Advances in Mixed Precision Algorithms: 2021 Edition

Abdelfattah, Ahmad; Anzt, Hartwig; Ayala, Angel; Boman, Erik G.; Carson, Erin; Cayrols, Sebastien; Cojean, Terry; Dongarra, Jack; Falgout, Robert D.; Gates, Mark; Gruetzmacher, Thomas; Higham, Nicholas J.; Kruger, Scott; Li, X; Lindquist, Neil; Liu, Y; Loe, Jennifer; Łuszczek, Piotr; Nayak, P. Pandurang; Osei-Kuffuor, Daniel; Pranesh, Srikara; Rajamanickam, Siva; Ribizel, Tobias; Smith, Brian T.; Świrydowicz, Kasia; Thomas, S; Tomov, Stanimire; Tsai, Yuh-Show; Yamazaki, Ichitaro; Yang, Ulrike Meier

doi:10.2172/1814447

Cited by 3 publications

(6 citation statements)

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“…Until recently, mathematical software was usually implemented in one fixed precision. This paradigm is starting to change [1,6,7,15,21,22,27,28] as we currently witness an expanding use of both high-precision (for enhanced accuracy) and low-precision (for enhanced speed) arithmetic; sometimes, different precisions may even be used for different steps in the same algorithm.…”

Section: Numerical Algorithmmentioning

confidence: 99%

“…The straightforward implementations, which we denote by ĝ and ĥ respectively, provide backward stable algorithms. However, in IEEE Standard 754 double-precision floating-point arithmetic, computing ĝ ○ ĥ results in the floating-point representation fl(1/3) of 1 3 . Since fl(1/3) ≠ 1 3 = g(h(x)), there is no real input x ′ such that (g ○ h)(x ′ ) = fl(1/3).…”

Section: Introductionmentioning

confidence: 99%

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When can forward stable algorithms be composed stably?

Beltrán

Noferini

Vannieuwenhoven

2023

IMA Journal of Numerical Analysis

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We state some widely satisfied hypotheses, depending only on two functions $g$ and $h$, under which the composition of a forward stable algorithm for $g$ and a forward stable algorithm for $h$ is a forward stable algorithm for the composition $g \circ h$. We show that the failure of these conditions can potentially lead to unstable algorithms. Finally, we list a number of examples to illustrate the new concepts and the usability of the results.

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Section: Numerical Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

When can forward stable algorithms be composed stably?

Beltrán

Noferini

Vannieuwenhoven

2023

IMA Journal of Numerical Analysis

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“…To get the full sum for d (3) , one should add the partial sums from M (0,0) , M (0,1) , and M (1,1) .…”

Section: With These Two Techniques the Required Number Of Rows For C ...mentioning

confidence: 99%

“…As fig. 8 showed, even in fp16 precision, the intermediate results of the tensor core are computed with fp32 numbers [14,1]. Given that the accuracy of im2tensor (fp16) is about the same as the mixed (fp16fp32) versions of "naive" and im2tensor, we can deduce that the accuracy is further limited by the storage type rather than the type used for intermediate computations.…”

mentioning

confidence: 96%

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Computing large 2D convolutions on GPU efficiently with the im2tensor algorithm

Seznec

Gac

Orieux

et al. 2022

J Real-Time Image Proc

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Mathematical Tools for Simulation of 3D Bioprinting Processes on High-Performance Computing Resources: The State of the Art

Carracciuolo,

D’Amora

2024

Applied Sciences

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Three-dimensional (3D) bioprinting belongs to the wide family of additive manufacturing techniques and employs cell-laden biomaterials. In particular, these materials, named “bioink”, are based on cytocompatible hydrogel compositions. To be printable, a bioink must have certain characteristics before, during, and after the printing process. These characteristics include achievable structural resolution, shape fidelity, and cell survival. In previous centuries, scientists have created mathematical models to understand how physical systems function. Only recently, with the quick progress of computational capabilities, high-fidelity and high-efficiency “computational simulation” tools have been developed based on such models and used as a proxy for real-world learning. Computational science, or “in silico” experimentation, is the term for this novel strategy that supplements pure theory and experiment. Moreover, a certain level of complexity characterizes the architecture of contemporary powerful computational resources, known as high-performance computing (HPC) resources, also due to the great heterogeneity of its structure. Lately, scientists and engineers have begun to develop and use computational models more extensively to also better understand the bioprinting process, rather than solely relying on experimental research, due to the large number of possible combinations of geometrical parameters and material properties, as well as the abundance of available bioprinting methods. This requires a new effort in designing and implementing computational tools capable of efficiently and effectively exploiting the potential of new HPC computing systems available in the Exascale Era. The final goal of this work is to offer an overview of the models, methods, and techniques that can be used for “in silico” experimentation of the physicochemical processes underlying the process of 3D bioprinting of cell-laden materials thanks to the use of up-to-date HPC resources.

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Advances in Mixed Precision Algorithms: 2021 Edition

Cited by 3 publications

References 0 publications

When can forward stable algorithms be composed stably?

When can forward stable algorithms be composed stably?

Computing large 2D convolutions on GPU efficiently with the im2tensor algorithm

Mathematical Tools for Simulation of 3D Bioprinting Processes on High-Performance Computing Resources: The State of the Art

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