Abstract
Vector spline techniques have been developed as general-purpose methods for vector field reconstruction. However, such vector splines involve high computational complexity, which precludes applications of this technique to many problems using large data sets. In this paper, we develop a fast multipole method for the rapid evaluation of the vector spline in three dimensions. The algorithm depends on a tree-data structure and two hierarchical approximations: an upward multipole expansion approximation and a downward local Taylor series approximation. In comparison with the CPU time of direct calculation, which increases at a quadratic rate with the number of points, the presented fast algorithm achieves a higher speed in evaluation at a linear rate. The theoretical error bounds are derived to ensure that the fast method works well with a specific accuracy. Numerical simulations are performed in order to demonstrate the speed and the accuracy of the proposed fast method.
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References
Amodei, L., Benbourhim, M. N.: A vector spline approximation with application to meteorology. In: Curves and surfaces (Laurent, P. J., Le Mehaute, A., Schumaker, L. L. eds.), pp. 5–10. Boston: Academic Press 1991.
Suter, D.: Motion estimation and vector splines. In: Proc. CVPR’94, Seattle WA, pp. 939–942, IEEE, June 1994. Available from http://www.batman.eng.monash.edu.au/suter_publications/finalcvpr.ps.gz.
Chen, F., Suter, D.: Elastic spline models for human cardiac motion estimation. IEEE Nonrigid and Articulated Motion Workshop, pp. 120–127, Puerto Rico, June 1997.
Amodei, L., Benbourhim, M. N.: A vector spline approximation. J. Approx. Theory67, 51–79 (1991).
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys.73, 325–348 (1987).
Carrier, J., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput.9, 669–686 (1988).
Graghicescu, C. I.: An efficient implementation of particle methods for the incompressible Euler equations. SIAM J. Numer. Anal.31, 1090–1108 (1994).
Ding, H., Karasawa, N., Goddard, W. A.: Atomic level simulations on a million particles: The cell multipole method for Coulomb and London nonbond interactions. J. Chem. Phys.97, 4309–4315 (1992).
McKenney, A.: An adaptation of the fast multipole method for evaluating layer potentials in two dimensions. Comput. Math. Appl.31, 33–57 (1996).
Christiansen, D., Perram, J. W., Petersen, H. G.: On the fast multipole method for computing the energy of periodic assemblies of charged and dipolar particles. J. Comput. Phys.107, 403–405 (1993).
Sølvason, D., Kolafa, J., Petersen, H. G., Perram, J. W.: A rigorous comparison of the Ewald method and the fast multipole method in two dimensions. Comput. Phys. Comm.87, 307–318 (1995).
Greengard, L., Rokhlin, V.: The rapid evaluation of potential fields in three dimensions. In: Vortex methods (Anderson, C., Greengards, C., eds.), pp. 121–141. Berlin Heidelberg New York Tokyo: Springer 1987.
Zhao, F.: An o(N) algorithm for three-dimensional N-body simulations. Technical Report 995, MIT Artificial Intelligence Laboratory, 1987.
Lustig, S. R.: Telescoping fast multipole methods using Chebyshev economization. J. Comp. Phys.122, 317–322 (1995).
Schmidt, K. E., Lee, M. A.: Implementing the fast multipole method in three dimensions. J. Stat. Phys.63, 1223–1235 (1991).
Zhao, F., Johnsson, S. L.: The parallel multipole method on the connection machine. SIAM J. Sci. Stat. Comput.12, 1420–1437 (1991).
Anderson, C. R.: An implementation of the fast multipole method without multipoles. SIAM J. Sci. Stat. Comput.13, 923–947 (1992).
Elliott, W. D., Board, J. A. Jr,: Fast Fourier transform accelerated fast multipole algorithm. SIAM J. Sci. Comput.17, 398–415 (1996).
Petersen, H. G., Sølvason, D., Perram, J. W., Smith, E. R.: Error estimates for the fast multipole method. I. the two-dimensional case. Proc. R. Soc. London Ser. A448, 389–400 (1995).
Petersen, H. G., Smith, E. R., Sølvason, D.: Error estimates for the fast multipole method. II. the three-dimensional case. Proc. London Ser. A448, 401–418 (1995).
Boyd, J. P.: Multipole expansions and pseudospectral cardinal functions: A new generalization of the fast Fourier transform. J. Comput. Phys.103, 184–186 (1992).
Brandt, A.: Multilevel computations of integral transforms and particle interactions with oscillatory kernels. Comput. Phys. Commun.65, 24–38 (1991).
Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math.44, 141–183 (1991).
Beatson, R. K., Newsam, G. N.: Fast evaluation of radial basis functions: 1. Comp. Math. Appl.24, 7–20 (1992).
Suter, D.: Fast evaluation of splines using Poisson formula. J. Appl. Sci. Comput.1, 70–87 (1994).
Beatson, R. K., Light, W. A.: Fast evaluation of radial basis functions. Methods for 2-dimensional polyharmonic splines. IMA J. Numer. Anal.17, 343–372 (1997).
Chen, F., Suter, D.: Fast evaluation of vector splines in two dimensions. The International Association for Mathematics and Computers in Simulation — IMACS1 469–474 (1997).
Macrobert, T. M.: Spherical harmonics. New York: Pergamon Press 1967.
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Chen, F., Suter, D. Fast evaluation of vector splines in three dimensions. Computing 61, 189–213 (1998). https://doi.org/10.1007/BF02684350
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DOI: https://doi.org/10.1007/BF02684350