# Effects of round-to-nearest and stochastic rounding in the numerical solution of the heat equation in low precision

Croci, M
Giles, M

30 October 2020

## Last Updated:

2021-11-12T03:27:24.957+00:00

## abstract:

Motivated by the advent of machine learning, the last few years saw the
return of hardware-supported low-precision computing. Computations with fewer
digits are faster and more memory and energy efficient, but can be extremely
susceptible to rounding errors. An application that can largely benefit from
the advantages of low-precision computing is the numerical solution of partial
differential equations (PDEs), but a careful implementation and rounding error
analysis are required to ensure that sensible results can still be obtained.
In this paper we study the accumulation of rounding errors in the solution of
the heat equation, a proxy for parabolic PDEs, via Runge-Kutta finite
difference methods using round-to-nearest (RtN) and stochastic rounding (SR).
We demonstrate how to implement the scheme to reduce rounding errors and we
derive \emph{a priori} estimates for local and global rounding errors. Let $u$
be the roundoff unit. While the worst-case local errors are $O(u)$ with respect
to the discretization parameters, the RtN and SR error behavior is
substantially different. We prove that the RtN solution is discretization,
initial condition and precision dependent, and always stagnates for small
enough $\Delta t$. Until stagnation, the global error grows like $O(u\Delta t^{-1})$. In contrast, we show that the leading order errors introduced by SR
are zero-mean, independent in space and mean-independent in time, making SR
resilient to stagnation and rounding error accumulation. In fact, we prove that
for SR the global rounding errors are only $O(u\Delta t^{-1/4})$ in 1D and are
essentially bounded (up to logarithmic factors) in higher dimensions.

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