离散标量和流形函数的多尺度拟插值

N. Sharon, Rafael Sherbu Cohen, H. Wendland
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引用次数: 0

摘要

我们解决了从任意分散位置给出的离散样本近似未知函数的问题。这个问题在数值科学中是必不可少的,在现代应用中也强调需要解决具有流形值的函数的情况。本文介绍并分析了标量函数和流形函数的基于核的拟插值和多尺度逼近的组合。当数据定义在无限网格上时,准插值为逼近问题提供了一个强大的工具,但当涉及到分散数据时,情况就更加复杂了。在这里,高阶准插值方案要么需要导数信息,要么在数值上变得不稳定。因此,本文主要研究拟插值与多尺度技术相结合的改进方法。本文的主要贡献如下:首先,我们介绍了标量函数的多尺度拟插值技术。其次,我们将展示如何使用将最小二乘算子移动到流形值设置来延续此技术。第三,给出了收敛拟插值也会导致收敛多尺度拟插值的数学证明。第四,我们提供了大量的数值证据,证明了多尺度拟插值比拟插值具有更好的收敛性。此外,我们还将提供一些例子,表明多尺度准插值方法为许多数据分析任务提供了强大的工具,例如去噪和异常检测。它对大量数据点和高维的情况特别有吸引力。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to the case of functions with manifold values. In this paper, we introduce and analyze a combination of kernel-based quasi-interpolation and multiscale approximations for both scalar- and manifold-valued functions. While quasi-interpolation provides a powerful tool for approximation problems if the data is defined on infinite grids, the situation is more complicated when it comes to scattered data. Here, higher-order quasi-interpolation schemes either require derivative information or become numerically unstable. Hence, this paper principally studies the improvement achieved by combining quasi-interpolation with a multiscale technique. The main contributions of this paper are as follows. First, we introduce the multiscale quasi-interpolation technique for scalar-valued functions. Second, we show how this technique can be carried over using moving least-squares operators to the manifold-valued setting. Third, we give a mathematical proof that converging quasi-interpolation will also lead to converging multiscale quasi-interpolation. Fourth, we provide ample numerical evidence that multiscale quasi-interpolation has superior convergence to quasi-interpolation. In addition, we will provide examples showing that the multiscale quasi-interpolation approach offers a powerful tool for many data analysis tasks, such as denoising and anomaly detection. It is especially attractive for cases of massive data points and high dimensionality.
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