论文标题
校正了分段平滑双变量功能的近似策略
Corrected approximation strategy for piecewise smooth bivariate functions
论文作者
论文摘要
给定域$ω$中平方网格的分段平滑函数$ f $的值，我们寻找分段自适应近似为$ f $。标准近似技术达到了域边界附近的近似顺序，并接近函数或其导数的跳跃奇异点的近曲线。这里使用的想法是，在边界附近或奇异曲线附近的行为是由跨边界和整个奇异性曲线的数据的某些差异的值完全表征和识别的。我们将这些值称为$ f $的签名。在本文中，我们旨在使用这些值来定义近似值。也就是说，我们寻找一个近似值，其签名与$ f $的签名匹配。在网格上给定的功能数据，假设该函数是分段平滑的，首先，识别函数的奇异性结构。例如，在二维情况下，我们发现了左右$ f $的平滑段之间的曲线的近似值。其次，同时我们找到了$ f $不同部分的近似值。从与给定网格相匹配的原理和函数的原理得出的方程式系统定义了第一阶段的近似值。第二阶段改进的近似是使用全局近似值对第一阶段近似中获得的误差进行构建的。
Given values of a piecewise smooth function $f$ on a square grid within a domain $Ω$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The idea used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example in the 2-D case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. An second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.
