  Historically, analysis for multiscale PDEs is largely unified while numerical
schemes tend to be equation-specific. In this paper, we propose a unified
framework for computing multiscale problems through random sampling. This is
achieved by incorporating randomized SVD solvers and manifold learning
techniques to numerically reconstruct the low-rank features of multiscale PDEs.
We use multiscale radiative transfer equation and elliptic equation with rough
media to showcase the application of this framework.
