Hierarchical approach for deriving a reproducible unblocked LU factorization

[EN] We propose a reproducible variant of the unblocked LU factorization for graphics processor units (GPUs). For this purpose, we build upon Level-1/2 BLAS kernels that deliver correctly-rounded and reproducible results for the dot (inner) product, vector scaling, and the matrix-vector product. In addition, we draw a strategy to enhance the accuracy of the triangular solve via iterative refinement. Following a bottom-up approach, we finally construct a reproducible unblocked implementation of the LU factorization for GPUs, which accommodates partial pivoting for stability and can be eventually integrated in a high performance and stable algorithm for the (blocked) LU factorization.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The simulations were performed on resources provided by the Swed-ish National Infrastructure for Computing (SNIC) at PDC Centre for High Performance Computing (PDC-HPC). This work was also granted access to the HPC resources of The Institute for Scientific Computing and Simulation financed by Region Ile-de-France and the project Equip@Meso (reference ANR-10-EQPX-29-01) overseen by the French National Agency for Research (ANR) as part of the Investissements d Avenir pro-gram. This work was also partly supported by the FastRelax (ANR-14-CE25-0018-01) project of ANR.Iakymchuk, R.; Graillat, S.; Defour, D.; Quintana-Orti, ES. (2019). Hierarchical approach for deriving a reproducible unblocked LU factorization. International Journal of High Performance Computing Applications. 33(5):791-803. https://doi.org/10.1177/1094342019832968S791803335Arteaga, A., Fuhrer, O., & Hoefler, T. (2014). Designing Bit-Reproducible Portable High-Performance Applications. 2014 IEEE 28th International Parallel and Distributed Processing Symposium. doi:10.1109/ipdps.2014.127Bientinesi, P., Quintana-Ortí, E. S., & Geijn, R. A. van de. (2005). Representing linear algebra algorithms in code: the FLAME application program interfaces. ACM Transactions on Mathematical Software, 31(1), 27-59. doi:10.1145/1055531.1055533Chohra, C., Langlois, P., & Parello, D. (2016). Efficiency of Reproducible Level 1 BLAS. Lecture Notes in Computer Science, 99-108. doi:10.1007/978-3-319-31769-4_8Collange, S., Defour, D., Graillat, S., & Iakymchuk, R. (2015). Numerical reproducibility for the parallel reduction on multi- and many-core architectures. Parallel Computing, 49, 83-97. doi:10.1016/j.parco.2015.09.001Demmel, J., & Hong Diep Nguyen. (2013). Fast Reproducible Floating-Point Summation. 2013 IEEE 21st Symposium on Computer Arithmetic. doi:10.1109/arith.2013.9Demmel, J., & Nguyen, H. D. (2015). Parallel Reproducible Summation. IEEE Transactions on Computers, 64(7), 2060-2070. doi:10.1109/tc.2014.2345391Dongarra, J. J., Du Croz, J., Hammarling, S., & Duff, I. S. (1990). A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1), 1-17. doi:10.1145/77626.79170Dongarra, J., Hittinger, J., Bell, J., Chacon, L., Falgout, R., Heroux, M., … Wild, S. (2014). Applied Mathematics Research for Exascale Computing. doi:10.2172/1149042Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR. ACM Transactions on Mathematical Software, 33(2), 13. doi:10.1145/1236463.1236468Haidar, A., Dong, T., Luszczek, P., Tomov, S., & Dongarra, J. (2015). Batched matrix computations on hardware accelerators based on GPUs. The International Journal of High Performance Computing Applications, 29(2), 193-208. doi:10.1177/1094342014567546Hida, Y., Li, X. S., & Bailey, D. H. (s. f.). Algorithms for quad-double precision floating point arithmetic. Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001. doi:10.1109/arith.2001.930115Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms. doi:10.1137/1.9780898718027Iakymchuk, R., Defour, D., Collange, S., & Graillat, S. (2015). Reproducible Triangular Solvers for High-Performance Computing. 2015 12th International Conference on Information Technology - New Generations. doi:10.1109/itng.2015.63Iakymchuk, R., Defour, D., Collange, S., & Graillat, S. (2016). Reproducible and Accurate Matrix Multiplication. Lecture Notes in Computer Science, 126-137. doi:10.1007/978-3-319-31769-4_11Kulisch, U., & Snyder, V. (2010). The exact dot product as basic tool for long interval arithmetic. Computing, 91(3), 307-313. doi:10.1007/s00607-010-0127-7Li, X. S., Demmel, J. W., Bailey, D. H., Henry, G., Hida, Y., Iskandar, J., … Yoo, D. J. (2002). Design, implementation and testing of extended and mixed precision BLAS. ACM Transactions on Mathematical Software, 28(2), 152-205. doi:10.1145/567806.567808Muller, J.-M., Brisebarre, N., de Dinechin, F., Jeannerod, C.-P., Lefèvre, V., Melquiond, G., … Torres, S. (2010). Handbook of Floating-Point Arithmetic. doi:10.1007/978-0-8176-4705-6Ogita, T., Rump, S. M., & Oishi, S. (2005). Accurate Sum and Dot Product. SIAM Journal on Scientific Computing, 26(6), 1955-1988. doi:10.1137/030601818Ortega, J. . (1988). The ijk forms of factorization methods I. Vector computers. Parallel Computing, 7(2), 135-147. doi:10.1016/0167-8191(88)90035-xRump, S. M. (2009). Ultimately Fast Accurate Summation. SIAM Journal on Scientific Computing, 31(5), 3466-3502. doi:10.1137/080738490Skeel, R. D. (1979). Scaling for Numerical Stability in Gaussian Elimination. Journal of the ACM, 26(3), 494-526. doi:10.1145/322139.322148Zhu, Y.-K., & Hayes, W. B. (2010). Algorithm 908. ACM Transactions on Mathematical Software, 37(3), 1-13. doi:10.1145/1824801.182481

