Radial basis functions under tension

We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces.     

Transfinite Thin Plate Spline Interpolation

Duchon’s method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo–Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo–Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo–Levi polysplines are based on the analysis of a new class of univariate exponential ℒ-splines satisfying adjoint natural end conditions.     

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Summary. Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Received March 14, 1995     

关于n维单形保多项式超限插值的表示问题

以R~n表示n维欧氏空间,Z_+~n是R~n中坐标均为非负整数的全体,e~s为Z_+~(n+1)中第s个坐标为1其余坐标为0的单位向量;π_d(R~n)为全次数不大于d的n元多项式全体,     

Multivariate interpolating (m, l, s)-splines

The multivariate interpolating (m, l, s)-splines are a natural generalization of Duchon's thin plate splines (TPS). More precisely, we consider the problem of interpolation with respect to some finite number of linear continuous functionals defined on a semi-Hilbert space and minimizing its semi-norm. The (m, l, s)-splines are explicitly given as a linear combination of translates of radial basis functions. We prove the existence and uniqueness of the interpolating (m, l, s)-splines and investigate some of their properties. Finally, we present some practical examples of (m, l, s)-splines for Lagrange and Hermite interpolation.     

Inverse and saturation theorems for radial basis function interpolation

While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline interpolation we also give certain saturation theorems.

     

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.     

Design with L-splines

We recently obtained a criterion to decide whether a given space of parametrically continuous piecewise Chebyshevian splines (i.e., splines with pieces taken from different Extended Chebyshev spaces) could be used for geometric design. One important field of application is the class of L-splines, that is, splines with pieces taken from the null space of some fixed real linear differential operator, generally investigated under the strong requirement that the null space should be an Extended Chebyshev space on the support of each possible B-spline. In the present work, we want to show the practical interest of the criterion in question for designing with L-splines. With this in view, we apply it to a specific class of linear differential operators with real constant coefficients and odd/even characteristic polynomials. We will thus establish necessary and sufficient conditions for the associated splines to be suitable for design. Because our criterion was achieved via a blossoming approach, shape preservation will be inherent in the obtained conditions. One specific advantage of the class of operators we consider is that hyperbolic and trigonometric functions can be mixed within the null space on which the splines are based. We show that this produces interesting shape effects.     

Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions

Thin plate spline methods provide an interpolant to values of a real function and are highly useful in many applications. The need for iterative procedures arises, since hardly any sparsity occurs in the linear system of interpolation equations. This paper considers a generalization of the iterative algorithm developed by Beatson, Goodsell and Powell. A proof of convergence of this method is given. It depends mainly on the remark that all the changes to the thin plate spline coefficients reduce a certain semi‐norm of the difference between the required interpolant and the current approximation. The analysis applies also to analogous algorithms for other radial basis functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A sinusoidal polynomial spline and its Bezier blended interpolant

Functional polynomials composed of sinusoidal functions are introduced as basis functions to construct an interpolatory spline. An interpolant constructed in this way does not require solving a system of linear equations as many approaches do. However there are vanishing tangent vectors at the interpolating points. By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant. The blending factor has the effect similar to the “tension” control of tension splines. Piecewise interpolants can be constructed in an analogous way as a connection of Bezier curve segments to achieve C¹ continuity at the connecting points. Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces. The interpolant can very efficiently be evaluated by tabulating the sinusoidal function.     

弹性薄板的样条梁函数解法

本文从三次及二次样条梁函数定义的四阶及三阶的广义梁的微分方程出发,由于采用了广义函数,可推导出连续荷载、间断荷载、集中荷载、集中弯矩等各种荷载及各种边界条件(简支、固支、自由)下的多项式梁函数.用最小势能原理推导弹性薄板变形曲面及应力,均获得精度较高的近似解.     

A class of piecewise interpolating functions based on barycentric coordinates

Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolant function by means of polynomial pieces and ensure that some regularity conditions are guaranteed at the break-points. In this work, we propose a novel class of piecewise interpolating functions whose expression depends on the barycentric coordinates and a suitable weight function. The underlying idea is to specialize to the 1D settings some aspects of techniques widely used in multi-dimensional interpolation, namely Shepard’s, barycentric and triangle-based blending methods. We show the properties of convergence for the interpolating functions and discuss how, in some cases, the properties of regularity that characterize the weight function are reflected on the interpolant function. Numerical experiments, applied to some case studies and real scenarios, show the benefit of our method compared to other interpolant models.     

Algorithms for cardinal interpolation using box splines and radial basis functions

Summary We describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of basis functions which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ²), the thin plate spline, and the multiquadric case give comparably equal and good results.     

On steady and large time solutions of the semi-discrete Moving Finite Element equations for one-dimensional diffusion problems

This paper considers how the Moving Finite Element (MFE) methodapproxim ates the steady and large time solutions of a familyof linear diffusion equations in one space dimension. In particular,it is shown that any steady solution to the Moving Finite Elementequations must satisfy the stationary equations for a best approximationto the steady solution of the PDE from the manifold of free-knotlinear splines, in some problem dependent norm. For the special case of the inhomogeneous linear heat equationit is also shown that, under certain conditions, the only steadyMFE solution is the unique global best fit to the true steadysolution, in the H¹ semi-norm. It is also demonstrated numericallythat these steady solutions are stable attractors. Finally,a numerical study of the large time solutions of the homogeneouslinear heat equation is undertaken and it is demonstrated thatthe MFE solutions appear to possess a rather novel temporalaccuracy property.     

Spatial Regression Models,Response Surfaces,and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.     

Martensen Splines

This paper is concerned with the construction of the fundamental functions associated with a two-point Hermite spline interpolation scheme used by Martensen in the context of the remainder of the Gregory quadrature rule. We derive both a recursive construction and an explicit representation in terms of the underlying B-Splines which can easily be deduced using Marsden’s identity. We can make use of these functions in order to introduce a local interpolation scheme which reproduces all splines. Finally, we examine the error of this interpolant to a sufficiently smooth function and realize that it behaves like in the case of splines of degree n. AMS subject classification (2000) 65D05, 65D07, 41A15     

Construction of interpolation splines minimizing semi-norm in W_{2}^{(m,m-1)}(0,1) space

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m?2 and e ^?x . Also some numerical results are presented.     

The bidimensional interproximation problem and mixed splines

Interproximation methods for surfaces can be used to construct a smooth surface interpolating some data points and passing through specified regions. In this paper we study the use of mixed splines, that is smoothing splines with additional interpolation constraints, to solve the interproximation problem for surfaces in the case of scattered data. The solution is obtained by solving a linear system whose structure can be improved by using “bell-shaped” thin plate splines.     

A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions

A meshless method is proposed for the numerical solution of the two space dimensional linear hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The new developed scheme uses collocation points and approximates the solution employing thin plate splines radial basis functions. Numerical results are obtained for various cases involving variable, singular and constant coefficients, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A class of spline functions for landmark‐based image registration

A class of spline functions, called Lobachevsky splines, is proposed for landmark‐based image registration. Analytic expressions of Lobachevsky splines and some of their properties are given, reasoning in the context of probability theory. Because these functions have simple analytic expressions and compact support, landmark‐based transformations can be advantageously defined using them. Numerical results point out accuracy and stability of Lobachevsky splines, comparing them with Gaussians and thin plate splines. Moreover, an application to a real‐life case (cervical X‐ray images) shows the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.