Numerical Integration Based on Bivariate Quartic Quasi-Interpolation Operators

In this paper,we propose a method to deal with numerical integral by using two kinds of C^2 quasi-interpolation operators on the bivariate spline space,and also dis- cuss the convergence properties and error estimates.Moreover,the proposed method is applied to the numerical evaluation of 2-D singular integrals.Numerical exper- iments will be carried out and the results will be compared with some previously published results.

The Bases of the Non-Uniform Cubic Spline Space S_(3)^(1,2)(Δ_(mn)^((2))

In this paper,the dimension of the nonuniform bivariate spline space S_(3)^(1,2)(Δ_(mn)^((2))is discussed based on the theory of multivariate spline space.Moreover,by means of the Conformality of Smoothing Cofactor Method,the basis ofS_(3)^(1,2)(Δ_(mn)^((2))composed of two sets of splines are worked out in the form of the values at ten domain points in each triangular cell,both of which possess distinct local supports.Furthermore,the explicit coefficients in terms of B-net are obtained for the two sets of splines respectively.

Sparse approximate solution of fitting surface to scattered points by MLASSO model

The goal of this paper is to achieve a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant(PSI) space and the l_1 norm minimization. In order to obtain different sparsity of the approximation solution, the problem is represented as a multilevel LASSO(MLASSO)model with different regularization parameters. The MLASSO model can be solved efficiently by the alternating direction method of multipliers. Numerical experiments indicate that compared to the AGLASSO model and the basic MBA algorithm, the MLASSO model can provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution. Moreover, the solution by the MLASSO model can reflect the regions of the underlying surface where high gradients occur.

The Cayley-Bacharach Theorem for Continuous Piecewise Algebraic Curves over Cross-cut Triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.

FITTING C1 SURFACES TO SCATTERED DATA WITH S 1 2 （△ m,n （2））

This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a Cl-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.