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A kernel-independent uniform fast multipole method based on barycentric rational interpolation

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Abstract

A kernel-independent uniform fast multipole method (UFMM) is presented for fast summation of particle interactions, where the kernel is approximated by using the Floater-Hormann (FH) rational interpolant at the equispaced grids. The proposed UFMM is stable and allows for reducing the cost of the moment-to-local translation (M2L) operators dramatically accelerated by fast Fourier transform (FFT). Moreover, the accuracy can be improved as the number of nodes increases. In addition, a modified smooth-UFMM for some sufficiently smooth kernels is considered, which has better performance than the originally smooth-UFMM. The efficiency and accuracy are illustrated by numerical examples arising from the method of the regularized Stokeslets (MRS) and inverse quadratic kernels.

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References

  1. Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987). https://doi.org/10.1016/0021-9991(87)90140-9

    Article  MathSciNet  MATH  Google Scholar 

  2. Hrycak, T., Rokhlin, V.: An improved fast multipole algorithm for potential fields. SIAM J. Sci. Comput. 19(6), 1804–1826 (1998). https://doi.org/10.1137/S106482759630989X

    Article  MathSciNet  MATH  Google Scholar 

  3. Yarvin, N., Rokhlin, V.: An improved fast multipole algorithm for potential fields on the line. SIAM J. Numer. Anal. 36(2), 629–666 (1999). https://doi.org/10.1137/S0036142997329232

    Article  MathSciNet  MATH  Google Scholar 

  4. Cecka, C., Darve, E.: Fourier-based fast multipole method for the Helmholtz equation. SIAM J. Sci. Comput. 35(1), A79–A103 (2013). https://doi.org/10.1137/11085774X

    Article  MathSciNet  MATH  Google Scholar 

  5. Engquist, B., Ying, L.: Fast directional multilevel algorithms for oscillatory kernels. SIAM J. Sci. Comput. 29(4), 1710–1737 (2007). https://doi.org/10.1137/07068583X

    Article  MathSciNet  MATH  Google Scholar 

  6. Darve, E.: The fast multipole method: Numerical implementation. J. Comput. Phys. 160(1), 195–240 (2000). https://doi.org/10.1006/jcph.2000.6451

    Article  MathSciNet  MATH  Google Scholar 

  7. Rokhlin, V.: Rapid solution of integral equations of scattering theory in two dimensions. J. Comput. Phys. 86(2), 414–439 (1990). https://doi.org/10.1016/0021-9991(90)90107-C

    Article  MathSciNet  MATH  Google Scholar 

  8. Rokhlin, V.: Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl. Comput. Harmon. Anal. 1(1), 82–93 (1993). https://doi.org/10.1006/acha.1993.1006

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu, Y., Rodin, G.: Fast solution method for three-dimensional Stokesian many-particle problems. Commun. Numer. Methods Eng. 16, 145–149 (2000). https://doi.org/10.1002/(SICI)1099-0887(200002)16:2<145::AID-CNM323>3.0.CO;2-E

    Article  MATH  Google Scholar 

  10. Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196 (2), 591–626 (2004). https://doi.org/10.1016/j.jcp.2003.11.021

    Article  MathSciNet  MATH  Google Scholar 

  11. Martinsson, P.G., Rokhlin, V.: An accelerated kernel-independent fast multipole method in one dimension. SIAM J. Sci. Comput. 29(3), 1160–1178 (2007). https://doi.org/10.1137/060662253

    Article  MathSciNet  MATH  Google Scholar 

  12. Fong, W., Darve, E.: The black-box fast multipole method. J. Comput. Phys. 228(23), 8712–8725 (2009). https://doi.org/10.1016/j.jcp.2009.08.031

    Article  MathSciNet  MATH  Google Scholar 

  13. Blanchard, P., Coulaud, O., Darve, E.: Fast hierarchical algorithms for generating Gaussian random fields. Research Report 8811 Inria Bordeaux Sud-Ouest (2015)

  14. Platte, R.B., Trefethen, L.N., Kuijlaars, A.B.J.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev. 53(2), 308–318 (2011). https://doi.org/10.1137/090774707

    Article  MathSciNet  MATH  Google Scholar 

  15. Cortez, R.: The method of regularized Stokeslets. SIAM J. Sci. Comput. 23(4), 1204–1225 (2001). https://doi.org/10.1137/S106482750038146X

    Article  MathSciNet  MATH  Google Scholar 

  16. Berrut, J.P., Klein, G.: Recent advances in linear barycentric rational interpolation. J. Comput. Appl. Math. 259(Part A), 95–107 (2014). https://doi.org/10.1016/j.cam.2013.03.044

    Article  MathSciNet  MATH  Google Scholar 

  17. Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles high rates of approximation. Numer. Math. 107(2), 315–331 (2007). https://doi.org/10.1007/s00211-007-0093-y

    Article  MathSciNet  MATH  Google Scholar 

  18. Bos, L., De Marchi, S., Hormann, K., Klein, G.: On the Lebesgue constant of barycentric rational interpolation at equidistant nodes. Numer. Math. 121(3), 461–471 (2012). https://doi.org/10.1007/s00211-011-0442-8

    Article  MathSciNet  MATH  Google Scholar 

  19. Klein, G.: Applications of linear barycentric rational interpolation. Phd thesis, University of Fribourg, Fribourg, Switzerland (2012)

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Acknowledgements

The authors are grateful for the referees’ helpful suggestions and insightful comments, which helped to improve the manuscript significantly.

Funding

This work was supported by the National Natural Science Foundation of China (No. 12271528). The first author is partly supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts029).

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Jiangli Liang & Shuhuang Xiang

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  1. Jiangli Liang
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  2. Shuhuang Xiang
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Correspondence to Shuhuang Xiang.

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Author contribution

Jiangli Liang: modeling, code development, conceptualization, literature survey. Shuhuang Xiang: conceptualization, analysis and technical writing.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Liang, J., Xiang, S. A kernel-independent uniform fast multipole method based on barycentric rational interpolation. Numer Algor 93, 1595–1611 (2023). https://doi.org/10.1007/s11075-022-01481-x

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