Reproducibility strategies for parallel preconditioned Conjugate Gradient

[EN] The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29% (ExBLAS) and 4% (lightweight) on 768 processes.To begin with, we would like to thank the reviewers for their thorough reading of the article as well as their valuable comments and suggestions. This research was partially supported by the European Union's Horizon 2020 research, innovation programme under the Marie Sklodowska-Curie grant agreement via the Robust project No. 842528 as well as the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the H2020 EC RIA Programme; in particular, the author gratefully acknowledges the support of Vicenc Beltran and the computer resources and technical support provided by BSC. The researchers from Universitat Jaume I (UJI) and Universidad Politecnica de Valencia (UPV) were supported by MINECO, Spain project TIN2017-82972-R. Maria Barreda was also supported by the POSDOC-A/2017/11 project from the Universitat Jaume I, Spain.Iakymchuk, R.; Barreda, M.; Wiesenberger, M.; Aliaga, JI.; Quintana Ortí, ES. (2020). Reproducibility strategies for parallel preconditioned Conjugate Gradient. Journal of Computational and Applied Mathematics. 371:1-13. https://doi.org/10.1016/j.cam.2019.112697S113371Lawson, C. L., Hanson, R. J., Kincaid, D. R., & Krogh, F. T. (1979). Basic Linear Algebra Subprograms for Fortran Usage. ACM Transactions on Mathematical Software, 5(3), 308-323. doi:10.1145/355841.355847Dongarra, J. J., Du Croz, J., Hammarling, S., & Duff, I. S. (1990). A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1), 1-17. doi:10.1145/77626.79170Demmel, J., & Nguyen, H. D. (2015). Parallel Reproducible Summation. IEEE Transactions on Computers, 64(7), 2060-2070. doi:10.1109/tc.2014.2345391Iakymchuk, R., Graillat, S., Defour, D., & Quintana-Ortí, E. S. (2019). Hierarchical approach for deriving a reproducible unblocked LU factorization. The International Journal of High Performance Computing Applications, 33(5), 791-803. doi:10.1177/1094342019832968Iakymchuk, R., Defour, D., Collange, S., & Graillat, S. (2016). Reproducible and Accurate Matrix Multiplication. Lecture Notes in Computer Science, 126-137. doi:10.1007/978-3-319-31769-4_11Rump, S. M., Ogita, T., & Oishi, S. (2009). Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest. SIAM Journal on Scientific Computing, 31(2), 1269-1302. doi:10.1137/07068816xBurgess, N., Goodyer, C., Hinds, C. N., & Lutz, D. R. (2019). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. IEEE Transactions on Computers, 68(7), 967-978. doi:10.1109/tc.2018.2855729D. Mukunoki, T. Ogita, K. Ozaki, Accurate and reproducible BLAS routines with Ozaki scheme for many-core architectures, in: Proc. International Conference on Parallel Processing and Applied Mathematics, PPAM2019, 2019, accepted.Ogita, T., Rump, S. M., & Oishi, S. (2005). Accurate Sum and Dot Product. SIAM Journal on Scientific Computing, 26(6), 1955-1988. doi:10.1137/030601818Kulisch, U., & Snyder, V. (2010). The exact dot product as basic tool for long interval arithmetic. Computing, 91(3), 307-313. doi:10.1007/s00607-010-0127-7Boldo, S., & Melquiond, G. (2008). Emulation of a FMA and Correctly Rounded Sums: Proved Algorithms Using Rounding to Odd. IEEE Transactions on Computers, 57(4), 462-471. doi:10.1109/tc.2007.70819Wiesenberger, M., Einkemmer, L., Held, M., Gutierrez-Milla, A., Sáez, X., & Iakymchuk, R. (2019). Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures. Computer Physics Communications, 238, 145-156. doi:10.1016/j.cpc.2018.12.006Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR. ACM Transactions on Mathematical Software, 33(2), 13. doi:10.1145/1236463.1236468J. Demmel, H.D. Nguyen, Fast reproducible floating-point summation, in: Proceedings of ARITH-21, 2013, pp. 163–172.Ozaki, K., Ogita, T., Oishi, S., & Rump, S. M. (2011). Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications. Numerical Algorithms, 59(1), 95-118. doi:10.1007/s11075-011-9478-1Carson, E., & Higham, N. J. (2018). Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions. SIAM Journal on Scientific Computing, 40(2), A817-A847. doi:10.1137/17m114081

