Multidimensional pseudo-spectral methods on lattice grids |
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Authors: | Hans Munthe-Kaas Tor Sørevik |
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Institution: | Dept. of Mathematics, University of Bergen, Johannes Brunsgt. 12, Bergen, Norway |
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Abstract: | When multidimensional functions are approximated by a truncated Fourier series, the number of terms typically increases exponentially with the dimension s. However, for functions with more structure than just being L2-integrable, the contributions from many of the Ns terms in the truncated Fourier series may be insignificant. In this paper we suggest a way to reduce the number of terms by omitting the insignificant ones. We then show how lattice rules can be used for approximating the associated Fourier coefficients, allowing a similar reduction in grid points as in expansion terms. We also show that using a lattice grid permits the efficient computation of the Fourier coefficients by the FFT algorithm. Finally we assemble these ideas into a pseudo-spectral algorithm and demonstrate its efficiency on the Poisson equation. |
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Keywords: | Multidimensional Fourier expansion Lattice rules Pseudo-spectral methods |
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