# Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)

Title | Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs) |

Publication Type | IMATI_Reports |

Month | April |

Year of Publication | 2017 |

Authors | Ernst, O.G., Sprungk B., and Tamellini L. |

Series Title | IMATI Report Series |

nr. | 17-10 |

Pagination | 30 |

City | Pavia |

Institution | CNR-IMATI |

Keywords | best-N -term approximation, lognormal diffusion coefficient, parameteric PDEs, Random PDEs, Sparse grids, stochastic collocation |

Abstract | We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general L2-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate |

URL | http://bibliograzia.imati.cnr.it/reports/bibrep17-10 |

Citation Key | bibrep17-10 |

Access Date | April 13, 2017 |