Seminars & Events

Mathematics and Computer Science Division Seminar
"Chebyshev Spectral Methods on Lattices for High-Dimensional Functions"

DATE: March 31, 2008
TIME: 10:30 am
SPEAKER: Xiaoyan Zeng, Illinois Institute of Technology
LOCATION: Building 221, Conference Room A216, Argonne National Laboratory
HOST: Mihai Anitescu

Description:
It is well-known that the number of nodes required to obtain a given accuracy using product algorithms increases exponentially with increasing dimension. Lattice methods are widely used in multiple integration to avoid the curse of the dimensionality. Recently some algorithms for approximation of periodic functions using lattice points have been developed by Li & Hickernell (2003), Kuo & Sloan & Wozniakowski(2005) and Zeng & Leung & Hickernell(2005).

In this paper, we present a numerical method for the approximation of non-periodic functions. The designs, x_i, considered here, in contrast to the classic ones, are half of the node sets of integration lattices with a cosine transformation. It is supposed that $f$ can be expressed by an absolutely convergent multidimensional Chebyshev series expansion. We approximate its Chebyshev coefficients at some selected nonnegative wavenumbers by a lattice rule and set the other Chebyshev coefficients to be zero. The approximation of f is then taken to be the Chebyshev expansion based on these approximate Chebyshev coefficients. The error of the approximation is discussed in a weighted L^2 norm. The convergence rate is related to the rate of the true Chebyshev coefficients' decay. We also provide the numerical simulation result of the $8$ dimensional borehole function with experimental convergence rate as O(N^{-1}).

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