Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, 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 insight 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 two-dimensional 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. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.

中文翻译：

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二维和三维分段平滑函数的两阶段逼近策略

给定域 $[0,1]^d$, $d=2,3$ 内的方形网格上的分段平滑函数 $f$ 的值，我们寻找 $f$ 的分段自适应逼近。标准逼近技术在域边界附近和函数或其导数的跳跃奇点曲线附近实现了减少的逼近阶数。这里使用的见解是边界附近或奇点曲线附近的行为完全由跨边界和跨奇点曲线的数据的某些差异的值来表征和识别。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，我们寻找一个签名与 $f$ 的签名匹配的近似值。给定网格上的函数数据，假设函数是分段光滑的，首先识别函数的奇点结构。例如，在二维情况下，我们找到了在 $f$ 的平滑段之间分离的曲线的近似值。其次，同时，我们找到了 $f$ 的不同部分的近似值。从匹配近似签名的原理和关于给定网格的函数导出的方程系统定义了第一阶段近似。使用在第一阶段近似中获得的误差的全局近似来构造第二阶段改进的近似。我们找到了 $f$ 不同部分的近似值。从匹配近似签名的原理和关于给定网格的函数导出的方程系统定义了第一阶段近似。使用在第一阶段近似中获得的误差的全局近似来构造第二阶段改进的近似。我们找到了 $f$ 不同部分的近似值。从匹配近似签名的原理和关于给定网格的函数导出的方程系统定义了第一阶段近似。使用在第一阶段近似中获得的误差的全局近似来构造第二阶段改进的近似。