  In this paper, we demonstrate the construction of generalized Rough
Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for
multiscale PDEs with rough coefficients. The optimal coarse basis can be
derived automatically by the randomization of the original PDEs with a proper
prior distribution and the conditional expectation given partial information on
edge or derivative measurements. We prove the (quasi)-optimal localization and
approximation properties of the obtained bases, and justify the theoretical
results with numerical experiments.
