Let the dimension n=200 be fixed. The problem I am interested in is sampling points in n-dimensional Euclidean space uniformly from the region $$ \sum_{i=1}^{n} x_{i}\leq 1, $$ where $0\leq x_{i}\leq 1$ for all $1\leq i\leq n$.
One naive approach is to sample n points uniformly from the unit cube and then reject the sample if the sum is greater than 1. But this is a very inefficient approach. By simple MonteCarlo simulations I am observing that the Probability of the event $\sum_{i=1}^{n} x_{i}\leq 1$ is less than $10^{-6}$.
So is there any efficient way to do this sampling?