A new algorithm for a recursive construction of the minimal interpolation space

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.     

A new algorithm for a recursive construction of the minimal interpolation space

In this work, we introduce a new interpolation algorithm, based on a recursive method for computing Lagrange interpolants. This algorithm allows to construct recursively the minimal interpolation space (see [1]) with respect to a finite set of points. We also extend this recursive method to the osculatory interpolation problem.     

Time-variant displacement structure and interpolation problems

Derives a new recursive solution for a general time-variant interpolation problem of the Hermite-Fejer type, based on a fast algorithm for the recursive triangular factorization of time-variant structured matrices. The solution follows from studying the properties of an associated cascade system and leads to a triangular array implementation of the recursive algorithm. The system can be drawn as a cascade of first-order lattice sections, where each section is composed of a rotation matrix followed by a storage element and a tapped-delay filter. Such cascades always have certain blocking properties, which can be made equivalent to the interpolation conditions. The authors also illustrate the application of the algorithm to problems in adaptive filtering, model validation, robust control, and analytic interpolation theory     

Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: an optimization approach

Over the last several years, a new theory of Nevanlinna-Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of "most interpolants" of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional H/sup /spl infin// controllers, we demonstrate the advantage of the proposed method.     

The map and blend scattered data interpolant on a sphere

The map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere. This technique consists of two different mappings of the sphere to planar domains, solving two corresponding scattered data interpolation problems on the planar domains, and then the interpolant on the sphere is formed by blending these two planar interpolants. If the user has software for solving the scattered data interpolation problem on a planar domain, only a small programming effort is needed to implement the map and blend interpolant on the sphere. Although any interpolant to scattered data on a planar domain can be used in our general technique, we use the multiquadric radial basis method.     

单线图动态规划最优布局成图技术

提出了一种基于动态规划算法得到布局最优解实现区域电网单线图生成的方法.根据电网空间数据构建拓扑模型,执行广度优先算法得到多个能构成连通图的邻接矩阵以及矩阵遍历序列,根据邻接矩阵宽度计算出能容纳全部设备的正方形范围,并建立了设备最小间距为优化目标的数学模型.提出了动态规划最优布局求解的算法,应用该算法求解布局最优解数组,最后按照最少交叉原则进行正交化处理.应用实例表明通过最优解布局的成图美观且高效.     

Stability-Preserving Rational Approximation Subject to Interpolation Constraints

A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted $H_2$ minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic interpolants.      

A new result on passivity preserving model reduction

The problem of model reduction with preservation of passivity is investigated. The approach is based on positive real interpolation, and is inspired by the similarity between Löwner and Pick matrices. The former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions. It follows that interpolation of the original set of data together with an appropriately defined mirror-image set of data yields automatically positive real interpolants. Subsequently, we show how this result can be implemented using a Krylov projection procedure. The ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large-scale systems.     

递归思想在指派问题中的运用

提出将矩阵进一步缩小的方法来求指派问题的解,用匈牙利解法求指派问题的解,然后用递归思想求解指派问题,并对两种方法的优缺点进行比较.通过对比可以知道,两种方法所得的结果是一样的.匈牙利算法的优点是算法时问复杂度较小,缺点是解题过程比较复杂.递归思想的优点是解题过程比较简单,缺点是算法时间复杂度较大.     

基于KRLS的非均匀采样非线性系统辨识

利用提升技术可将非均匀采样非线性系统离散化为一个多输入单输出传递函数模型,从而将系统输出表示为非均匀刷新非线性输入和输出回归项的线性参数模型,进一步基于非线性输入的估计或过参数化方法进行辨识.然而,当非线性环节结构未知或不能被可测非均匀输入参数化表示时,上述辨识方法将不再适用.为了解决这个问题,利用核方法将原始非线性数据投影到高维特征空间中使其线性可分,再对投影后的数据应用递推最小二乘算法进行辨识,提出基于核递推最小二乘的非均匀采样非线性系统辨识方法.此外,针对系统含有有色噪声干扰的情况,参考递推增广最小二乘算法的思想,利用估计残差代替不可测噪声,提出核递推增广最小二乘算法.最后,通过仿真例子验证所提算法的有效性.     

CNC系统中NURBS曲线插补及相关算法研究

为提高计算机数字控制（CNC）系统的轮廓控制精度,需解决系统控制软件样条直接输出的问题。基于Taylor公式得到了非均匀有理B样条（NURBS）曲线上两个相邻插补点参数间的递推关系。对于NURBS曲线插补过程中需要频繁计算的B样条基函数及其任意阶导数提出了一种分块矩阵连乘积形式的统一计算方法。应用数值计算方法解决了插补过程中曲线长度等的相关计算问题。实例运算表明所提出的方法可以应用到实际CNC系统中。     

自适应的三维地质建模插值加密方案

提出一种可适应各种地质条件的插值加密方案,结合设计模式的思想,把传统的插值功能分解为加密级别控制、构网、插值3个组件,解决以往插值相关功能耦合度过高、难以修改的问题,组件中的策略可以灵活修改或替换。给出一种自适应的构网策略,引进加密级别控制,解决加密尺度难以控制的问题,通过一个建模实例展示该方案的实际建模效果。     

