Rounding error using low precision approximate random variables

TL;DR

This paper investigates the impact of using low precision approximate random variables in stochastic differential equation simulations, providing error bounds and demonstrating significant potential speedups, especially with single and half precision.

Contribution

It introduces a model for rounding errors in low precision random variables and quantifies potential computational speedups in multilevel Monte Carlo methods.

Findings

Single precision can speed up computations by a factor of 7.

Half precision offers similar speedups at coarse levels, up to 10-12 times.

Error bounds justify the use of low precision in stochastic simulations.

Abstract

For numerical approximations to stochastic differential equations using the Euler-Maruyama scheme, we propose incorporating approximate random variables computed using low precisions, such as single and half precision. We propose and justify a model for the rounding error incurred, and produce an average case error bound for two and four way differences, appropriate for regular and nested multilevel Monte Carlo estimations. By considering the variance structure of multilevel Monte Carlo correction terms in various precisions with and without a Kahan compensated summation, we compute the potential speed ups offered from the various precisions. We find single precision offers the potential for approximate speed improvements by a factor of 7 across a wide span of discretisation levels. Half precision offers comparable improvements for several levels of coarse simulations, and even offers improvements by a factor of 10-12 for the very coarsest few levels.