Multigrid with Linear Storage Complexity

TL;DR

This paper introduces a multigrid method that computes PDE solutions using linear storage complexity, significantly reducing memory requirements for large-scale problems on supercomputers.

Contribution

It presents a novel multigrid algorithm that compresses solution and intermediate data to achieve linear storage complexity, enabling matrix-free PDE solutions with reduced memory footprint.

Findings

Achieves 4-12 bits per DoF for solutions on fine grids.

Reduces memory footprint by about an order of magnitude compared to existing methods.

Demonstrates applicability on two model PDE problems.

Abstract

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.