The Polynomial Chaos Expansion (PCE) methodology is widely used for uncertainty quantification of stochastic problems. The computational cost of PCE increases exponentially with the number of input uncertain variables (known as curse of dimensionality). Therefore, use of PCE for uncertainty quantification of industrial applications with large number of uncertain variables is challenging. In this paper, a novel methodology is presented for efficient uncertainty quantification of stochastic problems with large number of input random variables. The proposed method is based on PCE with combination of ℓ1-minimization and multifidelity methods. The developed method employs the ℓ1-minimization method to recover important coefficients of PCE using low-fidelity computations. The low-fidelity evaluations should be accurate enough to capture the physical trends well. After that the multifidelity PCE method is utilized to correct a subset of recovered coefficients using high-fidelity computations. A threshold parameter is defined in order to select the subset of recovered coefficients to be corrected. Two challenging analytical and CFD test cases namely, the Ackley function and the transonic RAE2822 airfoil with combined operational and geometrical uncertainties are considered to examine the performance of the methodology. It is shown that the proposed method can reproduce accurate results with much lower computational cost than the classical full Polynomial Chaos (PC), and ℓ1-minimization methods. It is observed that for the considered examples, the present method can achieve comparable accuracy with respect to the full PC and the ℓ1-minimization methods with significantly lower number of samples.