Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

Quentin Carayol; Francis Collino

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 2, page 371-394
  • ISSN: 0764-583X

Abstract

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We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave e i s ^ · v → in terms of spherical harmonics { Y ℓ , m ( s ^ ) } | m | ≤ ℓ ≤ ∞ . We consider the truncated series where the summation is performed over the ( ℓ , m ) 's satisfying | m | ≤ ℓ ≤ L . We prove that if v = | v → | is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies L + 1 2 ≃ v + C W 2 3 ( K ϵ - δ v γ ) v 1 3 where W is the Lambert function and C , K , δ , γ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.

How to cite

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Carayol, Quentin, and Collino, Francis. "Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series." ESAIM: Mathematical Modelling and Numerical Analysis 38.2 (2010): 371-394. <http://eudml.org/doc/194219>.

@article{Carayol2010,
abstract = { We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave $\{\rm e\}^\{i \hat\{s\} \cdot \vec\{v\}\}$ in terms of spherical harmonics $\\{ Y_\{\ell, m\}(\hat\{s\}) \\}_\{|m|\le \ell\le \infty\} $. We consider the truncated series where the summation is performed over the $(\ell,m)$'s satisfying $|m| \le \ell \le L$. We prove that if $v = |\vec\{v\}|$ is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies $L+\frac\{1\}\{2\} \simeq v + C W^\{\frac\{2\}\{3\}\}(K \epsilon^\{-\delta\} v^\gamma )\, v^\{\frac\{1\}\{3\}\}$ where W is the Lambert function and $C\,, K, \, \delta, \, \gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation. },
author = {Carayol, Quentin, Collino, Francis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Jacobi-Anger; fast multipole method; truncation error.; truncation error},
language = {eng},
month = {3},
number = {2},
pages = {371-394},
publisher = {EDP Sciences},
title = {Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series},
url = {http://eudml.org/doc/194219},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Carayol, Quentin
AU - Collino, Francis
TI - Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 2
SP - 371
EP - 394
AB - We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave ${\rm e}^{i \hat{s} \cdot \vec{v}}$ in terms of spherical harmonics $\{ Y_{\ell, m}(\hat{s}) \}_{|m|\le \ell\le \infty} $. We consider the truncated series where the summation is performed over the $(\ell,m)$'s satisfying $|m| \le \ell \le L$. We prove that if $v = |\vec{v}|$ is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies $L+\frac{1}{2} \simeq v + C W^{\frac{2}{3}}(K \epsilon^{-\delta} v^\gamma )\, v^{\frac{1}{3}}$ where W is the Lambert function and $C\,, K, \, \delta, \, \gamma$ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.
LA - eng
KW - Jacobi-Anger; fast multipole method; truncation error.; truncation error
UR - http://eudml.org/doc/194219
ER -

References

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