An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913731612508160 |
|---|---|
| author | Iserles, Arieh Maierhofer, Georg |
| author_facet | Iserles, Arieh Maierhofer, Georg |
| contents | The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just $\mathcal{O}(ν\logν)$ operations, where $ν$ is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals Iserles, Arieh Maierhofer, Georg Numerical Analysis The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just $\mathcal{O}(ν\logν)$ operations, where $ν$ is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples. |
| title | An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2404.11448 |