Multigrid Methods using Block Floating Point Arithmetic

TL;DR

This paper investigates the use of Block Floating Point arithmetic in multigrid methods for solving PDEs, demonstrating energy-efficient computations with minimal bit-widths while maintaining accuracy.

Contribution

It introduces algorithms for BFP arithmetic with maximal block sizes and presents a heuristic for multigrid methods that preserves accuracy without normalization overhead.

Findings

Some computations use as little as 4-bit integers.

Bit-width requirements are similar to standard floating point for target accuracy.

The proposed heuristic achieves discretization-error-accuracy efficiently.

Abstract

Block Floating Point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power, less chip area, and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy- and hardware-efficient integer arithmetic. While most existing applications of BFP arithmetic tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. This is sometimes also referred to as a scaled fixed-point format. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision. Using these algorithms, we study the asymptotic precision requirements to achieve discretization-error-accuracy. We demonstrate that some computations can be performed using as little as 4-bit integers, while the number of bits required to attain a certain target accuracy is similar to that of standard floating point arithmetic. Finally, we present a heuristic for full multigrid in BFP arithmetic based on saturation and truncation that still achieves discretization-error-accuracy without the need for expensive normalization steps of intermediate results.