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.      

The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. This leads to a new procedure for feedback control synthesis.      

Optimization-based computation of analytic interpolants of bounded complexity

This paper provides a unifying algorithm for computing any analytic interpolant of bounded complexity. Such computation can be performed by solving an optimization problem, due to a theorem by Georgiou and Lindquist. This optimization problem is numerically solvable by a continuation method. The proposed numerical algorithm is useful, among other cases, for designing a low-degree controller for a benchmark problem in robust control. The algorithm unifies previously developed algorithms for the Carathéodory extension and the Nevanlinna–Pick interpolation to one for more general interpolation problems.     

Superconvergence of Jacobi–Gauss-Type Spectral Interpolation

In this paper, we extend the study of superconvergence properties of Chebyshev-Gauss-type spectral interpolation in Zhang (SIAM J Numer Anal 50(5):2966–2985, 2012) to general Jacobi–Gauss-type interpolation. We follow the same principle as in Zhang (SIAM J Numer Anal 50(5):2966–2985, 2012) to identify superconvergence points from interpolating analytic functions, but rigorous error analysis turns out much more involved even for the Legendre case. We address the implication of this study to functions with limited regularity, that is, at superconvergence points of interpolating analytic functions, the leading term of the interpolation error vanishes, but there is no gain in order of convergence, which is in distinctive contrast with analytic functions. We provide a general framework for exponential convergence and superconvergence analysis. We also obtain interpolation error bounds for Jacobi–Gauss-type interpolation, and explicitly characterize the dependence of the underlying parameters and constants, whenever possible. Moreover, we provide illustrative numerical examples to show tightness of the bounds.     

A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

A polynomial interpolation based on a uniform grid yields the well-known Runge phenomenon, where maximum error is unbounded for functions with complex roots in the Runge zone. In this paper, we investigate the Runge phenomenon with the finite precision operation. We first show that the maximum error is bounded because of round-off errors inherent to the finite precision operation. Then we propose a simple truncation method based on the truncated singular value decomposition. The method consists of two stages: In the first stage, a new interpolating matrix is constructed using the assumption that the function is analytic. The new interpolating matrix is preconditioned using the statistical filter method. In the second stage, a truncation procedure is applied such that singular values of the new interpolating matrix are truncated if they are equal to or lower than a certain tolerance level. We generalize the method, by analyzing the singular vectors of both the original and new interpolation matrices based on the assumption in the first stage. We show that the structure of the singular vectors makes the first stage essential for an accurate reconstruction of the original function. Numerical examples show that exponential decay of the error can be achieved if an appropriate truncation is chosen.     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

Unit interpolation in H∞: Bounds of norm and degree of interpolants

This paper studies the problem of unit interpolation in H_∞. Specifically, bounds of the norm and degree of interpolants are obtained using the Nevanlinna-Pick interpolation theory. Also simpler bounds are given using the Poincaré metric. These results indicate that strong stabilization and simultaneous stabilization need a cautious approach if the system is ‘ill-conditioned’. In fact, it is demonstrated that the degree of stable compensator is not bounded for the class of plants with fixed degree. Also an example shows that there is a class of plants for which the sensitivity can be made small only if unstable compensators are used.     

Guaranteed Estimation Algorithms for Prediction and Interpolation of Random Processes

Prediction and interpolation of a process is investigated for known covariance functions of measurement error and perturbations belonging to a set of nonnegative-definite functions. Guaranteed estimation is studied, assuming that the guaranteed estimate is the best estimate of the parameters of the useful signal in the sense of the minimum of the mean-square error under the worst behavior of measurement errors and perturbations with covariance functions belonging to the set . Depending on the type of constraints, i.e., characteristics of the set , different approaches and methods are applied to guaranteed estimation. Papers concerned with this topic are reviewed. Prediction and interpolation are analytically investigated for the case in which covariance functions are bounded by the constraints imposed on their variance matrices. For the continuous case, the weight function and its equations are derived. Prediction and interpolation accuracies are evaluated and compared with the least-squares filters.     

基于边长的三维形状插值

形状插值在计算机图形学和几何处理中是一个极其重要而基础的问题,在计算机动画等领域有 着广泛应用。注意到在平面三角网格和三维四面体网格插值问题中,对边长平方插值等价于对回拉度量进行插 值,因此具有等距扭曲和共形扭曲同时有界的良好性质。通过将其推广至曲面三角网格,提出了一种完全基于 边长的曲面三角网格插值算法。给定边长,在重建网格阶段,使用牛顿法对边长误差能量进行优化。并且给出 了其海森矩阵的解析正定化形式,从而避免了高代价的特征值分解步骤。注意到四面体网格的边长平方插值结 果具有极低曲率,意味着只需少许修改即可将其压平从而嵌入三维空间。因此提出先将曲面三角网格四面体化, 再从四面体网格的插值结果提取表面。然后将这表面作为初始化用于边长误差能量的牛顿迭代,从而使得收敛 结果更加接近全局最优。在一系列三角网格上进行了实验,结果说明了本文方法比之前方法的边长误差更小, 且得到的结果还是有界扭曲的。     

Numerische Berechnung extremaler Fundamentalsysteme für Polynomräume über der Vollkugel

Extremal fundamental systems are nodal systems with a small Lebesgue-constant (≤dimension). An exchange algorithm for their calculation in case of polynomial interpolation over the unit ballB ^r is realized. ForB ³ and algebraic degree 6, a system consisting of 84 nodes and with a Lebesgue-constant <26 is documented, the corresponding cubature leading to a relative error of ≈10^?8 in case of analytic functions.     

