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1. Mixed Precision Randomized Low-Rank Approximation with GPU Tensor Cores

Author

Baboulin, Marc, Donfack, Simplice, Kaya, Oguz, Mary, Theo, Robeyns, Matthieu, Goos, Gerhard, Series Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Carretero, Jesus, editor, Shende, Sameer, editor, Garcia-Blas, Javier, editor, Brandic, Ivona, editor, Olcoz, Katzalin, editor, and Schreiber, Martin, editor

Published
2024
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2. Reduced-Precision and Reduced-Exponent Formats for Accelerating Adaptive Precision Sparse Matrix–Vector Product

Author

Graillat, Stef, Jézéquel, Fabienne, Mary, Theo, Molina, Roméo, Mukunoki, Daichi, Goos, Gerhard, Series Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Carretero, Jesus, editor, Shende, Sameer, editor, Garcia-Blas, Javier, editor, Brandic, Ivona, editor, Olcoz, Katzalin, editor, and Schreiber, Martin, editor

Published
2024
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3. Matrix-free construction of HSS representation using adaptive randomized sampling

Author

Gorman, Christopher, Chávez, Gustavo, Ghysels, Pieter, Mary, Théo, Rouet, François-Henry, and Li, Xiaoye Sherry

Subjects
Computer Science - Numerical Analysis, Computer Science - Mathematical Software
Abstract

We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to determine the maximum off-diagonal block rank. We develop both relative and absolute stopping criteria to determine the minimum dimension of the random projection matrix that is sufficient for the desired accuracy. Two strategies are discussed to adaptively enlarge the random sample matrix: repeated doubling of the number of random vectors, and iteratively incrementing the number of random vectors by a fixed number. The relative and absolute stopping criteria are based on probabilistic bounds for the Frobenius norm of the random projection of the Hankel blocks of the input matrix. We discuss parallel implementation and computation and communication cost of both variants. Parallel numerical results for a range of applications, including boundary element method matrices and quantum chemistry Toeplitz matrices, show the effectiveness, scalability and numerical robustness of the proposed algorithms., Comment: 24 pages, 4 figures, 20 references

Published
2018

7. Scaling is all you need: quantization of butterfly matrix products via optimal rank-one quantization

Author

Gribonval, Rémi, Mary, Theo, Riccietti, Elisa, Optimisation, Connaissances pHysiques, Algorithmes et Modèles (OCKHAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), Performance et Qualité des Algorithmes Numériques (PEQUAN), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), GDR ISIS project MOMIGS, and ANR-19-CHIA-0009,AllegroAssai,Algorithmes, Approximations, Parcimonie et Plongements pour l'IA(2019)

Subjects
butterfly, sparse matrices, Quantization, rank one, [INFO]Computer Science [cs], Floating point arithmetic
Abstract

We consider the problem of quantizing the factors of butterfly factorizations. We show how the properties of butterfly supports can be exploited to reduce the problem to the solution of a series of rank-one quantization problems. We thus propose an algorithm that, exploiting the intrinsic scaling invariance of the problem, gives the optimal solution of the rank-one quantization problem. We employ this algorithm as a building block in a heuristic procedure to solve the quantization problem of butterfly factors, which we show can be much more accurate that the naive round-to-nearest startegy of quantizing each factor independently.; On s'intéresse à la quantification des facteurs dans un produit de matrices creuses à structure papillon. On montre que les propriétés particulières es supports des facteurs papillon permettent d'aborder le problème comme une séquence de problème de quantification de rang un. On propose un algorithme optimal de quantification de rang un utilisant les invariances du problème par remise à l'échelle. Cet algorithme sert de brique de base dans une approche heuristique pour la quantification de produits papillon, qui s'avère bien plus précis qu'une approche naive basée sur la quantification de chaque facteur au plus proche voisin indépendamment.

