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Math. Softw."],"published-print":{"date-parts":[[2021,6,30]]},"abstract":"<jats:p>The use of mixed precision in numerical algorithms is a promising strategy for accelerating scientific applications. In particular, the adoption of specialized hardware and data formats for low-precision arithmetic in high-end GPUs (graphics processing units) has motivated numerous efforts aiming at carefully reducing the working precision in order to speed up the computations. For algorithms whose performance is bound by the memory bandwidth, the idea of compressing its data before (and after) memory accesses has received considerable attention. One idea is to store an approximate operator\u2013like a preconditioner\u2013in lower than working precision hopefully without impacting the algorithm output. We realize the first high-performance implementation of an adaptive precision block-Jacobi preconditioner which selects the precision format used to store the preconditioner data on-the-fly, taking into account the numerical properties of the individual preconditioner blocks. We implement the adaptive block-Jacobi preconditioner as production-ready functionality in the Ginkgo linear algebra library, considering not only the precision formats that are part of the IEEE standard, but also customized formats which optimize the length of the exponent and significand to the characteristics of the preconditioner blocks. Experiments run on a state-of-the-art GPU accelerator show that our implementation offers attractive runtime savings.<\/jats:p>","DOI":"10.1145\/3441850","type":"journal-article","created":{"date-parts":[[2021,4,26]],"date-time":"2021-04-26T07:05:44Z","timestamp":1619420744000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":22,"title":["Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software"],"prefix":"10.1145","volume":"47","author":[{"given":"Goran","family":"Flegar","sequence":"first","affiliation":[{"name":"Departamento de Ingenier\u00eda y Ciencia de Computadores, Universidad Jaime I, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hartwig","family":"Anzt","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology, Germany and University of Tennessee, Knoxville, Tennessee, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Terry","family":"Cojean","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique S.","family":"Quintana-Ort\u00ed","sequence":"additional","affiliation":[{"name":"Departamento de Inform\u00e1tica de Sistemas y Computadores, Universitat Polit\u00e8cnica de Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2021,4,26]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"Ahmad Abdelfattah Hartwig Anzt Erik Boman Erin Carson Terry Cojean Jack Dongarra Mark Gates Thomas Gruetzmacher Nicholas J. Higham Sherry Li Neil Lindquist Yang Liu Jennifer Loe Piotr Luszczek Pratik Nayak Sri Pranesh Siva Rajamanickam Tobias Ribizel Barry Smith Kasia Swirydowicz Stephen Thomas Stanimire Tomov Yaohung Tsai Ichitaro Yamazaki and Urike Meier Yang. 2020. A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic. SLATE Working Notes 15 ICL-UT-20-08."},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/3324989.3325719"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.21105\/joss.02260"},{"key":"e_1_2_1_4_1","volume-title":"Yuhsiang Mike Tsai, and Enrique S. Quintana-Ort\u00ed","author":"Anzt Hartwig","year":"2020","unstructured":"Hartwig Anzt, Terry Cojean, Goran Flegar, Fritz G\u00f6bel, Thomas Gr\u00fctzmacher, Pratik Nayak, Tobias Ribizel, Yuhsiang Mike Tsai, and Enrique S. Quintana-Ort\u00ed. 2020b. 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