  We propose a compressive spectral collocation method for the numerical
approximation of Partial Differential Equations (PDEs). The approach is based
on a spectral Sturm-Liouville approximation of the solution and on the
collocation of the PDE in strong form at randomized points, by taking advantage
of the compressive sensing principle. The proposed approach makes use of a
number of collocation points substantially less than the number of basis
functions when the solution to recover is sparse or compressible. Focusing on
the case of the diffusion equation, we prove that, under suitable assumptions
on the diffusion coefficient, the matrix associated with the compressive
spectral collocation approach satisfies the restricted isometry property of
compressive sensing with high probability. Moreover, we demonstrate the ability
of the proposed method to reduce the computational cost associated with the
corresponding full spectral collocation approach while preserving good accuracy
through numerical illustrations.
