I am seeking advice on methods for uniformly sampling from the space of increasing vectors within the interval $[0, 1]$. Specifically, I require an efficient algorithm that can handle high-dimensional vectors (e.g., dimensionality around 100), where each vector component satisfies $0 \leq x_1 \leq x_2 \leq \ldots \leq x_{100} \leq 1$.
I have attempted to use the hit-and-run sampling technique, applying linear constraints to ensure the monotonicity of the vector components. However, I've observed that this approach tends to converge very slowly in high dimensions and stay close to the initial guess
Does anyone have suggestions for alternative methods for uniform sampling? Thanks!