I am searching for methods which would allow me to simulate a random vector from its corresponding multivariate characteristic function. I assume that I do not have any additional information about the random vector except for an analytical expression of its characteristic function. I am not interested in the univariate case.
Of course, I am aware of the theoretical inversion formulas for multivariate characteristic functions. However, it seems to me that, already in the bivariate case, the inversion is numerically challenging (correct me if I am wrong). Moreover, if the target distribution has atoms, it seems even more challenging, if not impossible, to apply the inversion method in practice (again, correct me if I am wrong).
Is there a method to simulate a multivariate characteristic function without numerical inversion of the characteristic function?