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EFFICIENT SPECTRAL SPARSE GRID METHODS AND APPLICATIONS TO HIGH-DIMENSIONAL ELLIPTIC PROBLEMS
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DOI：10.1137/100787842.png)
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Abstract
En 中文
We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyak's construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev-Legendre-Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.
Keywords:
sparse grid
spectral method
high-dimensional problem
Chebyshev-Gauss-Lobatto quadrature
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