Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.     

带障碍的广义插值样条与带状态约束的最优控制

本文由样条的极值性质出发给同分算子插值样条（即广义插值样条）新的推导方法。用这种方法可推导出带障碍（即带不等式约束）的微分算子插值样条的解析性质,为简便计,本文以非负广义插值样条为例。最后,揭示了状态带不等式的最优控制解的必要性准则与带障碍的广义插值样条的联系。     

The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane

Summary In this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions. It turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed+1. Some numerical examples which conform the theoretical results are presented. Present address: (1. 7. 1988–31. 12. 1988) Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

The convergence of spline collocation for strongly elliptic equations on curves

Summary Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthdayThis work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the Stiftung Volkswagenwerk     

Likelihood ratio tests for goodness-of-fit of a nonlinear regression model

We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first-order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum-likelihood estimates are not as severely biased as the maximum-likelihood estimates in the penalized splines framework.     

Nonnegative splines, in particular of degree five

Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given. Received November 16, 1995 / Revised version received February 24, 1997     

A simple rational spline and its application to monotonic interpolation to monotonic data

Summary We shall consider a class of simple rational splines and their application to monotonic interpolation to monotonic data. Our method is situated between interpolation with the usual cubic splines and with monotone quadratic splines. A selection of numerical results is presented in Figs. 4–11.     

The Structural Characterization and Locally Supported Bases for Bivariate Super Splines

Super splines are bivariate splines defined on triangulations, where the smoothness enforced at the vertices is larger than the smoothness enforced across the edges. In this paper, the smoothness conditions and conformality conditions for super splines are presented. Three locally supported super splines on type-1 triangulation are presented. Moreover, the criteria to select local bases is also givenBy using local supported super spline function, a variation-diminishing operator is built. The approximation properties of the operator are also presented.     

Nonlinear general path models for degradation data with dynamic covariates

Degradation data have been widely used to estimate product reliability. Because of technology advancement, time‐varying usage and environmental variables, which are called dynamic covariates, can be easily recorded nowadays, in addition to the traditional degradation measurements. The use of dynamic covariates is appealing because they have the potential to explain more variability in degradation paths. We propose a class of general path models to incorporate dynamic covariates for modeling of degradation paths. Physically motivated nonlinear functions are used to describe the degradation paths, and random effects are used to describe unit‐to‐unit variability. The covariate effects are modeled by shape‐restricted splines. The estimation of unknown model parameters is challenging because of the involvement of nonlinear relationships, random effects, and shaped‐restricted splines. We develop an efficient procedure for parameter estimations. The performance of the proposed method is evaluated by simulations. An outdoor coating weathering dataset is used to illustrate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Summary In this paper the problem of smoothing a given data set by cubicC ²-splines is discussed. The spline may required to be convex in some parts of the domain and concave in other parts. Application of splines has the advantage that the smoothing problem is easily discretized. Moreover, the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained. Therefore the latter program is treated numerically much more easier than the original program. Further, the validity of a return-formula is of importance by which a minimizer of the orginal program is obtained from a maximizer of the dual program.The theoretical background of this general approach is discussed and, above all, the details for applying the strategy to the present smoothing problem are elaborated. Also some numerical tests are presented.     

Numerische Quadratur bei Projektionsverfahren

Summary In this paper we investigate the influence of the numerical quadrature in projection methods. In particular we derive conditions for the order of the quadrature formulas in finite element methods under which the order of convergence is not perturbed. It seems that this question has been discussed only for the Ritz method. There is an essential difference between this method on one side and the Galerkin and least squares methods on the other side. The methods using numerical integration are only in the latter case still projection methods. The resulting conditions for the quadrature formulas are often much weaker than those for the Ritz method. Numerical examples using cubic splines and polynomials show that the conditions derived are realistic. These examples also allow the comparison of some projection methods.

