  We propose, analyze, and test a penalty projection-based efficient and
accurate algorithm for the Uncertainty Quantification (UQ) of the
time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated
regimes. The algorithm uses the Els\"asser variables formulation and discrete
Hodge decomposition to decouple the stochastic MHD system into four
sub-problems (at each time-step for each realization) which are much easier to
solve than solving the coupled saddle point problems. Each of the sub-problems
is designed in a sophisticated way so that at each time-step the system matrix
remains the same for all the realizations but with different right-hand-side
vectors which allows saving a huge amount of computer memory and computational
time. Moreover, the scheme is equipped with ensemble eddy-viscosity and
grad-div stabilization terms. The stability of the algorithm is proven
rigorously. We prove that the proposed scheme converges to an equivalent
non-projection-based coupled MHD scheme for large grad-div stabilization
parameter values. We examine how Stochastic Collocation Methods (SCMs) can be
combined with the proposed penalty projection UQ algorithm. Finally, a series
of numerical experiments are given which verify the predicted convergence
rates, show the algorithm's performance on benchmark channel flow over a
rectangular step, and a regularized lid-driven cavity problem with high random
Reynolds number and magnetic Reynolds number.
