Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function for which the integral $\int f(x)g(x)dx<\infty$.


Are there known "efficient" quadrature rules specifically for computing the integral $\int f(x)g(x)dx$ under these assumptions?



Quadrature methods for high-dimensional Gaussian integration
You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be


* [A stochastic
algorithm for high-dimensional integrals over unbounded regions with
Gaussian weight](https://core.ac.uk/download/pdf/81977086.pdf) (1999)
* [Higher-Dimensional
Integration with Gaussian Weight for Applications in Probabilistic
Design](https://epubs.siam.org/doi/abs/10.1137/S1064827503426863) (2004)
* [Sparse Quadrature for High-Dimensional Integration with Gaussian Measure](https://www.researchgate.net/publication/323524125_Sparse_Quadrature_for_High-Dimensional_Integration_with_Gaussian_Measure) (2018)
* [Cubature formulae for the gaussian weight: Some old and new rules](https://etna.ricam.oeaw.ac.at/vol.53.2020/pp426-438.dir/pp426-438.pdf) (2020)


