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In this paper we show an approximation for [Formula: see text], where u is a fuzzy interval and [Formula: see text] is obtained from a continuous function f applying the Zadeh's extension principle. In this process, we used a decomposition for u and the linear spline function for the approximation of f. A bound for the error in the approximation is given. Also, some examples are presented.

This chapter describes the clustering ensemble method and the Kolmogorovs Spline Complex Network, in the context of adaptive dynamic modeling of time-variant multidimensional data. The purpose of the chapter is to give an introduction in these subjects and to stimulate a participation of both young and experienced researchers in a solution of challenging and important for theory and practice problems related to this area.

An optimization technique is proposed for the outline capture of planar images. The overall technique has various phases including extracting outlines of images, detecting corner points from the detected outline, and curve fitting. The idea of multilevel coordinate search has been used to optimize the shape parameters in the description of the generalized cubic spline introduced. The spline method ultimately produces optimal results for the approximate vectorization of the digital contour obtained from the generic shapes.

This paper presents a comparative study on the modeling of Electromagnetic (EM) waves using statistical and numerical techniques using MATLAB software. Authors focused only on amplitude modeling of EM waves at 0.25Hz frequency. The models using statistical and numerical techniques have been developed and sum square errors have been calculated and comparative studies have been made to find the best technique. First order and second order regressions were used for statistical models and polynomial curve fitting and spline interpolation were used for numerical technique. Results from these techniques were compared and we have found that spline interpolation gave the most fitted model to the EM wave amplitude.

Summary An algorithm for determining pressure-derivative functions from measured well-test data is presented. The pressure data are fit with a spline function that can account for curve shape constraints. The first and second derivatives of the resulting spline function are calculated analytically and can be used for type-curve matching. Two examples illustrate the performance of the algorithm.

A multivariable spline element analysis for a plate bending problem is presented. The bicubic spline functions are used to construct the bending moments and transverse displacements field. The spline element equations with multiple variables are derived based on Hellinger-Reissner principle. Some numerical results are given and compared with other methods.

A compact and efficient algorithm is developed for generating and evaluating a bicubic spline interpolating function. The method is useful for the numerical solution of heat transfer and fluid flow problems when the properties are a function of two variables. The explicit formulation allows a one-time calculation of sets of bicubic coefficients, which provide an interpolation of the dependent property over the domain of the independent variables. The method provides continuous first and second derivatives in both variables far the iterative solution of nonlinear problems.

In this paper, a numerical method based on non-polynomial spline functions is presented to solve the one-dimensional heat equation. The new method is unconditionally stable. The accuracy order of the new method is of O(k5+h4),where k and h denote the mesh parameters for t and x, respectively. The accuracy of the present method is much higher than previous known methods.

The key role and task of fire control simulation system in the finalized test for the gun servo system is analyzed. The composition of numeric fire control simulation system based on the Control Area Network (CAN) Bus is introduced. A new model based on the cubic spline function to construct guide function is provided to guarantee the smooth transition of velocity while the gun servo system starts or stops. Comparative analysis for the non-guidance, the sine function model guidance and the cubic spline function model guidance is completed via the simulation using MATLAB. The simulation results indicate that the spline guide function given in this paper can optimize stable and rapid track.

Special Issue: Recent Progress in Spline and Wavelet Approximation; International audience; We consider box spline quasi-interpolants based on local linear functionals of point evaluator and integral type. The approximations are easy to compute, and reproduce the whole spline space in question

This paper presents a formulation of a quadratic spline with periodic derivative that fits the first derivatives of a function at uniform mesh points and the function values at the end points of the interval. Error bounds for the function and its derivative are derived.

Use of the spline function in Milne's method for eigenvalue problems of Sturm-Liouville-type linear equations is found to provide a high-speed method for calculating eigenvalues.

Spline algorithms are evaluated for the non-linear, integro-differential equation describing the Hartree-Fock approximation for the He 1s21S ground state. The error in the energy decreases as h2K−2, where h is a grid parameter and K is the order of the spline. It is shown that for higher order splines, the method is fast and accurate, and contrary to the conclusion reached by Altenberger-Siczek and Gilbert, that spline methods are suitable for SCF atomic structure calculations. Accuracy and timing studies are presented as well as comparisons with other accurate procedures.