基于决策问题形式化的模型构造方法

在对目前决策模型构造方法的局限性进行分析的基础上,给出决策问题的形式化表示,提出一种基于决策问题形式化的模型构造方法,从描述模型与问题模型两个角度探讨了该方法构造模型的过程。并以实例论证了该方法的科学性与可行性。     

基于Lasso与RFE特征消除的RVM旋转机械故障预测

针对旋转机械故障诊断问题,提出一种基于相关向量机（RVM）的故障检测方法,RVM是一种用于回归和分类问题的贝叶斯稀疏核方法,其突出的优势是模型的稀疏性和预测的概率性。为进一步提高RVM模型的鲁棒性,减小样本数据中异常值对预测值的影响,针对Lasso方法进行特征选择时无法去除冗余特征的问题,提出以Lasso为底层算法的RFE递归特征消除方法去除样本数据集中无关特征和冗余特征。最后以工业环境下采集的数据作为样本集进行实验,同传统算法进行了比较,结果表明该方法在保持较高检测率的同时,提高了故障预测的时效性和稳定性。     

Fully automatic hp-adaptivity for Maxwell’s equations

I report on the development of a fully automatic hp-adaptive strategy for the solution of time-harmonic Maxwell equations. The strategy produces a sequence of grids that deliver exponential convergence for both regular and singular solutions. Given a (coarse) mesh, we refine it first globally in both h and p, and solve the problem on the resulting fine mesh. We consider then the projection-based interpolants of the fine mesh solution with respect to both current and next (to be determined) coarse grid, and introduce the interpolation error decrease rate equal to the difference of the old and new (coarse) mesh interpolation errors vs. number of degrees-of-freedom added. The optimal hp-refinements leading to the next coarse grid are then determined by maximizing the interpolation error decrease rate.     

基于近似偏最小一乘的闭环系统辨识新方法

本文基于近似最小一乘准则和主成分分析,针对反馈通道模型阶次低于前向通道模型阶次且反馈通道不存在噪声的闭环系统,进行了近似偏最小一乘递推辨识算法的推导.为解决最小一乘准则函数不可微的问题,本文算法用确定性可导函数近似代替残差绝对值.近似偏最小一乘辨识算法可以克服基于最小二乘准则的辨识算法在受到满足(SαS)分布的尖峰噪声干扰时残差平方项过大的缺点,具有目标函数可导,计算简单的优点.同时,通过主成分分析去除数据向量各元素之间的线性相关,可以得出模型参数的唯一解.仿真实验表明,本文算法可以对反馈通道模型阶次低于前向通道模型阶次的闭环系统进行直接辨识,抑制了尖峰噪声对辨识结果的影响,具有优良的稳健性,可以更好地应用于闭环系统辨识.     

Preconditioned conjugate gradients,radial basis functions,and Toeplitz matrices

Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires the solution of a dense linear system. Therefore, much attention has been given to iterative methods. In this paper, we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data. Thus, our method applies to the corresponding Toeplitz matrices. The number of iterations required to achieve a given tolerance is independent of the number of variables.     

How conservative is frozen-time interpolation?

It is shown in this note that for SISO systems under l² disturbances, when data commute approximately with the shift, the optimal interpolation (over all linear time-varying interpolants) can be approximated by the supremum of the frozen-time Hankel norms. This confirms the intuition that the frozen-time constructed optimal or suboptimal interpolants are in fact nearly optimal when data vary slowly.     

求解非线性期权定价模型的自适应小波同伦摄动技术

Leland 模型是在考虑交易费用的情况下,对 Black - Scholes 模型进行修改得到的非线性期权定价模型. 本文针对 Leland 模型,提出了一种求解非线性动力学模型的自适应多尺度小波同伦摄动法. 该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性期权定价模型方程自适应离散为非线性常微分方程组; 然后将用于求解非线性常微分方程组的同伦摄动技术和小波变换的动态过程相结合,构造了求解 Leland 模型的自适应数值求解方法. 数值模拟结果验证了该方法在数值精度和计算效率方面的优越性.     

Reducing control points in surface interpolation

Surface interpolation to rectangularly arranged points is an integral part of surface design and modeling in CAD/CAM and graphics. Using B-spline surfaces, the process involves curve interpolations through rows of data points and through columns of control points. This method, as well tuned as it is, proves inadequate for recent problems such as those of reverse engineering. Data acquisition devices, such as scanners, may be used to return rows of data points, but it's not guaranteed that each row contains the same number of points. The problem then arises of passing a smooth surface through these points (assuming that interpolation is justified, meaning the number of points isn't large). Since each row contains different numbers of points, regular data interpolation can't be used. One method to solve this problem is to interpolate each row with B-spline curves and to pass a smooth surface through these curves via surface skinning. While this is a legitimate solution, the number of control points tends to become prohibitively large, especially if the number of rows is large. This article addresses the problem of how to reduce the number of control points while maintaining precise interpolation. The idea is to give the knots some flexibility so that each row can be interpolated with as few new knots added as possible