  We present and analyze a novel wavelet-Fourier technique for the numerical
treatment of multidimensional advection-diffusion-reaction equations based on
the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin
technique with the compressed sensing approach, the proposed method is able to
approximate the largest coefficients of the solution with respect to a
biorthogonal wavelet basis. Namely, we assemble a compressed discretization
based on randomized subsampling of the Fourier test space and we employ sparse
recovery techniques to approximate the solution to the PDE. In this paper, we
provide the first rigorous recovery error bounds and effective recipes for the
implementation of the CORSING technique in the multi-dimensional setting. Our
theoretical analysis relies on new estimates for the local a-coherence, which
measures interferences between wavelet and Fourier basis functions with respect
to the metric induced by the PDE operator. The stability and robustness of the
proposed scheme is shown by numerical illustrations in the one-, two-, and
three-dimensional case.