Abstraction and refinement of mathematical functions toward SMT-based test-case generation

We propose a novel approach for generating test cases of software that includes mathematical functions, such as trigonometric functions, logarithmic functions, functions implemented as look-up tables with non-linear interpolation, and so on. A satisfiability modulo theories (SMT) solver is iteratively used to generate test cases in the scheme of bounded model checking. In the proposed method, mathematical functions are abstracted so that the derived formula can be easily treated using an SMT solver. The abstraction is refined adaptively based on the previous counterexamples. We also propose a general method to estimate an abstraction of a mathematical function by means of sampling and machine learning. Although the method proposed in this paper addresses mainly the topic of test-case generation, it is also applicable to ordinary bounded model checking under the assumption that the abstraction should be a correct over-approximation. We evaluated the proposed method by applying it to an example of embedded control software taken from the automotive industry. The experimental results show the usefulness of the proposed method.     

A State‐Space Based New Approach To The Directional Interpolation Problem

Directional interpolation plays an important role in robust control, system realization and model reduction. Several solutions to various directional interpolation problems have been proposed. In this paper, we consider the directional interpolation problem in a general setting and present a statespace based new approach to solving the problem. The solution is simple, and its exposition is as self‐contained as possible. We describe all the (strictly) bounded real rational matrix functions that satisfy the directional interpolation requirements by means of linear fractional transformation. Moreover, we give a necessary and sufficient condition for the interpolating function to be unique and show that the unique interpolating function is an inner (a co‐inner). The main procedures used to generate the interpolating function consist of standard matrix operations consisting of easy numerical computations, so the present solution is significant from the numerical viewpoint as well as the analytical viewpoint.     

The Intermittent Control Synchronization of Complex-Valued Memristive Recurrent Neural Networks with Time-Delays

In this paper, the intermittent control synchronization of complex-valued memristive recurrent neural networks with time-delays is investigated. As a generalization on the real-valued memristive recurrent neural networks, complex-valued memristive recurrent neural networks own more complicated properties. In complex-valued domain, bounded and analytic complex-valued activation functions do not exist. Some assumptions about activation functions in real-valued domain cannot be applied directly to complex-valued fields. By appropriate transformation, complex-valued memristive recurrent neural networks can be divided into real parts and imaginary parts, which can avoid discussing the bounded and analytic. In the framework of differential inclusion theory and Lyapunov method, sufficient criteria of intermittent control synchronization are established. Finally, a simulation is given to verify the validity and feasibility of the sufficient conditions.

     

Multipoint Taylor interpolation

Abstract We construct new multivariate polynomial interpolation schemes of Hermite type. The interpolant of a function is obtained by specifying suitable discrete differential conditions on the restrictions of the function to algebraic hypersurfaces. The least space of a finite-dimensional space of analytic functions plays an essential role in the definition of these differential conditions.     

有理[2m+1,2m]型分段插值样条

常见的较低次有理带单形状因子分段有理插值样条通过代数运算,可用Bernstein基函数等价表示,这类分段插值样条利用Hermite插值的方法推广到高次有理[2m+1,2m]型,样条的生成曲线满足Cm-连续,并给出了具体的Bern-stein基函数表示方法的表达式,其形式较为简单,最后分别讨论了这类有理插值的逼近阶与约束域及保单调等方面的形状因子的选取情况,并给出了例子分析。     

Cardinal interpolation with differences of tempered functions

In this paper, we investigate the existence and uniqueness of cardinal interpolants associated with functions arising from the k^th order iterated discrete Laplacian ^k applied to certain radial basis functions. In particular, we concentrate on determining, for a given radial function Φ, which functions ^kΦ give rise to cardinal interpolation operators which are both bounded and invertible ℓ₂ (Z³). In addition to solving the cardinal interpolation problem (CIP) associated with such functions ^kΦ, our approach provides a unified framework and simpler proofs for the CIP associated with polyharmonic splines and Hardy multiquadrics.     

矩形网格上二元矩阵切触有理插值的新方法

熟知的矩阵切触有理插值的方法都与连分式有关,不仅计算繁琐,而且难以避免出现“极点、不可达点”。用网格点构造有理插值基函数,用型值点构造具有承袭性的各阶矩阵插值算子,通过插值基函数与插值算子作线性运算,构造出二元矩阵各阶切触有理插值函数,有效避免了有理插值的“极点、不可达点”问题。若选择适当的参数,还可以任意降低插值函数的次数,数值例子表明了该方法简单、有效、实用性强。     

Geometric modeling of curves using weighted linear and circular segments

This paper presents a method for geometric modeling of curves. Linear and circular segments with weight functions are used for curve design. Similar weight functions are used for smoothing and interpolation. The curve can be composed of consecutive linear and circular segments with smoothing at knots. If an interpolating curve is necessary, then weighted linear and circular segments are used for blending. Smoothness of the designed curve depends on the chosen weight function. It is possible to design the curve of variable smoothness using different weight functions.     

Constructing Plane Spanners of Bounded Degree and Low Weight

Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.     

Bounded error parameter estimation: a sequential analytic centerapproach

A sequential analytic center approach for bounded error parameter estimation is proposed. The authors show that the analytic center minimizes the logarithmic average output error among all the estimates within the membership set and is a maximum likelihood estimator for a class of noise density functions which include parabolic densities and approximations of truncated Gaussian. They also show that the analytic center is easily computable for both offline and online problems with a sequential algorithm. The convergence proof of this sequential algorithm is obtained and, moreover, it is shown that the complexity in terms of the maximum number of Newton iterations is linear in the number of observed data points