Mixed precision and local error in ordinary differential equations
Résumé
In computational biology, many problems are modelled using individual-based or agent-based modelling. When expressed as ordinary differential equations (ODEs), these models lead to high-dimensional systems. The computational cost increases supra-linearly with the size of these systems. Accordingly, the solving of full-scale systems is intractable in terms of computational cost. Given that biological problems often have high empirical uncertainties, it could be possible to take advantage of the trade-off between computational speed and accuracy.
Mixed precision consists in using different levels of arithmetic precision within one computational task. Several fields already use this technique, such as geophysics [1] or machine learning [4]. Mixed precision methods have also attracted attention for problems involving differential equations [2]. Our study deals with a population of heterogeneous agents of size N, with N ≫ 1, where each agent is described by a state variable Xi ∈ R , which evolves according to an autonomous term (Fi, i = 1, ..., N) and a term accounting for complex pairwise interactions (Gij , i, j = 1, ..., N).
The evaluation of the N right-hand sides requires the sum of N nonlinear terms, leading to a O(N2) complexity. Reducing the precision used during the sum could accelerate the whole evaluation process by a considerable amount, as performed with iterative refinement solvers in [3]. To reduce the degradation of the global accuracy, mixing the precision inside the evaluations could
allow minimizing the impact of the numerical error due to the insertion of low precision [5]. We performed tests on two benchmarks, and our results show that as the size of the system increases,
the error introduced by low precision is absorbed by numerical compensation in high-dimensional systems. Moreover, the local error (in comparison with a double precision solver) measured by the
solver is more robust to low tolerances in the case of mixed precision than a full low precision solver.
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