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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?

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You may want to use a stochastic algorithm. Entry points to the literature (which is large) could be

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