Title data
Sharon, Nir ; Cohen, Rafael Sherbu ; Wendland, Holger:
On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions.
In: SIAM Journal on Scientific Computing.
Vol. 45
(2023)
Issue 5
.
- S. A2458 - A2482.
ISSN 1095-7197
DOI: https://doi.org/10.1137/22M1528306
Project information
Project title: |
Project's official title Project's id Multiskalen Approximationsverfahren für beliebig verteilte, Skalar- und Mannigfaltigkeit-wertige Daten / Multiscale Approximation Methods for Scattered Scalar- and Manifold-Valued Data No information |
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Project financing: |
Deutsche Forschungsgemeinschaft |
Abstract in another language
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.
Further data
Item Type: | Article in a journal |
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Refereed: | Yes |
Keywords: | multiscale approximation; manifold-valued functions; scattered data; quasi-interpolation |
Subject classification: | Mathematics Subject Classification Code: 65D15 65G99 68T09 43A99 |
Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics III (Applied and Numerical Analysis) > Chair Mathematics III (Applied and Numerical Analysis) - Univ.-Prof. Dr. Holger Wendland |
Result of work at the UBT: | Yes |
DDC Subjects: | 500 Science > 510 Mathematics |
Date Deposited: | 06 Mar 2024 08:04 |
Last Modified: | 06 Mar 2024 08:04 |
URI: | https://eref.uni-bayreuth.de/id/eprint/88799 |