Chong-Jun Li

Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54-65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.

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 l1 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.

A pseudo-heuristic parameter selection rule for l1-regularized minimization problems

Abstract This paper considers the regularization parameter determination of l 1 -regularized minimization problem. We solve the l 1 -regularized problem using iterative reweighted least squares (IRLS) which involves solving a linear system whose coefficient matrix has the form α M + ( 1 − α ) N ( α ∈ ( 0 , 1 ) ). The aim of this paper is to find an efficient and computationally inexpensive algorithm to both choose the regularization parameter and solve the l 1 -regularized problem. In order to achieve this, we propose an IRLS algorithm with adaptive regularization parameter selection based on a heuristic parameter determination rule—de Boor’s parameter selection criterion. Compared with some of the state-of-the-art algorithms and parameter selection rules, the numerical experiments show the efficiency and robustness of the proposed method.

The bivariate quadratic C1 spline spaces with stable dimensions on the triangulations

Abstract The dimension is a basic problem in the theory of the bivariate spline space. In general, the dimension of the bivariate spline space defined on the triangulation is unstable or with singularity. In this paper, we consider the dimensions of the bivariate quadratic C 1 spline spaces defined on the triangulations. Our main result is when the degree of each interior vertex of the non-degenerate triangulation is at least 6, the dimension of the corresponding bivariate quadratic C 1 spline space is stable and equal to the number of the boundary vertices plus 3. We also give an example to show that the non-degenerate condition is necessary.

The C1 and C2 quasi-Plateau problems☆

Abstract In this paper, we study the C k quasi-Plateau problem: to find the parametric surface of minimal area defined on a rectangular parametric domain among all the surfaces which are C k continuous on the border ( k = 1 , 2 ). An approach is proposed based on the Coons surface and the MRA (Multi-Resolution Analysis) formed by the B-spline of orders 3 and 4. The Coons surface is constructed firstly to satisfy the prescribed border conditions. Then replacing the area functional with the Dirichlet functional and using MRA, the problem reduces into solving a system of linear equations. The linear equations have simple and sparse structure. Finally, the method is concluded into six algorithms according to the different boundary conditions. Examples are provided to illustrate that the proposed method is effective and flexible.

Curve and surface fitting models based on the diagonalizable differential systems

Abstract Curve and surface fitting is an important problem in computer aided geometric design, including many methods, such as the B-spline method, the NURBS method and so on. However, many curves and surfaces in the natural or engineering fields need to be described by differential equations. In this paper, we propose a new curve and surface fitting method based on the homogeneous linear differential systems. In order to approximate general curves or surfaces well, the diagonalizable differential systems with variable coefficients are adopted, which have explicit solutions. The fitting algorithms are presented for curves and surfaces from discrete points. Some numerical examples show that the two algorithms can obtain good fitting accuracy as the B-spline method.