Bastien Vieublé | Searcharxiv

arXiv2026

Mixed-Precision Computing for Scientific Discovery: Formats, Co-Design, and Responsible Approximation

Reduced and mixed precision have moved from a niche optimization to a central design axis in scientific computing and engineering, driven by energy constraints, heterogeneous accelerators, and the convergence of simulation and machine learning. This paper organizes the landscape around seven coupled themes---number formats, floating-point emulation, emerging architectures, hardware/software co-design, relation to other approximations, software design, and precision as a multilevel resource ---and, for each theme, synthesizes the state of the art, future directions, and open questions. We emphasize \emph{energy per trusted solution} as the core objective, and we frame \say{recklessly responsible} computing as a pragmatic doctrine: exploit low precision aggressively, but with systematic detection, escalation, and certification pathways.

math.NA↗

arXiv2025

Mixed Precision General Alternating-Direction Implicit Method for Solving Large Sparse Linear Systems

In this article, we introduce a three-precision formulation of the General Alternating-Direction Implicit method (GADI) designed to accelerate the solution of large-scale sparse linear systems $Ax=b$. GADI is a framework that can represent many existing Alternating-Direction Implicit (ADI) methods. These methods are a class of linear solvers based on a splitting of $A$ such that the solution of the original linear system can be decomposed into the successive computation of easy-to-solve structured subsystems. Our proposed mixed precision scheme for GADI solves these subsystems in low precision to reduce the overall execution time while computing the residual and solution update in high precision to enable the solution to converge to high accuracy. We develop a rounding error analysis of mixed precision GADI that establishes the rates of convergence of the forward and backward errors to certain limiting accuracies. Our analysis also highlights the conditions on the splitting matrices under which mixed precision GADI is guaranteed to converge for a given set of precisions. We then discuss a systematic and robust strategy for selecting the GADI regularization parameter $α$, whose adjustment is critical for performance. Specifically, our proposed strategy makes use of a Gaussian Process Regression (GPR) model trained on a dataset of low-dimensional problems to initialize $α$. Finally, we proceed to a performance analysis of mixed precision GADI on an NVIDIA A100 GPU to validate our approach. Using low precision (Bfloat16 or FP32) to solve the subsystems, we obtain speedups of $2.6\times$, $1.7\times$, and $3.1\times$ over a full double precision GADI implementation on large-scale 2D, 3D convection-diffusion and complex reaction-diffusion problems (up to $1.3\times 10^{8}$ unknowns), respectively.

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