Numerical integration based on trivariate C 2 quartic spline quasi-interpolants

In this paper we consider the space generated by the scaled translates of the trivariate C ² quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals. We give weights and nodes of the above rules and we analyse their properties. Finally, some numerical tests and comparisons with other known integration formulas are presented.     

Shape preserving C2 cubic spline interpolation

A globally C² interpolatory cubic spline containing free parametersis derived and its properties established. Sufficient conditionsare given for choosing the parameters to control monotonicity;convexity is discussed. An algorithm is developed and testedon several examples.     

Properties of the Moreau-Yosida regularization of a piecewise C2 convex function

Received December 14, 1995 / Revised version received May 7, 1998 Published online October 21, 1998     

Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

^** Email: sorokina{at}math.uga.edu^*** Corresponding author. Email: zeilfeld{at}euklid.math.uni-mannheim.de We describe an approximating scheme based on cubic C¹ splineson type-6 tetrahedral partitions using data on volumetric grids.The quasi-interpolating piecewise polynomials are directly determinedby setting their Bernstein–Bézier coefficientsto appropriate combinations of the data values. Hence, eachpolynomial piece of the approximating spline is immediatelyavailable from local portions of the data, without using prescribedderivatives at any point of the domain. The locality of themethod and the uniform boundedness of the associated operatorprovide an error bound, which shows that the approach can beused to approximate and reconstruct trivariate functions. Simultaneously,we show that the derivatives of the quasi-interpolating splinesyield nearly optimal approximation order. Numerical tests withup to 17 x 10⁶ data sites show that the method can be used forefficient approximation.     

Inverses for a class of banded matrices and applications to piecewise cubic approximation

We give a simple criterion for the invertibility of a class of banded matrices that arise in the approximation by piecewise cubic polynomials. We also give a formula for the inverse in terms of the powers of a 2 × 2 matrix. We present sample applications of these results to interpolation and eigenvalue problems. As a side result wee find that the Gaussian points are best.     

C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

Product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes

Summary In this paper we consider the approximate evaluation of , whereK(x) is a fixed Lebesgue integrable function, by product formulas of the form based on cubic spline interpolation of the functionf.Generally, whenever it is possible, product quadratures incorporate the bad behaviour of the integrand in the kernelK. Here, however, we allowf to have a finite number of jump discontinuities in [a, b]. Convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

Local cardinal interpolation by C2 cubic B2-splines with a tunable shape parameter

A $C^2$ cubic local interpolating B2-spline, controllable by a shape parameter, is introduced and its properties are analyzed. An algorithm for the automatic selection of the free parameter is developed and tested on several examples. Finally, a two-phase subdivision scheme for its efficient evaluation at dyadic points is presented.     

Superconvergent recovery of gradients of piecewise linear finite element approximations on non-uniform mesh partitions

We propose the use of an averaging scheme, which recovers gradients from piecewise linear finite element approximations on the (1 + α˜)—regular triangular elements to gradients of the weak solution of a second-order elliptic boundary value problem in the 2-dimensional space. The recovered gradients, from mid-points of element edges, are superconvergent estimates of the true gradients. This work is an extension of Levine [Levine, IMA J. Numer. Anal. 5 , 407 (1985)] and follows some of the ideas therein. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:169–192, 1998     

An algorithm for least-squares fitting of cubic spline surfaces to functions on a rectilinear mesh over a rectangle

In this paper a method is presented for fitting, in the least-squares sense, a bivariate cubic spline function to values of a dependent variable, specified at points on a rectangular grid in the plane of the independent variables. Products of B-splines are used to represent the bicubic splines. The coefficients in this representation are determined by solving a set of one-dimensional least-squares problems.     

Monotone interpolation of order 3 by C2 cubic splines

We propose a local solution for the problem of interpolatingmonotone data by monotone C² cubic splines with two additionalknots per interval, on an arbitrary partition, and with an approximationof order 3. Our paper extends recent works by Archer & LeGruyer and Pruess.     

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation,which applies piecewise cubic polynomials(P_3) and possesses O(h~2) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with O(h2) accuracy in the broken H~2 norm. Besides, a discrete strengthened Miranda-Talenti estimate(▽_h~2·,▽_h~2·) =(?h·, ?h·), which is usually not true for nonconforming finite element spaces, is proved.The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.     

Sufficient conditions for the comonotone interpolation of cubic C 2-splines

We consider the problem of interpolation of a function under the condition of the preservation of the nature of its piecewise monotonicity. We give sufficient conditions for the comonotone interpolation by a classical cubic C ²-spline in the representation based on the expansion of its first derivative in a basis consisting of B-splines. These conditions allow to determine whether the soobtained spline is comonotone without solving the interpolation problem.     

New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C² edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle‐shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2^?j, each element has an envelope that is aligned along a “ridge” of length 2^?j/2 and width 2^?j. We prove that curvelets provide an essentially optimal representation of typical objects f that are C² except for discontinuities along piecewise C² curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n‐term partial reconstruction f obtained by selecting the n largest terms in the curvelet series obeys This rate of convergence holds uniformly over a class of functions that are C² except for discontinuities along piecewise C² curves and is essentially optimal. In comparison, the squared error of n‐term wavelet approximations only converges as n^?1 as n → ∞, which is considerably worse than the optimal behavior. © 2003 Wiley Periodicals, Inc.     

C1-continuous time integration based on cubic Hermite interpolation

We discuss a C¹-continuous time integration method based on piecewise cubic Hermite approximation. This method, denoted as p2-scheme, belongs to a class of one-step integration methods derived recently [1]. It exhibits a convergence rate of order four and shows properties similar to variational integrators, such as an excellent long-term energy preservation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)