I have to implement a numerical algorithm that can calculate following integral as fast as possible: $$ I=\iint_{\mathbb{R}^2}dx \,dy\, e^{-\left(x^2+y^2\right)} f\left(x,y\right) $$ where the integrand $f(x,y)$ is quite expensive in terms of computation.
If it were 1-D case, I would have used Gauss-Hermite Quadrature, which gives a good enough approximation with $n\sim 10$. $$ \int_{\mathbb{R}}dx\,e^{-x^2} f(x)\simeq \sum_{m} w_m\, f\left(x_m\right) $$
I am inclined to believe that doing this twice would do the trick: $$ I\simeq \sum_{m}\sum_{n} w_m\,w_n\, f\left(x_m,y_n\right) $$ but I couldn't find any reference for it.
Questions.
- Is this the correct way? Is there any published materials I could look for on this question?
- Is there a better way to handle this?