| Issue |
ESAIM: M2AN
Volume 52, Number 2, March–April 2018
|
|
|---|---|---|
| Page(s) | 631 - 657 | |
| DOI | https://doi.org/10.1051/m2an/2018012 | |
| Published online | 11 July 2018 | |
Sparse quadrature for high-dimensional integration with Gaussian measure★
Institute for Computational Engineering & Sciences, The University of Texas at Austin,
Stop C0200,
Austin,
TX
78712 USA
* Corresponding author: e-mail: peng@ices.utexas.edu
Received:
20
June
2017
Accepted:
8
February
2018
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.
Mathematics Subject Classification: 65C20 / 65D30 / 65D32 / 65N12 / 65N15 / 65N21
Key words: Uncertainty quantification / high-dimensional integration / curse of dimensionality / convergence analysis / Gaussian measure / sparse grids / a priori construction / a posteriori construction
© EDP Sciences, SMAI 2018
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