I want to compute $ f(x) = \int\nolimits_{-\infty}^{\Lambda x} \phi_q(y; \mu, \Sigma)\ dy $ where $p \ll q$, $x \in \mathcal{R}^p$, $\Lambda \in \mathcal{R}^{q \times p}$, $\mu \in \mathcal{R}^q$, $\Sigma \in \mathcal{R}^{q \times q}$ is positive-definite, $\operatorname{rank}(\Lambda) = p$, and $\phi_q(y; \mu, \Sigma)$ is the pdf of the $q$-dimensional normal distribution with parameters $\mu, \Sigma$.
Is there a way to represent $f(x)$ in a way that allows it to be computed or approximated more efficiently than to compute a $q$-dimensional Gaussian CDF? How about if $\mu = 0$?
More generally, what are the sorts of tools used to attack problems like this?