Most engineering problems contain a large number of input random variables, and thus their Polynomial Chaos Expansion (PCE) suffers from the curse of dimensionality. This issue can be tackled if the polynomial chaos representation is sparse. In the current study, the compressed sensing theory is employed to reconstruct the sparse representation of polynomial chaos expansion of challenging stochastic problems. The sparse recovery problem is solved using the Orthogonal Matching Pursuit (OMP). The Leave-One-Out (LOO) cross-validation is employed for the estimation of truncation error in the OMP procedure. In contrast to previous studies, which mainly focused on the random variables with uniform or Gaussian distributions, this paper applies the ℓ₁-minimization technique to arbitrarily distributed random variables. The orthogonal polynomials are constructed using the Gram–Schmidt orthogonalization method. Two challenging analytical test functions namely, Isighami and corner-peak, and three CFD problems namely, the two-dimensional heat diffusion problem with stochastic thermal diffusivity, the transonic RAE2822 airfoil with operational and geometrical uncertainties and the fully-developed turbulent channel flow with random turbulence model coefficients are considered to examine the performance of the methodology. The numerical results of the developed method are compared with the results of Monte Carlo (MC) simulation and regression-based polynomial chaos expansion. It is demonstrated that the ℓ₁-minimization method can be successfully applied to arbitrarily distributed uncertainties. Results show that the method can reproduce the sparse PCE with much lower computational time than the classical full Polynomial Chaos (PC) method. For the problems considered in the current study, for the same accuracy, the number of required samples for the sparse PC representation is significantly reduced.