Multigrid Solver With Super-Resolved Interpolation

TL;DR

This paper introduces a novel multigrid solver that integrates a super-resolution GAN as the interpolation operator, enhancing convergence speed for solving elliptic PDEs in physics and engineering.

Contribution

The paper presents the first integration of a super-resolution GAN with multigrid algorithms, improving their efficiency over traditional cubic spline interpolation.

Findings

GAN-based interpolation accelerates convergence

Improved accuracy in multiscale PDE solutions

Enhanced performance over standard methods

Abstract

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.