Published
2023

8. Optimal quantization of rank-one matrices in floating-point arithmetic---with applications to butterfly factorizations

Author

Gribonval, Rémi, Mary, Theo, Riccietti, Elisa, Optimisation, Connaissances pHysiques, Algorithmes et Modèles (OCKHAM), Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut Rhône-Alpin des systèmes complexes (IXXI), École normale supérieure de Lyon (ENS de Lyon)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Lumière - Lyon 2 (UL2)-Université Jean Moulin - Lyon 3 (UJML), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), Performance et Qualité des Algorithmes Numériques (PEQUAN), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and ANR-19-CHIA-0009,AllegroAssai,Algorithmes, Approximations, Parcimonie et Plongements pour l'IA(2019)

Subjects
floating-point arihmetic, sparse matrices, low-rank approximations, numerical linear algebra, rank-one matrices, [INFO]Computer Science [cs], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], low precision quantization, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], butterfly factorizations
Abstract

We consider the problem of optimally quantizing rank-one matrices to low precision floating-point arithmetic. We first explain that the naive strategy of separately quantizing the two rank-one factors can be far from optimal, and we provide worst case error bounds to support this observation. We characterize the optimal solution as the quantization of suitably scaled factors of the rank-one matrix and we develop an algorithm of tractable complexity to find the optimal scaling parameters. Using random rank-one matrices, we show experimentally that our algorithm can significantly reduce the quantization error. We then apply this algorithm to the quantization of butterfly factorizations, a fundamental tool that appears in many fast linear transforms. We show how the properties of butterfly supports can be exploited to approach the problem via a series of rank-one quantization problems and we employ our algorithm as a building block in a heuristic procedure to quantize a product of butterfly factors. We show that, despite being only heuristic, this strategy can be much more accurate than quantizing each factor independently or, equivalently, can achieve storage reductions of up to 30% with no loss of accuracy.

Published
2023

9. COMMUNICATION AVOIDING BLOCK LOW-RANK PARALLEL MULTIFRONTAL TRIANGULAR SOLVE WITH MANY RIGHT-HAND SIDES.

Author

AMESTOY, PATRICK, BOITEAU, OLIVIER, BUTTARI, ALFREDO, GEREST, MATTHIEU, JÉZÉQUEL, FABIENNE, LÉXCELLENT, JEAN-YVES, and MARY, THEO

Subjects
NUMERICAL solutions for linear algebra, PATTERNS (Mathematics), SPARSE matrices, LOW-rank matrices
Abstract

Block low-rank (BLR) compression can significantly reduce the memory and time costs of parallel sparse direct solvers. In this paper, we investigate the performance of the BLR triangular solve phase, which we observe to be underwhelming when dealing with many right-hand sides (RHS). We explain that this is because the bottleneck of the triangular solve is not in accessing the BLR LU factors, but rather in accessing the RHS, which are uncompressed. Motivated by this finding, we propose several new hybrid variants, which combine the right-looking and left-looking communication patterns to minimize the number of accesses to the RHS. We confirm via a theoretical analysis that these new variants can significantly reduce the total communication volume. We assess the impact of this reduction on the time performance on a range of real-life applications using the MUMPS solver, obtaining up to 20% time reduction. [ABSTRACT FROM AUTHOR]

Published
2024
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14. Mixed precision low-rank approximations and their application to block low-rank LU factorization.

Author

Amestoy, Patrick, Boiteau, Olivier, Buttari, Alfredo, Gerest, Matthieu, Jézéquel, Fabienne, L'Excellent, Jean-Yves, and Mary, Theo

Subjects
SINGULAR value decomposition, NUMERICAL solutions for linear algebra, FACTORIZATION, FRACTIONS, ARITHMETIC, LOW-rank matrices, SPARSE matrices, LUTETIUM compounds
Abstract