Reproducibility of parallel preconditioned conjugate gradient in hybrid programming environments

[EN] The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the other relies upon floating-point expansions and, hence, expands the intermediate precision. Instead of converting the entire solver into its ExBLAS-related implementation, we identify those parts that violate reproducibility/non-associativity, secure them, and combine this with the sequential executions. These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these approaches on two modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of less than 37.7% on 768 cores.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially supported by the European Union's Horizon 2020 research, innovation program under the Marie Sklodowska-Curie grant agreement via the Robust project No. 842528 as well as the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the H2020 EC RIA Programme; in particular, the author gratefully acknowledges the support of Vicenc comma Beltran and the computer resources and technical support provided by BSC. The researchers from Universitat Jaume I (UJI) and Universitat Polit ' ecnica de Valencia (UPV) were supported by MINECO project TIN2017-82972-R. Maria Barreda was also supported by the POSDOC-A/2017/11 project from the Universitat Jaume I.Iakymchuk, R.; Barreda Vayá, M.; Graillat, S.; Aliaga, JI.; Quintana Ortí, ES. (2020). Reproducibility of parallel preconditioned conjugate gradient in hybrid programming environments. International Journal of High Performance Computing Applications. 34(5):502-518. https://doi.org/10.1177/1094342020932650S502518345Aliaga, J. I., Barreda, M., Flegar, G., Bollhöfer, M., & Quintana-Ortí, E. S. (2017). Communication in task-parallel ILU-preconditioned CG solvers using MPI + OmpSs. Concurrency and Computation: Practice and Experience, 29(21), e4280. doi:10.1002/cpe.4280Bailey, D. H. (2013). High-precision computation: Applications and challenges [Keynote I]. 2013 IEEE 21st Symposium on Computer Arithmetic. doi:10.1109/arith.2013.39Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., … van der Vorst, H. (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. doi:10.1137/1.9781611971538Burgess, N., Goodyer, C., Hinds, C. N., & Lutz, D. R. (2019). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. IEEE Transactions on Computers, 68(7), 967-978. doi:10.1109/tc.2018.2855729Carson, E., & Higham, N. J. (2018). Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions. SIAM Journal on Scientific Computing, 40(2), A817-A847. doi:10.1137/17m1140819Collange, S., Defour, D., Graillat, S., & Iakymchuk, R. (2015). Numerical reproducibility for the parallel reduction on multi- and many-core architectures. Parallel Computing, 49, 83-97. doi:10.1016/j.parco.2015.09.001Dekker, T. J. (1971). A floating-point technique for extending the available precision. Numerische Mathematik, 18(3), 224-242. doi:10.1007/bf01397083Demmel, J., & Hong Diep Nguyen. (2013). Fast Reproducible Floating-Point Summation. 