SUMMARY A spline finite-strip method capable of analysing the buckling of prismatic thin-walled structural members under arbitrary loading is presented. It utilizes cubic B3-splines as displacement functions in the longitudinal direction and polynomials in the transverse direction. Numerical integration is used to determine the geometric stiffness matrix of the strips. Two examples which demonstrate the accuracy and versatility of the method are presented.

A new numerical method of determining potentiometric titration end-points is presented. It consists in calculating the coefficients of approximative spline functions describing the experimental data (e.m.f., volume of titrant added). The end-point (the inflection point of the curve) is determined by calculating zero points of the second derivative of the approximative spline function. This spline function, unlike rational spline functions, is free from oscillations and its course is largely independent of random errors in e.m.f. measurements. The proposed method is useful for direct analysis of titration data and especially as a basis for construction of microcomputer-controlled automatic titrators.

This preliminary investigation deals with the use of spline functions for smoothing spectral data. Three variant implementations are outlined and some tentative results are presented.

This paper presents a class of numerical method for the approximate solution of ordinary differential equations with retarded argument. These methods are essentially based on the spline functions. The study of existence and uniqueness are considered for such methods. Numerical examples and comparisons with other methods are given.

International audience; In this correspondence, we propose a new approach to scale-space filtering using a box spline representation of multidimensional signals. The use of box splines is motivated by their ability to handle complex geometries better than tensor-product B-splines. The box spline we use is defined by a set of vectors invariant under the multiplication by a sampling matrix. We show that such a box spline satisfies a dilation equation which is the basis for the scale-space filtering we propose. Several numerical applications in 2D conclude the correspondence.

We present a general framework for a higher-order spline level-set (HLS) method and apply this to smooth surface constructions. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solve this partial differential equation using a C2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. We provide two demonstrative smooth surface construction examples of our HLS method. The first is the construction of a smooth surface model (an implicit solvation interface) of bio-molecules in solvent, given their individual atomic coordinates and solvated radii. The second is the smooth surface reconstruction from a cloud of points generated from a 3D surface scanner.

A distributed information model of spline finite strip method based on the Multi-agent system is raised, and its mathematics expression is given to describe the operational situation of each Agents. The characters of the splines can be identified and the request from the CAX system can be processed through the Spline Agents' intercourse and collaboration in the model. Finally, a simple pass design of cold roll forming is taken as an example to illustrate the application of the mathematical model.

In this paper, we develop a systematic method for detecting the extrema of bivariate spline functions and of their derivatives. It is assumed that the splines are constituted by employing normalized, uniform B-splines as the basis functions, and the detection procedure fully utilizes the spline properties. All the extrema can be found except those with singular Hessian matrix. By numerical examples, we demonstrate the effectiveness of the method.

The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models (GLMM). We consider the asymptotic rates such that Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a full Bayesian view point by imposing a priori distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coeffocients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov Chain Monte Carlo (MCMC) results with their asymptotic approximation in a simulation study. ispartof: DTEW - KBI_0733 pages:1-39 status: published

In this paper, we study an effective quintic polynomial spline method for numerical solution, and first order to fifth order numerical derivatives of the analytic solution at the knots for a class of sixth order two-point boundary value problems. Our new method is based on a quintic spline interpolation problem. It is easy to implement and is able to provide sixth order accurate numerical results at the knots. Numerical tests show that our method is very practical and effective.

This paper, resulting from research collaboration with the UK National Physical Laboratory, is the first to present successfully a simple method for controlling the location parameters in univariate spline approximations. Traditional highly non-linear approaches are avoided by considering the parameters to be a function of a given density model. We present a number of density models for a range of data types, such as dominant local variability. This paper delivers to a scientific discipline applying polynomial spline approximations to recover discrete data to a high level of accuracy a method which avoids the need to construct complicated mathematical models.

In this paper, we present a systematic method for determining an optimal smoothing parameter, an important issue in the design of smoothing splines. The optimal parameter is obtained as the minimizer of the so-called cross-validation function, and the global minimum is found very efficiently using extrema detection method of spline functions. Compared to the conventional search algorithm, the computational amount is reduced dramatically. The effectiveness is verified by a numerical example.

