I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to sample $\vec{x}$ uniformly. $A$'s dimension is about $100 \times 10000$

I am well aware that rejection-sampling and MCMC sampling can theoretically solve this problem. However, I have already implemented both approaches in programming, and neither of these two methods performs well enough. This is because the dimension of my sampling space usually goes up to 10000, and rejection sampling simply throws away too many points and MCMC is taking forever to converge. Therefore, I'm desperate to try new methods. Many thanks in advance!! (please do not provide answers using rejection sampling; methods that already have open-source programming implementations are favored)