Rapid Prolate Pseudospectral Differentiation and Interpolation with the Fast Multipole Method

Pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials. However, the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix-vector products, which, if evaluated directly, entail O(N2) memory requirement and computational complexity (where N is the number of unknowns utilized for discretization and interpolation). In this work we show that the fast multipole method (FMM) can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to O(N). Example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver.

[1]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[2]  Richard Baltensperger,et al.  Spectral Differencing with a Twist , 2002, SIAM J. Sci. Comput..

[3]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[4]  L. Trefethen,et al.  Stability of the method of lines , 1992, Spectra and Pseudospectra.

[5]  D. Slepian.  Some comments on Fourier analysis, uncertainty and modeling , 1983 .

[6]  J. Boyd.  Multipole expansions and pseudospectral cardinal functions: A new generalization of the fast fourier transform , 1992 .

[7]  L. Carin,et al.  Pseudospectral method based on prolate spheroidal wave functions for frequency-domain electromagnetic simulations , 2005, IEEE Transactions on Antennas and Propagation.

[8]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[9]  L. Trefethen.  Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations , 1996 .

[10]  V. Rokhlin,et al.  Prolate spheroidal wavefunctions, quadrature and interpolation , 2001 .

[11]  L. Trefethen,et al.  Stiffness of ODEs , 1993 .

[12]  Leslie Greengard,et al.  The numerical solution of the N-body problem , 1990 .

[13]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[14]  Lloyd N. Trefethen,et al.  Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues , 1990 .

[15]  Gilbert G. Walter,et al.  Prolate Spheroidal Wave Functions and Wavelets , 2006 .

[16]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[17]  V. Rokhlin,et al.  Fast algorithms for polynomial interpolation, integration, and differentiation , 1996 .

[18]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[19]  John P. Boyd,et al.  Algorithm 840: computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions---prolate elements , 2005, TOMS.

[20]  L. Trefethen,et al.  An Instability Phenomenon in Spectral Methods , 1987 .

[21]  G. Beylkin,et al.  Wave propagation using bases for bandlimited functions , 2005 .

[22]  J. Boyd.  Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms , 2004 .

[23]  L. Greengard.  The Rapid Evaluation of Potential Fields in Particle Systems , 1988 .

[24]  Jan S. Hesthaven,et al.  Spectral Methods Based on Prolate Spheroidal Wave Functions for Hyperbolic PDEs , 2005, SIAM J. Numer. Anal..

[25]  C. Bouwkamp.  On Spheroidal Wave Functions of Order Zero , 1947 .

[26]  Leslie Greengard,et al.  Spectral integration and two-point boundary value problems , 1991 .