Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation

Peter J. Prins; Sander Wahls

doi:10.1109/ICASSP.2018.8461708

Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation

Team Sander Wahls

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

10 Citations (Scopus)

130 Downloads (Pure)

Abstract

Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.

Original language	English
Title of host publication	Proceedings 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
Place of Publication	Piscataway, NJ, USA
Publisher	IEEE
Pages	4524-4528
ISBN (Electronic)	978-1-5386-4658-8, 978-1-5386-4657-1
ISBN (Print)	978-1-5386-4659-5
DOIs	https://doi.org/10.1109/ICASSP.2018.8461708
Publication status	Published - 2018
Event	2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018: Signal Processing and Artificial Intelligence: Changing the World - Calgary Telus Convention Center (CTCC), Calgary, Canada Duration: 15 Apr 2018 → 20 Apr 2018 https://2018.ieeeicassp.org

Conference

Conference	2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018
Country/Territory	Canada
City	Calgary
Period	15/04/18 → 20/04/18
Internet address	https://2018.ieeeicassp.org

Bibliographical note

Accepted Author Manuscript

Keywords

Fourier transforms
Differential equations
Signal processing algorithms
Boundary conditions
Scattering
Computational complexity
Europe
Non-linear Fourier transform
exponential splittings
Korteweg-de Vries equation

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10.1109/ICASSP.2018.8461708

Higher order exponential splittings for the fast non-linear Fourier transform of the Korteweg-de Vries equationAccepted author manuscript, 492 KBLicence: Other

Cite this

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title = "Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation",

abstract = "Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.",

keywords = "Fourier transforms, Differential equations, Signal processing algorithms, Boundary conditions, Scattering, Computational complexity, Europe, Non-linear Fourier transform, exponential splittings, Korteweg-de Vries equation",

author = "Prins, {Peter J.} and Sander Wahls",

note = "Accepted Author Manuscript; 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018 : Signal Processing and Artificial Intelligence: Changing the World ; Conference date: 15-04-2018 Through 20-04-2018",

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doi = "10.1109/ICASSP.2018.8461708",

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booktitle = "Proceedings 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)",

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}

Prins, PJ & Wahls, S 2018, Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation. in Proceedings 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, Piscataway, NJ, USA, pp. 4524-4528, 2018 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2018, Calgary, Canada, 15/04/18. https://doi.org/10.1109/ICASSP.2018.8461708

Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation. / Prins, Peter J.; Wahls, Sander.
Proceedings 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Piscataway, NJ, USA: IEEE, 2018. p. 4524-4528.

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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AU - Wahls, Sander

N1 - Accepted Author Manuscript

PY - 2018

Y1 - 2018

N2 - Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.

AB - Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.

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KW - Differential equations

KW - Signal processing algorithms

KW - Boundary conditions

KW - Scattering

KW - Computational complexity

KW - Europe

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KW - exponential splittings

KW - Korteweg-de Vries equation

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BT - Proceedings 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Y2 - 15 April 2018 through 20 April 2018

ER -

Higher order exponential splittings for the fast Non-linear Fourier Transform of the Korteweg-De Vries equation

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Keywords

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