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Bounds on Projections onto Bivariate Polynomial Spline Spaces with Stable Bases
 Constr. Approx
, 2002
"... . We derive L1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2  and ` 2 approximation by splines on quasiuniform triangulations. x1. Introduction Let X ` L1(\O ..."
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Cited by 19 (3 self)
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. We derive L1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2  and ` 2 approximation by splines on quasiuniform triangulations. x1. Introduction Let X ` L1(\Omega\Gamma be a linear space defined a set\Omega with polygonal boundary. Suppose h\Delta; \Deltai is a semidefinite innerproduct on X with associated seminorm k \Delta k. We assume that hf; gi = 0, whenever fg = 0 on \Omega\Gamma (1:1) kfk kgk, whenever jf(x)j jg(x)j for all x 2\Omega : (1:2) Suppose S ` X is a linear space of polynomial splines (bivariate piecewise polynomials) defined on a regular triangulation 4 of\Omega (two triangles intersect only at a common vertex or along a common edge). We assume that S is a Hilbert space with respect to h\Delta; \Deltai. Let P : X ! S be the projection of X onto S defined by the minimization problem kf \Gamma Pfk = min s2S kf \Gamm...
The multivariate spline method for numerical solution of partial differential equations and scattered data fitting
"... Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of nsimplices in the Euclidean space R^n. We present a straightforward method for using these spline functions to numerically solve elliptic partial differential equations such as Poisson and bi ..."
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Cited by 18 (14 self)
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Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of nsimplices in the Euclidean space R^n. We present a straightforward method for using these spline functions to numerically solve elliptic partial differential equations such as Poisson and biharmonic equations and fit given scattered data. This method does not require constructing macroelements or locally supported basis functions nor computing the dimension of the finite element spaces or spline spaces. We have implemented the method in MATLAB using multivariate splines in R² and R³. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the method.
Spherical Splines for Data Interpolation and Fitting
 SIAM J. Scientific Computing
, 2005
"... Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numer ..."
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Cited by 10 (6 self)
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Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines. Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D17
Multivariate Splines for Data Fitting and Approximation
"... Abstract. Methods for scattered data fitting using multivariate splines will be surveyed in this paper. Existence, uniqueness, and computational algorithms for these methods, as well as their approximation properties will be discussed. Some applications of multivariate splines for data fitting will ..."
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Cited by 9 (7 self)
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Abstract. Methods for scattered data fitting using multivariate splines will be surveyed in this paper. Existence, uniqueness, and computational algorithms for these methods, as well as their approximation properties will be discussed. Some applications of multivariate splines for data fitting will be briefly explained. Some new research initiatives of scattered data fitting will be outlined. Given a set of scattered data, e.g., {(xi, yi, zi), i = 1, · · · , N}, we need to find a smooth function or surface S such that S(xi, yi) = zi, i = 1, · · · , N, if zi are very accurate measurements or
C¹ Quintic Splines on Type4 Tetrahedral Partitions
 ADVANCES IN COMP. MATH
, 2004
"... Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type4) partition and associated trivariate C¹ quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is a ..."
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Cited by 9 (4 self)
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Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type4) partition and associated trivariate C¹ quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.
Computing Bivariate Splines in Scattered Data Fitting and the Finiteelement Method
"... Abstract. A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at one time. Typically this involves solving a system of linear equations. Examples include several wellknown methods for fitting scattered data, suc ..."
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Cited by 4 (2 self)
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Abstract. A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at one time. Typically this involves solving a system of linear equations. Examples include several wellknown methods for fitting scattered data, such as the minimal energy, leastsquares, and penalized leastsquares methods. Finiteelement methods for solving boundaryvalue problems are also of this type. It is shown here that these types of globallydefined splines can be efficiently computed, provided we work with spline spaces with stable local minimal determining sets. Bivariate splines defined over triangulations are important tools in several application areas including scattered data fitting and the numerical solution of boundaryvalue problems by the finite element method. Methods for computing spline approximations fall into two classes:
Energy minimization method for scattered data Hermite interpolation
 APPLIED NUMERICAL MATHEMATICS
, 2007
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A Domain Decomposition Method for Computing Bivariate Spline Fits of Scattered Data
"... Abstract. A domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy, discrete leastsquares, and penalized leastsquares splines is described. The method is based on splitting the domain into smaller domains, solving the associated smalle ..."
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Cited by 1 (1 self)
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Abstract. A domain decomposition method for solving large bivariate scattered data fitting problems with bivariate minimal energy, discrete leastsquares, and penalized leastsquares splines is described. The method is based on splitting the domain into smaller domains, solving the associated smaller fitting problems, and combining the coefficients to get a global fit. Explicit error bounds are established for how well our locally constructed spline fits approximate the global fits. Some numerical examples are given to illustrate the effectiveness of the method. Suppose f is a smooth function defined on a domain Ω in IR 2 with polygonal boundary. Given the values {fi: = f(xi, yi)} nd i=1 of f at some set of scattered points
A Bivariate Spline Approach for Image Enhancement
, 2010
"... A new approach for image enhancement is developed in this paper. It is based on bivariate spline functions. By choosing a local region from an image to be enhanced, one can use bivariate splines with minimal surface area to enhance the image by reducing noises, smoothing the contrast, e.g. smoothing ..."
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A new approach for image enhancement is developed in this paper. It is based on bivariate spline functions. By choosing a local region from an image to be enhanced, one can use bivariate splines with minimal surface area to enhance the image by reducing noises, smoothing the contrast, e.g. smoothing wrinkles and removing stains or damages from images. To establish this approach, we first discuss its mathematical aspects: the existence, uniqueness and stability of the splines of minimal surface area and propose an iterative algorithm to compute the fitting splines of minimal surface area. The convergence of the iterative solutions will be established. In addition, the fitting splines of minimal surface area are convergent as the size of triangulation goes to zero in theory. Finally, several numerical examples are shown to demonstrate the effectiveness of this new approach. 1