\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft. Motivated by the demand in machine learning, modern computer hardware is increasingly supporting reduced precision floating-point arithmetic, which provides advantages in speed, energy, and memory usage over single and double precision. Given the availability of such hardware, mixed precision algorithms that work in single or double precision but carry out part of a computation in half precision are now of great interest for general scientific computing tasks. Because of the limited range of half precision arithmetic, in which positive numbers lie between 6 \times 10-8 and 7 \times 10 4 , a straightforward rounding of single or double precision data into half precision can lead to overflow, underflow, or subnormal numbers being generated, all of which are undesirable. We develop an algorithm for converting a matrix from single or double precision to half precision. It first applies two-sided diagonal scaling in order to equilibrate the matrix (that is, to ensure that every row and column has \infty-norm 1), then multiplies by a scalar to bring the largest element within a factor \theta \leq 1 of the overflow level, and finally rounds to half precision. The second step ensures that full use is made of the limited range of half precision arithmetic, and \theta must be chosen to allow sufficient headroom for subsequent computations. We apply the new algorithm to GMRES-based iterative refinement (GMRES-IR), which solves a linear system Ax = b with single or double precision data by LU factorizing A in half precision and carrying out iterative refinement with the correction equations solved by GMRES preconditioned with the low precision LU factors. Previous implementations of this algorithm have used a crude conversion to half precision that our experiments show can cause slow convergence of GMRES-IR for badly scaled matrices or failure to converge at all. The new conversion algorithm computes \infty-norms of rows and columns of the matrix and its cost is negligible in the context of LU factorization. We show that it leads to faster convergence of GMRES-IR for badly scaled matrices and thereby allows a much wider class of problems to be solved. \bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs. diagonal scaling, half precision arithmetic, fp16, overflow, underflow, subnormal numbers, iterative refinement, linear system, mixed precision, GMRES, preconditioning \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs. 65F05, 65F08, 65F35, 65F10 \bfD \bfO \bfI. 10.1137/18M1229511 1. Introduction. The landscape of scientific computing is changing, because of the growing availability and usage of low precision floating-point arithmetic. The 2008 revision of IEEE standard 754 introduced a 16-bit floating point format, known as half precision (fp16) [19]. Although defined only as a storage format, it has been widely adopted for computing, and is supported by the NVIDIA P100 and V100 GPUs and the AMD Radeon Instinct MI25 GPU. On such ha...

As parallel computers approach the exascale, power efficiency in Highperformance computing (HPC) systems is of increasing concern. Exploiting both, the hardware features, and algorithms is an effective solution to achieve power efficiency, and address the energy constraints in modern and future HPC systems. In this work, we present a novel design and implementation of an energy efficient solution for dense linear system of equations, which are at the heart of largescale HPC applications. The proposed energy efficient linear system solvers are based on two main components: (1) iterative refinement techniques, and (2) reduced precision computing features in the modern accelerators and co-processors. While most of the energy efficiency approaches aim to reduce the consumption with a minimal performance penalty, our method improves both, the performance and the energy-efficiency. Compared to highly optimised linear system solvers, our kernels are up to 2× faster to deliver the same accuracy solution, and reduce the energy consumption up to half on Intel KNL architectures. By using efficiently the tensor cores available in the NVIDIA V100 PCIe GPUs, the speedups can be up to 4× with more than 80% reduction on the energy consumption.

A current trend in high-performance computing is to decompose a large linear algebra problem into batches containing thousands of smaller problems, that can be solved independently, before collating the results. To standardize the interface to these routines, the community is developing an extension to the BLAS standard (the batched BLAS), enabling users to perform thousands of small BLAS operations in parallel whilst making efficient use of their hardware. We discuss the benefits and drawbacks of the current batched BLAS proposals and perform a number of experiments, focusing on a general matrix-matrix multiplication (GEMM), to explore their affect on the performance. In particular we analyze the effect of novel data layouts which, for example, interleave the matrices in memory to aid vectorization and prefetching of data. Utilizing these modifications our code outperforms both MKL1 CuBLAS2 by up to 6 times on the self-hosted Intel KNL (codenamed Knights Landing) and Kepler GPU architectures, for large numbers of double precision GEMM operations using matrices of size 2 × 2 to 20 × 20.The authors would like to thank The University of Tennessee for the use of their computational resources. This research was funded in part from the European Union’s Horizon 2020 research and innovation programme under the NLAFET grant agreement No. 671633.Peer ReviewedPostprint (published version