We introduce a novel approach to exploit mixed precision arithmetic for low-rank approximations. Our approach is based on the observation that singular vectors associated with small singular values can be stored in lower precisions while preserving high accuracy overall. We provide an explicit criterion to determine which level of precision is needed for each singular vector. We apply this approach to block low-rank (BLR) matrices, most of whose off-diagonal blocks have low rank. We propose a new BLR LU factorization algorithm that exploits the mixed precision representation of the blocks. We carry out the rounding error analysis of this algorithm and prove that the use of mixed precision arithmetic does not compromise the numerical stability of the BLR LU factorization. Moreover, our analysis determines which level of precision is needed for each floating-point operation (flop), and therefore guides us toward an implementation that is both robust and efficient. We evaluate the potential of this new algorithm on a range of matrices coming from real-life problems in industrial and academic applications. We show that a large fraction of the entries in the LU factors and flops to perform the BLR LU factorization can be safely switched to lower precisions, leading to significant reductions of the storage and expected time costs, of up to a factor three using fp64, fp32, and bfloat16 arithmetics. [ABSTRACT FROM AUTHOR]

Published
2023
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17. Adversarial attacks via Sequential Quadratic Programming

Author

Beuzeville, Théo, Buttari, Alfredo, Gratton, Serge, Mary, Theo, Ulker, Erkan, Beuzeville, Théo, and Artificial and Natural Intelligence Toulouse Institute - - ANITI2019 - ANR-19-P3IA-0004 - P3IA - VALID

Subjects
[INFO.INFO-AI] Computer Science [cs]/Artificial Intelligence [cs.AI]
Abstract

Deep neural networks (DNN) achieve state-of-the-art performance in many machine learning tasks and in various types of applications. Their efficiency in solving complex problems has led to apply deep learning techniques in safety-critical tasks such as autonomous driving or medicine. However their sensitivity to adversarial attacks, rounding errors, or quantization processes raises concerns and has lead to high interest in finding new approaches to make them more robust. In this work we propose a novel approach for the construction of adversarial attacks which relies on a local Sequential Quadratic Programming (SQP) strategy. These attacks use second order information to achieve competitive performance compared with existing state-of-the-art approaches. We present numerical results that support our theoretical findings and illustrate the relevance of our approach on well-known datasets.

Published
2022

22. Solving block low-rank linear systems by LU factorization is numerically stable.

Author

Higham, Nicholas J and Mary, Theo

Subjects
*NUMERICAL solutions for linear algebra, *LINEAR systems, *FACTORIZATION, *FLOATING-point arithmetic, *LOW-rank matrices, *MATRIX decomposition
Abstract

Block low-rank (BLR) matrices possess a blockwise low-rank property that can be exploited to reduce the complexity of numerical linear algebra algorithms. The impact of these low-rank approximations on the numerical stability of the algorithms in floating-point arithmetic has not previously been analysed. We present rounding error analysis for the solution of a linear system by LU factorization of BLR matrices. Assuming that a stable pivoting scheme is used, we prove backward stability: the relative backward error is bounded by a modest constant times |$\varepsilon $|⁠ , where the low-rank threshold |$\varepsilon $| is the parameter controlling the accuracy of the blockwise low-rank approximations. In addition to this key result, our analysis offers three new insights into the numerical behaviour of BLR algorithms. First, we compare the use of a global or local low-rank threshold and find that a global one should be preferred. Second, we show that performing intermediate recompressions during the factorization can significantly reduce its cost without compromising numerical stability. Third, we consider different BLR factorization variants and determine the update–compress–factor variant to be the best. Tests on a wide range of matrices from various real-life applications show that the predictions from the analysis are realized in practice. [ABSTRACT FROM AUTHOR]

Published
2022
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23. Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods.

