  The multigrid algorithm is an efficient numerical method for solving a
variety of elliptic partial differential equations (PDEs). The method damps
errors at progressively finer grid scales, resulting in faster convergence
compared to standard iterative methods such as Gauss-Seidel. The prolongation,
or coarse-to-fine interpolation operator within the multigrid algorithm lends
itself to a data-driven treatment with ML super resolution, commonly used to
increase the resolution of images. We (i) propose the novel integration of a
super resolution generative adversarial network (GAN) model with the multigrid
algorithm as the prolongation operator and (ii) show that the GAN-interpolation
improves the convergence properties of the multigrid in comparison to cubic
spline interpolation on a class of multiscale PDEs typically solved in physics
and engineering simulations.
