Research: A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic – HPCwire

Within the past years, hardware vendors have started designing low precision special function units in response to the demand of the machine learning community and their demand for high compute power in low precision formats. Also, server-line products are increasingly featuring low-precision special function units, such as the Nvidia tensor cores in the Oak Ridge National Laboratorys Summit supercomputer, providing more than an order of magnitude of higher performance than what is available in IEEE double precision.

At the same time, the gap between the compute power on the one hand and the memory bandwidth on the other hand keeps increasing, making data access and communication prohibitively expensive compared to arithmetic operations. Having the choice between ignoring the hardware trends and continuing the traditional path, and adjusting the software stack to the changing hardware designs, the Department of Energys Exascale Computing Project decided for the aggressive step of building a multiprecision focus effort to take on the challenge of designing and engineering novel algorithms exploiting the compute power available in low precision and adjusting the communication format to the application-specific needs.

To start the multiprecision focus effort, we have written a survey of the numerical linear algebra community and summarized all existing multiprecision knowledge, expertise, and software capabilities in this landscape analysis report. We also include current efforts and preliminary results that may not yet be considered mature technology, but have the potential to grow into production quality within the multiprecision focus effort. As we expect the reader to be familiar with the basics of numerical linear algebra, we refrain from providing a detailed background on the algorithms themselves but focus on how mixed- and multiprecision technology can help to improve the performance of these methods and present highlights of application significantly outperforming the traditional fixed precision methods.

This report covers low precision BLAS operations, solving systems of linear systems, least squares problems, eigenvalue computations using mixed precision. These are demonstrated with dense and sparse matrix computations and direct and iterative methods. The ideas presented try to exploit low precision computations for the bulk of the compute time and then use mathematical techniques to enhance the accuracy of the solution to bring it to full precision accuracy with less time to solution.

On modern architectures, the performance of 32-bit operations is often at least twice as fast as the performance of 64-bit operations. There are two reasons for this. Firstly, a 32-bit floating point arithmetic rate of execution is usually twice as fast as a 64-bit floating point arithmetic on most modern processors. Secondly, the number of bytes moved through the memory system is halved. It may be possible to care out the computation in lower precision, say 16-bit operations.

One approach exploiting the compute power in low precision is motivated by the observation that in many cases, a single precision solution of a problem can be refined to the point where double precision accuracy is achieved. The refinement can be accomplished, for instance, by means of the Newtons algorithm (see Equation (1)) which computes the zero of a function f (x) according to the iterative formula:

In general, we would compute a starting point and f (x) in single precision arithmetic, and the refinement process will be computed in double precision arithmetic. If the refinement process is cheaper than the initial computation of the solution, then double precision accuracy can be achieved nearly at the same speed as the single precision accuracy.

Stunning results can be achieved. In Figure 1, we are comparing the solution of a general system of linear equations using a dense solver on an Nvidia V100 GPU comparing the performance for 64-, 32-, and 16-bit floating point operations for the factorization and then using refinement techniques to improve the solution for the 32- and 16-bit solution to what was achieved using 64-bit factorization.

The survey report presents much more detail on the methods and approaches using these techniques, see https://www.icl.utk.edu/files/publications/2020/icl-utk-1392-2020.pdf.

Author Bio Hartwig Anzt

Hartwig Anzt is a Helmholtz-Young-Investigator Group leader at the Steinbuch Centre for Computing at the Karlsruhe Institute of Technology (KIT). He obtained his PhD in Mathematics at the Karlsruhe Institute of Technology, and afterwards joined Jack Dongarras Innovative Computing Lab at the University of Tennessee in 2013. Since 2015 he also holds a Senior Research Scientist position at the University of Tennessee. Hartwig Anzt has a strong background in numerical mathematics, specializes in iterative methods and preconditioning techniques for the next generation hardware architectures. His Helmholtz group on Fixed-point methods for numerics at Exascale (FiNE) is granted funding until 2022. Hartwig Anzt has a long track record of high-quality software development. He is author of the MAGMA-sparse open source software package managing lead and developer of the Ginkgo numerical linear algebra library, and part of the US Exascale computing project delivering production-ready numerical linear algebra libraries.

Author Bio Jack Dongarra

Jack Dongarra received a Bachelor of Science in Mathematics from Chicago State University in 1972 and a Master of Science in Computer Science from the Illinois Institute of Technology in 1973. He received his PhD in Applied Mathematics from the University of New Mexico in 1980. He worked at the Argonne National Laboratory until 1989, becoming a senior scientist.He now holds an appointment as University Distinguished Professor of Computer Science in the Computer Science Department at the University of Tennessee, has the position of a Distinguished Research Staff member in the Computer Science and Mathematics Division at Oak Ridge National Laboratory (ORNL), Turing Fellow in the Computer Science and Mathematics Schools at the University of Manchester, and an Adjunct Professor in the Computer Science Department at Rice University.

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