2013 IEEE 21st Symposium on Computer Arithmetic. doi:10.1109/arith.2013.9Demmel, J., & Nguyen, H. D. (2015). Parallel Reproducible Summation. IEEE Transactions on Computers, 64(7), 2060-2070. doi:10.1109/tc.2014.2345391Dongarra, J. J., Du Croz, J., Hammarling, S., & Duff, I. S. (1990). A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1), 1-17. doi:10.1145/77626.79170Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., & Zimmermann, P. (2007). MPFR. ACM Transactions on Mathematical Software, 33(2), 13. doi:10.1145/1236463.1236468Hida, Y., Li, X. S., & Bailey, D. H. (s. f.). Algorithms for quad-double precision floating point arithmetic. Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001. doi:10.1109/arith.2001.930115Hunold, S., & Carpen-Amarie, A. (2016). Reproducible MPI Benchmarking is Still Not as Easy as You Think. IEEE Transactions on Parallel and Distributed Systems, 27(12), 3617-3630. doi:10.1109/tpds.2016.2539167IEEE Computer Society (2008) IEEE Standard for Floating-Point Arithmetic. Piscataway: IEEE Standard, pp. 754–2008.Kulisch, U., & Snyder, V. (2010). The exact dot product as basic tool for long interval arithmetic. Computing, 91(3), 307-313. doi:10.1007/s00607-010-0127-7Kulisch, U. (2013). Computer Arithmetic and Validity. doi:10.1515/9783110301793Lawson, C. L., Hanson, R. J., Kincaid, D. R., & Krogh, F. T. (1979). Basic Linear Algebra Subprograms for Fortran Usage. ACM Transactions on Mathematical Software, 5(3), 308-323. doi:10.1145/355841.355847Lutz, D. R., & Hinds, C. N. (2017). High-Precision Anchored Accumulators for Reproducible Floating-Point Summation. 2017 IEEE 24th Symposium on Computer Arithmetic (ARITH). doi:10.1109/arith.2017.20Mukunoki, D., Ogita, T., & Ozaki, K. (2020). Reproducible BLAS Routines with Tunable Accuracy Using Ozaki Scheme for Many-Core Architectures. Lecture Notes in Computer Science, 516-527. doi:10.1007/978-3-030-43229-4_44Nguyen, H. D., & Demmel, J. (2015). Reproducible Tall-Skinny QR. 2015 IEEE 22nd Symposium on Computer Arithmetic. doi:10.1109/arith.2015.28Ogita, T., Rump, S. M., & Oishi, S. (2005). Accurate Sum and Dot Product. SIAM Journal on Scientific Computing, 26(6), 1955-1988. doi:10.1137/030601818Ozaki, K., Ogita, T., Oishi, S., & Rump, S. M. (2011). Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications. Numerical Algorithms, 59(1), 95-118. doi:10.1007/s11075-011-9478-1Priest, D. M. (s. f.). Algorithms for arbitrary precision floating point arithmetic. [1991] Proceedings 10th IEEE Symposium on Computer Arithmetic. doi:10.1109/arith.1991.145549Rump, S. M., Ogita, T., & Oishi, S. (2008). Accurate Floating-Point Summation Part I: Faithful Rounding. SIAM Journal on Scientific Computing, 31(1), 189-224. doi:10.1137/050645671Rump, S. M., Ogita, T., & Oishi, S. (2009). Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest. SIAM Journal on Scientific Computing, 31(2), 1269-1302. doi:10.1137/07068816xRump, S. M., Ogita, T., & Oishi, S. (2010). Fast high precision summation. Nonlinear Theory and Its Applications, IEICE, 1(1), 2-24. doi:10.1587/nolta.1.2Saad, Y. (2003). Iterative Methods for Sparse Linear Systems. doi:10.1137/1.9780898718003Wiesenberger, M., Einkemmer, L., Held, M., Gutierrez-Milla, A., Sáez, X., & Iakymchuk, R. (2019). Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures. Computer Physics Communications, 238, 145-156. doi:10.1016/j.cpc.2018.12.00

