Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

Authors:
Jie Shen;Haijun Yu
Affiliations:
shen7@purdue.edu;hyu@lsec.cc.ac.cn
Venue:
SIAM Journal on Scientific Computing
Year:
2010

Citing 10
Cited 3

The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points

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Efficient spectral-Galerkin method II.: direct solvers of second- and fourth-order equations using Chebyshev polynomials

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The solution of multidimensional real Helmholtz equations on Sparse Grids

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Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method

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Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation

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Superfast Multifrontal Method for Large Structured Linear Systems of Equations

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Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross

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Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II. Unbounded Domains

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Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions

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Array-representation integration factor method for high-dimensional systems

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Abstract

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.