Bivariate hierarchical Hermite spline quasi-interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.     

Quadratic spline interpolation with coinciding interpolation and spline grids

In this paper the quadratic spline interpolation with coinciding interpolation and spline grids for continuous functions is considered. The theorems mainly concern error estimations which allow to formulate a convergence statement. To get such results it is assumed that the function to be interpolated is suitably smooth or possesses a special behavior. A best approximation property and a statement about the solution of boundary value problems using quadratic spline functions are added.     

Discrete cubic spline interpolation

Defining equations, a best approximation property, and error bounds are given for a discrete cubic spline interpolant. Furthermore the distance between two cubic spline interpolants is estimated, and numerical examples are provided.     

Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L ₂.     

Successive polynomial spline function approximation

The use of successive polynomial spline approximation is established as a method of improving the accuracy of estimates of derivatives of periodic functions approximated by interpolating odd order splines defined on a uniformly spaced set of data points. For the various configurations possible with this multiple-approximation method, bounds for the leading error terms are explicitly given. In particular, for the quintic spline, the variety of approximation sequences is described in detail.     

High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem

We use the generalized L1 approximation for the Caputo fractional derivative, the second-order fractional quadrature rule approximation for the integral term, and a classical Crank-Nicolson alternating direction implicit (ADI) scheme for the time discretization of a new two-dimensional (2D) fractional integro-differential equation, in combination with a space discretization by an arbitrary-order orthogonal spline collocation (OSC) method. The stability of a Crank-Nicolson ADI OSC scheme is rigourously established, and error estimate is also derived. Finally, some numerical tests are given.     

Stability of spline approximation methods for multidimensional pseudodifferential operators

The present paper provides stability considerations of spline approximation methods for multidimensional singular operators. This paper should be regarded as a first step in establishing spline approximation methods for pseudodifferential operators on manifolds.     

Dynamic programming and bicubic spline interpolation

Dynamic programming techniques were used to obtain the spline approximation for a function with prescribed values on the knot points along a line. Extending this procedure to two dimensions, the bicubic spline approximation defined over a two-dimensional region is obtained in this paper employing the methods of dynamic programming. A regular rectangular region as well as a region with irregular boundaries can be handled by this method, avoiding the difficulties of large storage and high dimensionality.     

Another approach to the quintic spline

A set of end conditions for interpolatory quintic spline is derived by use of integration. These end conditions are only in terms of function values at the knots (data), and rise to O(h⁶) spline approximation.     

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing-as a model problem-image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.     

Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.     

On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations

A boundary value problem for linear partial differential algebraic systems of equations with multiple characteristic curves is examined. It is assumed that the pencil of matrix functions associated with this system is smoothly equivalent to a special canonic form. The spline collocation is used to construct for this problem a difference scheme of an arbitrary approximation order with respect to each independent variable. Sufficient conditions are found for this scheme to be absolutely stable.     

On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.     

A note on mean square spline approximation

A method of obtaining the mean-square spline approximation by the use of dynamic programming is indicated.     

Cubic spline functions and initial value problems

A numerical process is presented which provides a cubic spline function approximation for the solution of initial value problems in ordinary differential equations. With interpolate cubic spline functions we can achieveO(h ⁴) convergence.     

Local spline approximation methods for singular product integration

The purpose of this paper is to propose and study local spline approximation methods for singular product integration, for which; i) the precision degree is the highest possible using spline approximation; ii) the nodes can be assumed equal to arbitrary points, where the integrand function f is known; iii) the number of the requested evaluations of f at the nodes is low; iv) a satis factory convergence theory can be proved. Work sponsored by “Ministero dell' University” and CNR of Italy     

A multilevel univariate cubic spline quasi‐interpolation and application to numerical integration

Quasi‐interpolation is very important in the study of the approximation theory and applications. In this paper, a multilevel univariate quasi‐interpolation scheme with better smoothness using cubic spline basis on uniform partition of bounded interval is proposed. Moreover, its application to numerical integration is presented. Furthermore, some examples are given and show that the presented method is easy and valid. Copyright © 2010 John Wiley & Sons, Ltd.     

Quadrature-free spline method for two-dimensional Navier-Stokes equation

In this paper, a quadrature-free scheme of spline method for two-dimensional Navier- Stokes equation is derived, which can dramatically improve the efficiency of spline method for fluid problems proposed by Lai and Wenston（2004）. Additionally, the explicit formulation for boundary condition with up to second order derivatives is presented. The numerical simulations on several benchmark problems show that the scheme is very efficient.     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

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.