Abstract
This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.
Highlights
In the classical problem of variational spline interpolation one looks for a C2 curve sthat satisfies the problem ∫ minimize{ ′′ |s (t)|2 dt : s ∈ S, s(ti) = yi, i ··
In the paper [13], we focused on the sparest representation of interpolatory basis in truncated powers for scattered data, and studied one kind of interpolation for scattered data by bi-cubic polynomial natural spline, such that the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal
Theory 6 and Theory 7 show that bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval
Summary
In the paper [13], we focused on the sparest representation of interpolatory basis in truncated powers for scattered data, and studied one kind of interpolation for scattered data by bi-cubic polynomial natural spline, such that the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). The experiments show that our PUIEM is adaptive and outperforms the bi-cubic polynomial natural spline interpolations of four kinds and the thin plate spline in the interpolation results for scattered data. The paper is organized as follows: in Section 2, we recall the problems of bi-cubic natural spline interpolations for scattered data with four kinds of boundary conditions and study their solutions.
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