Reproducibility Strategies for Parallel Preconditioned Conjugate Gradient

The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29 % (ExBLAS) and 4 % (lightweight) on 768 processes

Étude de la propagation acoustique en milieu complexe par des réseaux de neurones profonds

Abstract : Predicting the propagation of aerocoustic noise is a challenging task in the presence of complex mean flows and geometry installation effects. The design of future quiet propul- sion systems requires tools that are able to perform many accurate evaluations with a low computational cost. Analytical models or hybrid numerical approaches have tradition- ally been employed for that purpose. However, such methods are typically constrained by simplifying hypotheses that are not easily relaxed. Thus, the main objective of this thesis is to develop and validate novel methods for the fast and accurate prediction of aeroacoustic propagation in complex mean flows and geometries. For that, data-driven deep convolutional neural networks acting as auto-regressive spatio-temporal predictors are considered. These surrogates are trained on high-fidelity data, generated by direct aeroacoustic numerical solvers. Such datasets are able to model complex flow phenomena, along with complex geometrical parameters. The neural network is designed to substitute the high-fidelity solver at a much lower computational cost once the training is finished, while predicting the time-domain acoustic propagation with sufficient accuracy. Three test cases of growing complexity are employed to test the approach, where the learned surrogate is compared to analytical and numerical solutions. The first one corresponds to the two-dimensional propagation of Gaussian pulses in closed domains, which allows understanding the fundamental behavior of the employed convolution neural networks. Second, the approach is extended in order to consider a variety of boundary conditions, from non-reflecting to curved reflecting obstacles, including the reflection and scattering of waves at obstacles. This allows the prediction of acoustic propagation in configurations closer to industrial problems. Finally, the effects of complex mean flows is investigated through a dataset of acoustic waves propagating inside sheared flows. These applications highlight the flexibility of the employed data-driven methods using convolutional neural networks. They allow a significant acceleration of the acoustic predictions, while keeping an adequate accuracy and being also able to correctly predict the acoustic propagation outside the range of the training data. For that, prior knowledge about the wave propa- gation physics is included during and after the neural network training phase, allowing an increased control over the error performed by the surrogate. Among this prior knowledge, the conservation of physics quantities and the correct treatment of boundary conditions are identified as key parameters that improve the surrogate predictions.Prédire la propagation du bruit aéroacoustique est une tâche difficile en présence d’écoulements moyens complexes et d’effets géométriques d’installation. La conception des futurs systèmes de propulsion silencieux appelle au développement d’outils capables d’effectuer de nombreuses évaluations avec une faible erreur et un faible coût de calcul. Traditionnellement, des modèles analytiques ou des approches numériques hybrides ont été utilisés à cette fin. Cependant, ces méthodes sont généralement contraintes par des hypothèses simplificatrices qui ne sont pas facilement assouplies. Ainsi, l’objectif principal de cette thèse est de développer et de valider de nouvelles méthodes pour la prédiction rapide et précise de la propagation aéroacoustique dans des écoulements moyens et des géométries complexes. Pour cela, des réseaux de neurones profonds à convolution, entraînés sur des données, et agissant comme prédicteurs spatio-temporels sont considérés. Ces modèles par substitution sont entraînés sur des données de haute fidélité, générées par des solveurs numériques aérocoustiques directs. De telles bases de données sont capables de modéliser des phénomènes d’écoulement, ainsi que des paramètres géométriques complexes. Le réseau de neurones est conçu pour remplacer le solveur haute fidélité à un coût de calcul beaucoup plus faible une fois la phase d’entraînement terminée, tout en prédisant la propagation acoustique dans le domaine temporel avec une précision suffisante. Trois cas de test, de complexité croissante, sont utilisés pour tester l’approche, où le substitut appris est comparé à des solutions analytiques et numériques. Le premier cas correspond à la propagation acoustique bidimensionnelle dans des domaines fermés, où des sources impulsionnelles Gaussiennes sont considérées. Ceci permet de comprendre le comportement fondamental des réseaux de neurones à convolution étudiés. Deuxièmement, l’approche est étendue afin de prendre en compte une variété de conditions aux limites, notamment des conditions aux limites non réfléchissantes et des obstacles réfléchissants de géométrie arbitraire, modélisant la réflexion et la diffusion des ondes acoustiques sur ces obstacles. Cela permet de prédire la propagation acoustique dans des configurations plus proches des problématiques industrielles. Enfin, les effets des écoulements moyens complexes sont étudiés à travers une base de données d’ondes acoustiques qui se propagent à l’intérieur d’écoulements cisaillés. Ces applications mettent en évidence la flexibilité des méthodes basées sur les données, utilisant des réseaux de neurones à convolution. Ils permettent une accélération significative des prédictions acoustiques, tout en gardant une précision adéquate et en étant également capables de prédire correctement la propagation acoustique en dehors de la gamme de paramètres des données d’apprentissage. Pour cela, des connaissances préalables sur la physique de propagation des ondes sont incluses pendant et après la phase d’apprentissage du réseau de neurones, permettant un contrôle accru sur l’erreur effectuée par le substitut. Parmi ces connaissances préalables, la conservation des grandeurs physiques et le traitement correct des conditions aux limites sont identifiés comme des paramètres clés qui améliorent les prédictions du modèle proposé

