On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917754843430912 |
|---|---|
| author | Harbrecht, Helmut Kempf, Rüdiger Multerer, Michael |
| author_facet | Harbrecht, Helmut Kempf, Rüdiger Multerer, Michael |
| contents | In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces Harbrecht, Helmut Kempf, Rüdiger Multerer, Michael Numerical Analysis Machine Learning In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics. |
| title | On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2408.11389 |