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Hermite Spectral Method with Hyperbolic Cross Approximations to High-dimensional Parabolic PDEs

Xue Luo Beihang University Stephen S.-T. Yau Tsinghua University

Numerical Analysis and Scientific Computing mathscidoc:1701.25004

SIAM J. NUMER. ANAL., 51, (6), 3186-3212, 2013
It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the "curse of dimensionality" in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized hyperbolic cross approximations has been shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic PDEs with HC approximations has been investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.
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@inproceedings{xue2013hermite,
  title={Hermite Spectral Method with Hyperbolic Cross Approximations to High-dimensional Parabolic PDEs},
  author={Xue Luo, and Stephen S.-T. Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170129110500559603132},
  booktitle={SIAM J. NUMER. ANAL.},
  volume={51},
  number={6},
  pages={3186-3212},
  year={2013},
}
Xue Luo, and Stephen S.-T. Yau. Hermite Spectral Method with Hyperbolic Cross Approximations to High-dimensional Parabolic PDEs. 2013. Vol. 51. In SIAM J. NUMER. ANAL.. pp.3186-3212. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170129110500559603132.
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