Reproducible and Accurate Matrix Multiplication

International audienceDue to non-associativity of floating-point operations and dynamic scheduling on parallel architectures, getting a bit-wise reproducible floating-point result for multiple executions of the same code on different or even similar parallel architectures is challenging. In this paper, we address the problem of reproducibility in the context of matrix multiplication and propose an algorithm that yields both reproducible and accurate results. This algorithm is composed of two main stages: a filtering stage that uses fast vectorized floating-point expansions in conjunction with error-free transformations; an accumulation stage based on Kulisch long accumulators in a high-radix carry-save representation. Finally, we provide implementations and performance results in parallel environments like GPUs

Reproducible and Accurate Matrix Multiplication

International audienceDue to non-associativity of floating-point operations and dynamic scheduling on parallel architectures, getting a bit-wise reproducible floating-point result for multiple executions of the same code on different or even similar parallel architectures is challenging. In this paper, we address the problem of reproducibility in the context of matrix multiplication and propose an algorithm that yields both reproducible and accurate results. This algorithm is composed of two main stages: a filtering stage that uses fast vectorized floating-point expansions in conjunction with error-free transformations; an accumulation stage based on Kulisch long accumulators in a high-radix carry-save representation. Finally, we provide implementations and performance results in parallel environments like GPUs

Reproducible and Accurate Matrix Multiplication for GPU Accelerators

Due to non-associativity of floating-point operations and dynamic scheduling on parallel architectures, getting a bitwise reproducible floating-point result for multiple executions of the same code on different or even similar parallel architectures is challenging. In this paper, we address the problem of reproducibility in the context of matrix multiplication and propose an algorithm that yields both reproducible and accurate results. This algorithm is composed of two main stages: a filtering stage that uses fast vectorized floating-point expansions in con-junction with error-free transformations; an accumulation stage based on Kulisch long accumulators in a high-radix carry-save representation. Finally, we provide implementations and performance results in parallel environments like GPUs

Reproducible and Accurate Matrix Multiplication for GPU Accelerators

Due to non-associativity of floating-point operations and dynamic scheduling on parallel architectures, getting a bitwise reproducible floating-point result for multiple executions of the same code on different or even similar parallel architectures is challenging. In this paper, we address the problem of reproducibility in the context of matrix multiplication and propose an algorithm that yields both reproducible and accurate results. This algorithm is composed of two main stages: a filtering stage that uses fast vectorized floating-point expansions in con-junction with error-free transformations; an accumulation stage based on Kulisch long accumulators in a high-radix carry-save representation. Finally, we provide implementations and performance results in parallel environments like GPUs

Reproducible and Accurate Matrix Multiplication for High-Performance Computing

International audienceOn modern multi-core, many-core, and heterogeneous architectures, floating-point computations may become non-deterministic and thus non-reproducible mainly due to non-associativity of floating-point operations. We introduce an algorithm to compute a product of two floating-point matrices that delivers reproducible results with the best possible accuracy. Our multi-level algorithm relies on fast vectorized floating-point expansions and as well as superaccumulators in a high-radix carry-save representation. We present implementations on recent Intel Xeon Phi accelerators and both AMD and NVIDIA GPUs