和变化遥感监测提供技术参考,同时为国土资源管理部门提供决策支持。     

应用Calerkin方法辨识一维抛物型偏微分方程及其边界条件中的参数

本文提出了一种一维抛物型偏微分方程及其边界条件中定常参数的辨识方法.这一方法将所研究的偏微分方程初-边值问题转化为具有已知初值的常微分方程组问题,然后再利用最优化方法将参数估算出来.数值仿真与实验验证都表明这一辨识方法是可行的.     

Determining the Upper Bound of the Spectral Norm of a Random Matrix and Its Application to Diagnosis of the Kalman Filter

An upper confidence bound for the spectral norm of a random matrix A(k) R ^n×m that consists of normally distributed random variables with zero mathematical expectation is found in the paper. Based on the theoretical results obtained, an approach is proposed to dynamic diagnosis of the Kalman filter via an innovation sequence.     

Binet-Cauchy Kernels on Dynamical Systems and its Application to the Analysis of Dynamic Scenes

We propose a family of kernels based on the Binet-Cauchy theorem, and its extension to Fredholm operators. Our derivation provides a unifying framework for all kernels on dynamical systems currently used in machine learning, including kernels derived from the behavioral framework, diffusion processes, marginalized kernels, kernels on graphs, and the kernels on sets arising from the subspace angle approach. In the case of linear time-invariant systems, we derive explicit formulae for computing the proposed Binet-Cauchy kernels by solving Sylvester equations, and relate the proposed kernels to existing kernels based on cepstrum coefficients and subspace angles. We show efficient methods for computing our kernels which make them viable for the practitioner. Besides their theoretical appeal, these kernels can be used efficiently in the comparison of video sequences of dynamic scenes that can be modeled as the output of a linear time-invariant dynamical system. One advantage of our kernels is that they take the initial conditions of the dynamical systems into account. As a first example, we use our kernels to compare video sequences of dynamic textures. As a second example, we apply our kernels to the problem of clustering short clips of a movie. Experimental evidence shows superior performance of our kernels. Parts of this paper were presented at SYSID 2003 and NIPS 2004.     

Is it Possible to Study Chaotic and ARCH Behaviour Jointly? Application of a Noisy Mackey–Glass Equation with Heteroskedastic Errors to the Paris Stock Exchange Returns Series

Most recent empirical works that apply sophisticated statistical proceduressuch as a correlation-dimension method have shown that stock returns arehighly complex. The estimated correlation dimension is high and there islittle evidence of low-dimensional deterministic chaos. Taking the complexbehaviour in stock markets into account, we think it is more robust than thetraditional stochastic approach to model the observed data by a nonlinearchaotic model disturbed by dynamic noise. In fact, we construct a model havingnegligible or even zero autocorrelations in the conditional mean, but a richstructure in the conditional variance. The model is a noisy Mackey–Glassequation with errors that follow a GARCH(p,q) process. This model permits usto capture volatility-clustering phenomena. Its characteristic is thatvolatility clustering is interpreted as an endogenous phenomenon. The mainobjective of this article is the identification of the underlying process ofthe Paris Stock Exchange returns series CAC40. To this end, we apply severaldifferent tests to detect longmemory components and chaotic structures.Forecasting results for the CAC40 returns series, will conclude this paper.