Leveraging Mixed Precision in Exponential Time Integration Methods

TL;DR

This work addresses the challenge of leveraging mixed-precision hardware for exponential time integration by proposing two complementary approaches: (i) a reformulated exponential Rosenbrock--Euler method with a tunable parameter $\gamma_n$ that enhances tolerance to inexact $\varphi$-function evaluations, achieving an improved local error of $\|e_1\|_2 = O(h_0^3 + \epsilon h_0^2)$, and (ii) a mixed-precision Krylov method for phi-function products using an inexact/incomplete Arnoldi process with adaptive precision switching. The paper demonstrates, through both analysis and advection–diffusion–reaction PDE experiments, that the reformulated scheme preserves second-order accuracy under low-precision phi-functions, and that smart switching in the Krylov Arnoldi method yields accuracy comparable to double precision while reducing computational effort. Collectively, these contributions enable efficient, accurate exponential time integration on modern hardware. The practical impact is improved solver performance for stiff ODEs and PDE discretizations when using contemporary accelerators with limited precision, without sacrificing accuracy.

Abstract

The machine learning explosion has created a prominent trend in modern computer hardware towards low precision floating-point operations. In response, there have been growing efforts to use low and mixed precision in general scientific computing. One important area that has received limited exploration is time-integration methods, which are used for solving differential equations that are ubiquitous in science and engineering applications. In this work, we develop two new approaches for leveraging mixed precision in exponential time integration methods. The first approach is based on a reformulation of the exponential Rosenbrock--Euler method allowing for low precision computations in matrix exponentials independent of the particular algorithm for matrix exponentiation. The second approach is based on an inexact and incomplete Arnoldi procedure in Krylov approximation methods for computing matrix exponentials and is agnostic to the chosen integration method. We show that both approaches improve accuracy compared to using purely low precision and offer better efficiency than using only double precision when solving an advection-diffusion-reaction partial differential equation.

Figures (4)

• Figure 1: The reformulated exponential Rosenbrock--Euler method \ref{['eq:mixed_precision_exp_Euler']} maintains second order convergence despite using single precision for $\varphi$-functions, while the traditional form \ref{['eq:exp_Euler']} stagnates at the accuracy of the $\varphi$-function.
• Figure 2: With half precision $\varphi$-functions, the reformulated exponential Rosenbrock--Euler method \ref{['eq:mixed_precision_exp_Euler']} achieves a minimum error approximately ten times smaller than that of the traditional form \ref{['eq:exp_Euler']}.
• Figure 3: The reformulated exponential Rosenbrock--Euler (RERE) method \ref{['eq:mixed_precision_exp_Euler']} achieves a much lower error than the standard exponential Rosenbrock--Euler (ERE) when using KIOPS with only half precision matrix-vector products via Algorithm \ref{['alg:IOP_Arnoldi']}. Using mixed precision matrix-vector products in KIOPS via Algorithm \ref{['alg:IOP_Arnoldi_Smart']} significantly improves the error for all methods. exprk4s6 is unable to converge at all with Algorithm \ref{['alg:IOP_Arnoldi']} and only half precision matrix-vector products while using Algorithm \ref{['alg:IOP_Arnoldi_Smart']} enables it to run and obtain a reasonably accurate solution.
• Figure 4: In the case of the reformulated exponential Rosenbrock--Euler (RERE) and standard exponential Rosenbrock--Euler (ERE) methods, using the mixed precision IOP Arnoldi (Algorithm \ref{['alg:IOP_Arnoldi_Smart']}) in KIOPS reduced the number of effective matrix-vector products compared to the double precision version. With exprk4s6, the mixed precision KIOPS was more efficient until the error reached $~10^{-10}$.