In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions on the exactness and boundedness of univariate quadrature rules as well as on the regularity assumptions on the parametric functions with respect to the parameters, we prove that the convergence of the sparse quadrature error is independent of the number of the parameter dimensions. Moreover, we propose both an a priori and an a posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates.
Accepté le :
DOI : 10.1051/m2an/2018012
Mots clés : Uncertainty quantification, high-dimensional integration, curse of dimensionality, convergence analysis, Gaussian measure, sparse grids, a priori construction, a posteriori construction
@article{M2AN_2018__52_2_631_0,
author = {Chen, Peng},
title = {Sparse quadrature for high-dimensional integration with {Gaussian} measure},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {631--657},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018012},
mrnumber = {3834438},
zbl = {06966736},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018012/}
}
TY - JOUR AU - Chen, Peng TI - Sparse quadrature for high-dimensional integration with Gaussian measure JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 631 EP - 657 VL - 52 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018012/ DO - 10.1051/m2an/2018012 LA - en ID - M2AN_2018__52_2_631_0 ER -
%0 Journal Article %A Chen, Peng %T Sparse quadrature for high-dimensional integration with Gaussian measure %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 631-657 %V 52 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018012/ %R 10.1051/m2an/2018012 %G en %F M2AN_2018__52_2_631_0
Chen, Peng. Sparse quadrature for high-dimensional integration with Gaussian measure. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 631-657. doi : 10.1051/m2an/2018012. http://www.numdam.org/articles/10.1051/m2an/2018012/
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