Author

Buttari, Alfredo, Huber, Markus, Leleux, Philippe, Mary, Theo, Rüde, Ulrich, and Wohlmuth, Barbara

Subjects
MULTIGRID methods (Numerical analysis), GRIDS (Cartography), DEGREES of freedom
Abstract

Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to 1011 degrees of freedom on a petascale supercomputer using 43,200 processes. [ABSTRACT FROM AUTHOR]

Published
2022
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28. Distributed Solvers: Direct and Iterative Methods for EoCoE (LSSC 2017)

Author

Amestoy, Patrick, Buttari, Alfredo, Jolivet, Pierre, L'Excellent, Jean-Yves, Leleux, Philippe, Mary, Theo, Ruiz, Daniel, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Optimisation des ressources : modèles, algorithmes et ordonnancement (ROMA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS)-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-École normale supérieure - Lyon (ENS Lyon), Institut de Biologie Computationnelle (IBC), Institut National de la Recherche Agronomique (INRA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Barcelona Supercomputing Center - Centro Nacional de Supercomputacion (BSC - CNS), Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Society for Industrial and Applied Mathematics (SIAM), and Sozopol municipality

Subjects
Energy oriented Center of Excellencefor computing applications (EoCoE), [INFO]Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2017

29. Comparison of BLR and HSS low-rank formats in multifrontal solvers: theory and practice, talk @SIAM CSE'17

Author

Amestoy, Patrick, Buttari, Alfredo, Ghysels, Pieter, L'Excellent, Jean-Yves, Li, Xiaoye Sherry, Mary, Theo, Rouet, François-Henry, Algorithmes Parallèles et Optimisation (IRIT-APO), Institut de recherche en informatique de Toulouse (IRIT), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Optimisation des ressources : modèles, algorithmes et ordonnancement (ROMA), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de l'Informatique du Parallélisme (LIP), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Laboratoire de l'Informatique du Parallélisme (LIP), Université de Lyon-Université de Lyon-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Lawrence Berkeley National Laboratory [Berkeley] (LBNL), and Society for Industrial and Applied Mathematics ( SIAM )

Subjects
[INFO]Computer Science [cs], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2017

38. Crisis management and navy organizational culture

Author

Callahan, Mary P., Wirtz, James J., Hoeksema, Mary Theo, Callahan, Mary P., Wirtz, James J., and Hoeksema, Mary Theo

Abstract

In recent years, the Navy has been plagued by a series of non- operational events that have attracted negative attention from the media and intense scrutiny from political leaders, special interest groups and the public. Too often, Navy responses to this scrutiny have been counterproductive. This study suggests that the Navy could benefit from a broader definition of what constitutes an organizational crisis. Also, a more complex approach to crisis management is needed, especially when the potential for damage to the institutional image is high. This thesis proposes a formalized system of crisis management. It also presents critical case studies to illustrate how organizational culture can serve as an impediment to constructive crisis management. The cases reviewed include the Naval Academy's handling of the 1989 Gwen Dreyer sexual harassment and hazing incident, the 1992 Electrical Engineering 311 exam cheating scandal, and the critical editorial written by faculty member James Barry and published by The Washington Post in March 1996. My research concludes that the key to effective crisis management is an organizational culture which institutionalizes the use of ethical and constructive practices, http://archive.org/details/crisismanagement109458177, Lieutenant, United States Navy, Approved for public release; distribution is unlimited.

Published
2012

39. Crisis management and navy organizational culture

Author

Hoeksema, Mary Theo, Callahan, Mary P., and Wirtz, James J.

Subjects
organizational culture, Crisis management, Naval Academy
Abstract

In recent years, the Navy has been plagued by a series of non- operational events that have attracted negative attention from the media and intense scrutiny from political leaders, special interest groups and the public. Too often, Navy responses to this scrutiny have been counterproductive. This study suggests that the Navy could benefit from a broader definition of what constitutes an organizational crisis. Also, a more complex approach to crisis management is needed, especially when the potential for damage to the institutional image is high. This thesis proposes a formalized system of crisis management. It also presents critical case studies to illustrate how organizational culture can serve as an impediment to constructive crisis management. The cases reviewed include the Naval Academy's handling of the 1989 Gwen Dreyer sexual harassment and hazing incident, the 1992 Electrical Engineering 311 exam cheating scandal, and the critical editorial written by faculty member James Barry and published by The Washington Post in March 1996. My research concludes that the key to effective crisis management is an organizational culture which institutionalizes the use of ethical and constructive practices http://archive.org/details/crisismanagement109458177 Lieutenant, United States Navy Approved for public release; distribution is unlimited.

