论文标题
关于分段平滑函数近似的正规化校正方法
On a regularization-correction approach for the approximation of piecewise smooth functions
论文作者
论文摘要
当近似功能具有奇异性时，线性近似方法患有Gibbs振荡。 ENO-SR分辨率是避免振荡并完全准确的局部方法，但出现了大约规律性的丧失。本文的目的是引入一种具有完全准确性和规律性的属性的新方法。为了获得它，我们提出了一个三阶段算法：首先，通过减去适当的非平滑数据序列来平滑数据；然后，将所选的高阶线性近似运算符应用于平滑的数据，最后，通过使用第一阶段中使用的非平滑元素校正平滑近似值，可以恢复具有适当奇异性结构的近似值。我们使用第二阶段的4点细分算法应用于点值数据和细胞平均数据的两种方法。使用所提出的方法，我们能够以高精度构建近似值，具有很高的分段规律性，并且在存在不连续性的情况下没有扩散或振荡。
Linear approximation approaches suffer from Gibbs oscillations when approximating functions with singularities. ENO-SR resolution is a local approach avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper singularity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. We apply this approach to both cases of point-value data and of cell-average data, using the 4-point subdivision algorithm in the second stage. Using the proposed approach we are able to construct approximations with high precision, with high piecewise regularity, and without diffusion nor oscillations in the presence of discontinuities.
