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?
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 (1999)
Higher-Dimensional Integration with Gaussian Weight for Applications in Probabilistic Design (2004)
Sparse Quadrature for High-Dimensional Integration with Gaussian Measure (2018)
Cubature formulae for the gaussian weight: Some old and new rules (2020)
