How do I compute the following?

$$ \mathrm{Prob}\left( \sum_{i=1}^N x_i >1\right) = ?$$

where $x_i \sim \mathrm{Log}\mathcal{-N}(\mu_i, \sigma_i^2)$.

AFAIK, we do not know how the sum of log-normal distributions is distributed, so what I have in mind is to pick an appropriate interval for $x$, sample uniformly from there, and calculate sum of $x_i$s. If it's greater than 1, then I count it as 1. Am I doing it right? Is this the "Monte Carlo Method"? Are there any (more appropriate) other methods you can suggest? I wanted to get some experts' words to be sure. Thanks.