A generalized cubic spline technique for identification of multivariable systems

The application of cubic splines to the identification of time-invariant systems is considered. The use of splines, initially proposed by Bellman in 1971, has been extended to the multidimensional case. In addition, the effects of noise on the identification procedure are considered and techniques are presented for improving the identification accuracy. A general spline technique, used in conjunction with a Kalman estimation procedure, has been developed for identifying physical systems described by a set of first-order differential equations. This method has been found to be superior to the exponential fitting technique proposed by Prony and to other finite-difference methods.     

Univariate cubic L 1 interpolating splines based on the first derivative and on 5-point windows: analysis, algorithm and shape-preserving properties

In this paper, univariate cubic L ₁ interpolating splines based on the first derivative and on 5-point windows are introduced. Analytical results for minimizing the local spline functional on 5-point windows are presented and, based on these results, an efficient algorithm for calculating the spline coefficients is set up. It is shown that cubic L ₁ splines based on the first derivative and on 5-point windows preserve linearity of the original data and avoid extraneous oscillation. Computational examples, including comparison with first-derivative-based cubic L ₁ splines calculated by a primal affine algorithm and with second-derivative-based cubic L ₁ splines, show the advantages of the first-derivative-based cubic L ₁ splines calculated by the new algorithm.     

Discretized Lavrent’ev regularization for the autoconvolution equation

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.     

几种有理插值函数的逼近性质

1　引　　言在曲线和曲面设计中,样条插值是有用的和强有力的工具.不少作者已经研究了很多种类型的样条插值[1,2,3,4].近些年来,有理插值样条,特别是三次有理插值样条,以及它们在外型控制中的应用,已有了不少工作[5,6,7].有理插值样条的表达式中有某些参数,正是由于这些参数,有理插值样条在外型控制中充分显示了它的灵活性;但也正是由于这些参数,使它的逼近性质的研究增加了困难.因此,关于有理插值样条的逼近性质的研究很少见诸文献.本文在第二节首先叙述几种典型的有理插值样条,其中包括分母为一次、二次的三次有理插值样条和仅基于函数值…     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

Adaptive Order Selection for Spline Smoothing

Computational methods are presented for spline smoothing that make it practical to compute smoothing splines of degrees other than just the standard cubic case. Specifically, an order n algorithm is developed that has conceptual and practical advantages relative to classical methods. From a conceptual standpoint, the algorithm uses only standard programming techniques that do not require specialized knowledge about spline functions, methods for solving sparse equation systems or Kalman filtering. This allows for the practical development of methods for adaptive selection of both the level of smoothing and degree of the estimator. Simulation experiments are presented that show adaptive degree selection can improve estimator efficiency over the use of cubic smoothing splines. Run-time comparisons are also conducted between the proposed algorithm and a classical, band-limited, computing method for the cubic case.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

Local cubic splines on non-uniform grids and real-time computation of wavelet transform

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise.     

Methods for solving singular boundary value problems using splines: a review

This paper surveys and reviews papers of spline solution of singular boundary value problems. Among a number of numerical methods used to solve two-point singular boundary value problems, spline methods provide an efficient tool. Techniques collected in this paper include cubic splines, non-polynomial splines, parametric splines, B-splines and TAGE method.     

On cubic formulas related to the mixed Hermite splines

A cubic formula containing partial integrals is considered on a class of functions of two variables. It is shown that the integral of a mixed Hermite spline gives the best cubic formula for the given class. The coincidence of cubic formulas, which are exact for odd and even mixed Hermite splines, is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.45, No. 4, pp. 579–581, April, 1993.     

Orthogonal splines based on B-splines — with applications to least squares,smoothing and regularisation problems

Orthogonal linear and cubic splines are introduced, based on a simple recurrence procedure using B-splines. Stable formulae are obtained for explicit least squares approximation. An application to the smoothing of noisy data is given, in which the approximation is essentially a second integral of an orthogonal linear spline and this leads to an efficient solution procedure. An application to the regularisation of integral transforms, and specifically to the finite moment problem, is also described.     

Approximation properties of the spline fit

Different topics in connection with spline fit are discussed in this paper. In particular, an example is given showing non-convergence of splines, and further some error bounds of cubic spline interpolation are proved.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

An O(h6) cubic spline interpolating procedure for harmonic functions

An O(h⁶) method for the interpolation of harmonic functions in rectangular domains is described and analyzed, The method is based on an earlier cubic spline technique [N. Papamichael and J.R. Whiteman, BIT 14 , 452–459 (1974)], and makes use of recent results concerning the a posteriori correction of interpolatory cubic splines.     

Discrete cubic spline interpolation

Summary In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

Fitting data using optimal Hermite type cubic interpolating splines

In this work we obtain a new optimal property for cubic interpolating splines of Hermite type applied to data-fitting problems. The existence and uniqueness of the Hermite type cubic spline with minimal quadratic oscillation in average are proved.     

Bivariate spline method for numerical solution of steady state Navier–Stokes equations over polygons in stream function formulation

We use the bivariate spline finite elements to numerically solve the steady state Navier–Stokes equations. The bivariate spline finite element space we use in this article is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the steady state Navier–Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth‐order equation, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in H²(Ω) of the nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The Galerkin method with C¹ cubic splines is implemented in MATLAB. Our numerical experiments show that the method is effective and efficient. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 147–183, 2000     

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.     

An Analysis of Two Algorithms for Shape-Preserving Cubic Spline Interpolation

The use of polynomial splines as a basis for the interpolationof discrete data can be theoretically justified by a minimumprinciple. It is natural to apply this principle also if shapepreserving is added as a constraint, although the constructionprocess is then nonlinear. We discuss two algorithms for theconstruction of the cubic spline interpolant under the constraintof positivity or monotonicity, and give a detailed convergenceanalysis. Numerical tests illustrate that analysis.