Published
1997

40. Crisis management and navy organizational culture

Author

Callahan, Mary P., Wirtz, James J., Hoeksema, Mary Theo, Callahan, Mary P., Wirtz, James J., and Hoeksema, Mary Theo

Abstract

In recent years, the Navy has been plagued by a series of non- operational events that have attracted negative attention from the media and intense scrutiny from political leaders, special interest groups and the public. Too often, Navy responses to this scrutiny have been counterproductive. This study suggests that the Navy could benefit from a broader definition of what constitutes an organizational crisis. Also, a more complex approach to crisis management is needed, especially when the potential for damage to the institutional image is high. This thesis proposes a formalized system of crisis management. It also presents critical case studies to illustrate how organizational culture can serve as an impediment to constructive crisis management. The cases reviewed include the Naval Academy's handling of the 1989 Gwen Dreyer sexual harassment and hazing incident, the 1992 Electrical Engineering 311 exam cheating scandal, and the critical editorial written by faculty member James Barry and published by The Washington Post in March 1996. My research concludes that the key to effective crisis management is an organizational culture which institutionalizes the use of ethical and constructive practices

Published
1997

41. Stochastic Rounding: Implementation, Error Analysis, and Applications

Author

Croci, Matteo, Fasi, Massimiliano, Higham, Nicholas J., Mary, Theo, Mikaitis, Mantas, Croci, Matteo, Fasi, Massimiliano, Higham, Nicholas J., Mary, Theo, and Mikaitis, Mantas

Abstract

Stochastic rounding randomly maps a real number to one of the two nearest values in a finite precision number system. First proposed for use in computer arithmetic in the 1950s, it is attracting renewed interest. If used in floating-point arithmetic in the computation of the inner product of two vectors of length n, it yields an error bounded by \sqrt(n)u with high probability, where u is the unit roundoff, which is not necessarily the case for round to nearest. A particular attraction of stochastic rounding is that, unlike round to nearest, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity are lost. We survey stochastic rounding, covering its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, including deep learning and the numerical solution of differential equations.

42. Mixed Precision Algorithms in Numerical Linear Algebra

Author

Higham, Nicholas J., Mary, Theo, Higham, Nicholas J., and Mary, Theo

Abstract

Today's floating-point arithmetic landscape is broader than ever. While scientific computing has traditionally used single precision and double precision floating-point arithmetics, half precision is increasingly available in hardware and quadruple precision is supported in software. Lower precision arithmetic brings increased speed and reduced communication and energy costs, but it produces results of correspondingly low accuracy. Higher precisions are more expensive but can potentially provide great benefits, even if used sparingly. A variety of mixed precision algorithms have been developed that combine the superior performance of lower precisions with the better accuracy of higher precisions. Some of these algorithms aim to provide results of the same quality as algorithms running in a fixed precision but at a much lower cost; others use a little higher precision to improve the accuracy of an algorithm. This survey treats a broad range of mixed precision algorithms in numerical linear algebra, both direct and iterative, for problems including matrix multiplication, matrix factorization, linear systems, least squares, eigenvalue decomposition, and singular value decomposition. We identify key algorithmic ideas, such as iterative refinement, adapting the precision to the data, and exploiting mixed precision block fused multiply--add operations. We also describe the possible performance benefits and explain what is known about the numerical stability of the algorithms. This survey should be useful to a wide community of researchers and practitioners who wish to develop or benefit from mixed precision numerical linear algebra algorithms.

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