We develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The sparse Fast Fourier Transform (sFFT) detects the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. Our uniform sFFT (usFFT) does this w.r.t. the stochastic domain simultaneously for every node of a finite element mesh in the spatial domain and creates a suitable approximation space for all spatial nodes by joining the detected frequency sets. This strategy allows for a faster and more efficient computation, than just using other algorithms, like for example the sFFT, for each node separately. We then test the usFFT for different, high-dimensional examples using affine and lognormal random coefficients in the PDE problems. The results are significantly better than when using given standard frequency sets and the algorithm does not require any critical a priori information about the solution.