I have a question about the uncertainty propagation of constrained random variables and I hope you can kindly advise me.

To be specific, all the input random variables \boldsymbol{x} are uniformly distributed in [x_{i,min},x_{i,max}]. I was trying to use PCE to propagate uncertainty. However, not all experimental designs in this hypercube space are feasible, which means those variables are constrained, see the picture at the bottom. In this case, are there any UQ techniques I can use to model the input-output relationship? Moreover, I want to obtain an explicit expression for this relationship.

Any help will be appreciated. Thank you very much.

Moreover, I am wondering if I can build a kriging model if I can obtain enough experimental designs. If so, how can I generate enough samples which will satisfy the constraints?

I think the easiest way to get samples following given constraints is to use rejection sampling.
However, I think you then need to be a bit careful when using PCE, as these selected samples may no longer follow the original uniform distribution you used to sample from the hypercube.

Thanks very much for your reply. Following your advice, I have quickly gone through the idea of rejection sampling. However, I still have a few questions, and I would be grateful if you could kindly provide further clarification.

As far as I understand, in rejection sampling, we first need to know the target distribution in advance. However, in my case, I only know that all the random variables follow uniform distributions, subject to certain constraints. Consequently, it seems that the only viable option is Monte Carlo sampling, followed by testing each sample to see if it satisfies the constraints. If the sample does satisfy the constraints, we can accept it; if not, we must reject it. I want to confirm if this is the method you mentioned. If so, could this still be classified as rejection sampling?

Moreover, as for the surrogate modeling, do you think it is appropriate to use some other techniques not requiring distributions (such as Gaussian Process or neural network)?