     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Error control process in function interpolation using statistical spline model

In this paper, we propose an algebraic method based on solving some inequalities of polynomial type to control the error value of interpolation formulas whose residue depends on a monic polynomial. This method then leads to construct some piecewise approximations (splines) of statistical type, which are based on a specific partition of the main interval. In other words, in this model of spline, approximate criteria are considered fixed and sub-intervals corresponding to criteria are derived as accurately as possible. In this sense, some statistical concepts such as expected value, variance measure, skewness and kurtosis coefficients are also inserted into the definition of statistical splines. Finally, a numerical results section is separately given to confirm all results in the paper.     

Lösung von gewöhnlichen Differentialgleichungen mit nichtlinearen Splines

Zusammenfassung In dieser Arbeit werden nichtlineare Splines zur Lösung von Anfangswertaufgaben bei gewöhnlichen Differentialgleichungen herangezogen. In der Nähe von Singularitäten besitzen z.B. verallgemeinerte rationale Splines mit variablen Exponenten gute Approximationseigenschaften. Bei Polynomsplines können Konvergenzaussagen hergeleitet werden, indem Äquivalenz dieser Verfahren mit gewissen linearen Mehrschrittverfahren gezeigt wird. In dieser Arbeit behandeln wir den nichtlinearen Fall, indem wir die lokalen Fehler in den Knoten direkt verfolgen. Einige numerische Beispiele zeigen die Güte dieser Verfahren insbesondere bei solchen Lösungen, die sehr steil anwachsen oder sogar im betrachteten Intervall singulär werden.

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Solution of ordinary differential equations with nonlinear splines

Summary We consider the technique of using nonlinear splines to solve the initial value problem of ordinary differential equations. It is known, for example, that generalized rational splines with variable exponents yield good approximations to the exact solution in the neighborhood of a singularity. In the case of polynomial splines, convergence results may be derived by demonstrating the equivalence of the method to linear multistep methods. This sort of analysis has been done by many authors. In this paper we treat the nonlinear case and are able to prove convergence by directly estimating the local errors at interior knots. Some computational examples are given which illustrate the power of the method near a singularity.

     

An algorithm for computing constrained smoothing spline functions

Summary The problem of computing constrained spline functions, both for ideal data and noisy data, is considered. Two types of constriints are treated, namely convexity and convexity together with monotonity. A characterization result for constrained smoothing splines is derived. Based on this result a Newton-type algorithm is defined for computing the constrained spline function. Thereby it is possible to apply the constraints over a whole interval rather than at a discrete set of points. Results from numerical experiments are included.     

Boundary element collocation methods using splines with multiple knots

Summary. We extend the theory of boundary element collocation methods by allowing reduced inter-element smoothness (or in other words, by allowing trial functions that are splines with multiple knots). Our convergence analysis is based on a recurrence relation for the Fourier coefficients of the numerical solution, and so is restricted to uniform grids on smooth, closed curves. Superconvergence is possible with special choices of the collocation points. Numerical experiments with a model problem confirm the convergence rates predicted by our theory. Received September 19, 1995     

Algorithm for constructing range restricted histosplines

This paper is concerned with numerical methods in range restricted histopolation. The proposal is to apply splines on refined grids. The ratios of the added split points are considered to be parameters. In this way, by choosing suitable spline classes, range restricted histosplines can always be constructed if the restrictions are compatible with the given histogram. We offer an algorithm for solving the bivariate problem on a rectangular grid which utilizes univariate results as well as tensor product techniques. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bivariate spline interpolation at grid points

Summary. We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples. Received October 5, 1992 / Revised version received May 13, 1994     

A method of adding–removing knots for solving smoothing problems with obstacles

We study a method of adding–removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.     

A topological spline system approach to even degree polynomial splines

The existence of minimum norm properties for even degree polynomial splines, analogous to the. ones known for odd degree splines, is investigated within the framework of the theory of

topological spline systems. It is shown that such properties cannot exist for even degree splines interpolating functions halfway between the partition points. For another class of even

degree spline functions, however, which hterpolate the local integrals of given functions with respect to the partitions, the seeked minimum norm properties can be proved. This is carried out

by first investigating a generalized problem within the theory of spline systems and then deriving corresponding conclusions. As a corollary the existence of spline systems with respect to differential operators of fractional degree is obtained.