Abstract-We present a high-performance GPU kernel with a substantial speedup over vendor libraries for very small matrix computations. In addition, we discuss most of the challenges that hinder the design of efficient GPU kernels for small matrix algorithms. We propose relevant algorithm analysis to harness the full power of a GPU, and strategies for predicting the performance, before introducing a proper implementation. We develop a theoretical analysis and a methodology for high-performance linear solvers for very small matrices. As test cases, we take the Cholesky and LU factorizations and show how the proposed methodology enables us to achieve a performance close to the theoretical upper bound of the hardware. This work investigates and proposes novel algorithms for designing highly optimized GPU kernels for solving batches of hundreds of thousands of small-size Cholesky and LU factorizations. Our focus on efficient batched Cholesky and batched LU kernels is motivated by the increasing need for these kernels in scientific simulations (e.g., astrophysics applications). Techniques for optimal memory traffic, register blocking, and tunable concurrency are incorporated in our proposed design. The proposed GPU kernels achieve performance speedups vs. CUBLAS of up to 6× for the factorizations, using double precision arithmetic on an NVIDIA Pascal P100 GPU.

International audienceAs the computational power of high performance computing (HPC) systems continues to increase by using a huge number of cores or specialized processing units, HPC applications are increasingly prone to faults. In this paper, we present a new class of numerical fault tolerance algorithms to cope with node crashes in parallel distributed environments. This new resilient scheme is designed at application level and does not require extra resources, i.e., computational unit or computing time, when no fault occurs. In the framework of iterative methods for the solution of sparse linear systems, we present numerical algorithms to extract relevant information from available data after a fault, assuming a separate mechanism ensures the fault detection. After data extraction, a well chosen part of missing data is regenerated through interpolation strategies to constitute meaningful inputs to restart the iterative scheme. We have developed these methods, referred to as Interpolation-Restart techniques, for Krylov subspace linear solvers. After a fault, lost entries of the current iterate computed by the solver are interpolated to define a new initial guess to restart the Krylov method. A well suited initial guess is computed by using the entries of the faulty iterate available on surviving nodes. We present two interpolation policies that preserve key numerical properties of well-known linear solvers, namely the monotonic decrease of the A-norm of the error of the conjugate gradient or the residual norm decrease of GMRES. The qualitative numerical behavior of the resulting scheme have been validated with sequential simulations, when the number of faults and the amount of data losses are varied. Finally, the computational costs associated with the recovery mechanism have been evaluated through parallel experiments

As the computational power of high performance computing (HPC) systems continues to increase by using a huge number of CPU cores or specialized processing units, extreme-scale applications are increasingly prone to faults. Consequently, the HPC community has proposed many contributions to design resilient HPC applications. These contributions may be system-oriented, theoretical or numerical. In this study we consider an actual fully-featured parallel sparse hybrid (direct/iterative) linear solver, MAPHYS, and we propose numerical remedies to design a resilient version of the solver. The solver being hybrid, we focus in this study on the iterative solution step, which is often the dominant step in practice. We furthermore assume that a separate mechanism ensures fault detection and that a system layer provides support for setting back the environment (processes, . . . ) in a running state. The present manuscript therefore focuses on (and only on) strategies for recovering lost data after the fault has been detected (a separate concern beyond the scope of this study), once the system is restored (another separate concern not studied here). The numerical remedies we propose are twofold. Whenever possible, we exploit the natural data redundancy between processes from the solver to perform exact recovery through clever copies over processes. Otherwise, data that has been lost and no longer available on any process is recovered through a so-called interpolation-restart mechanism. This mechanism is derived from [1] by carefully taking into account the properties of the target hybrid solver. These numerical remedies have been implemented in the MAPHYS parallel solver so that we can assess their efficiency on a large number of processing units (up to 12, 288 CPU cores) for solving large-scale real-life problems.

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