Sparse Quadrature for High-Dimensional Integration with Gaussian Measure
Provides theoretical guarantees for sparse quadrature in high-dimensional integration under Gaussian measure, benefiting computational statistics and uncertainty quantification.
The paper proves dimension-independent convergence rate O(N^{-s}) for sparse quadrature integration of high-dimensional functions with Gaussian measure, and proposes a-priori and a-posteriori construction schemes validated by numerical experiments.
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 of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate $O(N^{-s})$, where $N$ is the number of indices, and $s$ 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.