An explicit quasi-interpolation scheme based on C1 quartic splines on type-1 triangulations
ABSTRACT We describe a new scheme based on quartic C1-splines on type-1 triangulations approximating regularly distributed data. The quasi-interpolating splines are directly determined by setting the Bernstein–Bézier coefficients of the splines to appropriate combinations of the given data values. Each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Moreover, the operator interpolates the given data values at all the vertices of the underlying triangulation. Since the Bernstein–Bézier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield nearly-optimal approximation order for smooth functions. The order is known to be best possible for these spaces. Numerical tests confirm the theoretical behavior and show that the approach leads to functional surfaces of high visual quality.
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ABSTRACT: In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants, which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the space P m of polynomials of total degree at most m, we first propose a general method to determine a new differential quasi-interpolation operator Q D r which is exact on P m+r . Q D r uses the values of the function to be approximated at the points involved in the linear functional defining Q d as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C 1 B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolant Q d . We estimate with small constants the quasi-interpolation errors f −Q D r [f] and f Q d [f] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.Journal of Computational and Applied Mathematics 11/2013; 252. DOI:10.1016/j.cam.2013.01.015 · 1.08 Impact Factor
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ABSTRACT: In this paper, we describe the construction of a suitable normalized B-spline representation for bivariate C1C1 cubic super splines defined on triangulations with a Powell–Sabin refinement. The basis functions have local supports, they form a convex partition of unity, and every spline is locally controllable by means of control triangles. As application, discrete and differential quasi-interpolants of optimal approximation order are developed and numerical tests for illustrating theoretical results are presented.Mathematics and Computers in Simulation 05/2014; 99:108–124. DOI:10.1016/j.matcom.2013.04.020 · 0.86 Impact Factor