Three different strategies to determine the tension parameters pi of the exponential spline (or spline under tension) are discussed. A first heuristic strategy is based on the knowledge of the interpolating cubic spline and the pi-values are proposed in order to eliminate undesired inflection points. Convexity or monotonicity of the interpolant cannot be guaranteed. A convexity preserving C2 — interpolant (if possible) is constructed by solving a constrained nonlinear optimization problem for the tension parameters pi. This second strategy characterizes an ‘optimal’ set of pi-values. The optimization problem is the base to derive ‘a priori’ estimates for the pi in a third strategy. The convexity arguments are supplemented by monotonicity constraints. The performance of all strategies is demonstrated in several examples.

This article describes a new approach to estimate F0 curves using B-spline and Spline models characterized by a knot sequence and associated control points The free parameters of the model are the number of knots and their location The free-knot placement, which is a NP-hard problem, is done using a global MLE (Maximum Likelihood Estimation) within a simulated-annealing strategy Experiments are conducted in a speech processing context on a 7000 syllables french corpus We estimate the two challenging models for increasing values of the number of free parameters We show that a B-spline model provides a slightly better improvement than the Spline model in terms of RMS error.

This paper considers a problem of designing optimal smoothing spline surfaces employing normalized uniform B-splines as the basis functions. Assuming that the data is obtained by sampling some surface with noises, an expression for optimal smoothing surfaces is derived when the number of data becomes infinity. Then, under certain condition, we present the convergent properties of optimal smoothing spline surface. Moreover, they are extended to the case of periodic spline surfaces. The results are applied to the problem of contour modeling of moving deformable objects, and the effectiveness is examined by numerical and experimental studies.

We review recent developments concerning refinable, multivariate piecewise polynomials with compact support. We first consider uniform meshes, box splines, box spline wavelets and a generalisation of box splines called multi-box splines. Our next topic is spline functions on general triangulations, including continuous linear spline wavelets and hierarchical bases for C1 splines based on macro-elements. Similar types of spline functions are then studied for meshes gained from triangulating a mesh of quadrilaterals.

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

This paper discusses how Beta spline curve with geometric continuity is influenced by the change of parameters beta1 and beta2, offers the range of parameters beta1 and beta2 and presents the algorithm principles of bibeta spline curved surface that is controlled by double parameters. Based on all above, a freedom surface modeling system (FSMS) is designed and some examples are verified and analyzed

Converting point samples and/or triangular meshes to a more compact spline representation for arbitrarily topology is both desirable and necessary for computer vision and computer graphics. This paper presents a C1 manifold interpolatory spline that can exactly pass through all the vertices and interpolate their normals for data input of complicated topological type. Starting from the Powell-Sabin spline as a building block, we integrate the concepts of global parametrization, affine atlas, and splines defined over local, open domains to arrive at an elegant, easy-to-use spline solution for complicated datasets. The proposed global spline scheme enables the rapid surface reconstruction and facilitates the shape editing and analysis functionality.

This paper deals with a very important practical problem of constructing the “best” spline approximation to a curve or a surface. Needless to say, this problem has many practical application, especially in cases of data modeling and data fitting. In this paper we are dealing with curves, in one- and multi- dimensional cases, and, in particular, with closed curves. The main methods of this paper are methods of (generalized) Pade approximations and orthogonal polynomials on the interval and the unit circle, and related methods of mechanical quadrature formulas. For references to the approaches to the spline reconstruction of a data curve, where a variety of “fitting” figures of merit is used, see [KH], [MW], [MP], [W], [M95], [C95], [M97], [MM], [LM94], [Z].

A new class of Smooth Transition Autoregressive(STAR) models, based on cubic spline type transition functions, has been recently introduced by the authors of this paper and subjected to comparison with models based on the traditional logistic functions. A very high degree of similarity between these two classes of models has been demonstrated.

A rational cubic C2 spline curve has been described which has point tension weights for manipulating the shape of the curve. The spline is presented in both interpolatory and local support basis form, and the effect of the weights on these representations is analysed.