Effect of Mixed Precision Computing on H-Matrix Vector Multiplication in BEM Analysis

Ooi, Rise; Iwashita, Takeshi; Fukaya, Takeshi; Ida, Akihiro; Yokota, Rio

doi:10.1145/3368474.3368479

Cited by 5 publications

(4 citation statements)

References 31 publications

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“…We thus end up with a partitioning of the SVD as defined by (2.17)- (2.18). We note that this partitioning is similar to the Method 3 proposed in [30]. Our analysis justifies the use of this partitioning and gives a precise rule to define the p groups depending on the singular values and on the precisions.…”

Section: Mixed Precision Low-rank Approximationssupporting

confidence: 59%

Mixed precision low-rank approximations and their application to block low-rank LU factorization

Amestoy

Boiteau

Buttari

et al. 2022

IMA Journal of Numerical Analysis

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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.

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Section: Mixed Precision Low-rank Approximationssupporting

confidence: 59%

Mixed precision low-rank approximations and their application to block low-rank LU factorization

Amestoy

Boiteau

Buttari

et al. 2022

IMA Journal of Numerical Analysis

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“…Thus, performance enhancements can be obtained by performing compute-intensive portions in FP32, provided the error incurred is within reason. Similar methods have been successfully used in other domains of computational science as well 40,41 . For the class of high dimensional dynamical systems considered in this paper, it is proposed to evaluate the nested summations, arising out of the coupling terms, involved in the function evaluation in FP32 (see Eq.…”

Section: Utilizing Mixed Fp32-fp64 Computationsmentioning

confidence: 99%

Cost-effective massive computational speedups in simulations of high dimensional dynamical systems

Biswas,

Gupta

2023

Preprint

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High dimensional dynamical systems involve a large number of coupled, generally nonlinear, differential equations that are computationally intensive to simulate, thus requiring expensive computational resources. This paper presents a novel strategy for numerically integrating such systems by utilizing cheaper gaming GPUs and mixed-precision computations. This is facilitated by exploiting the higher single-precision floating-point throughput of gaming GPUs and their relatively cheaper costs which enable packing multiple gaming GPUs into a single compute node. Appropriately parallelizing the algorithm and using mixed-precision computation enables massive computational speedup without severely compromising on accuracy. The potential of these developments is illustrated through a representative high dimensional dynamical system which shows a massive speedup over serial execution, even exceeding 1000-fold for larger-scale systems. These developments have the potential to make a substantial impact on the simulation and exploration of dynamical systems at massive scales without the need for expensive hardware resources.

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“…Note that this approach is applicable not only to the SVD but also to other types of rank-revealing decompositions, such as QR factorization with column pivoting. Ooi et al (2020) propose three different methods to introduce mixed precision arithmetic in the product of a low-rank matrix with a vector. Their method 3 is similar to the representation (12.1), which they use with fp64 and fp32 arithmetics.…”

Section: Svd With Rapidly Decaying Singular Valuesmentioning

confidence: 99%

“…This can be the case, for example, of SVDs or rank-revealing factorizations. In fact, the mixed precision truncated SVD approaches described in Section 12.2 (Amestoy et al 2021a, Ooi et al 2020 are precisely based on this property: rounding errors introduced by converting singular vectors to lower precision are demagnified by the associated singular value, and so the precision of each vector should be selected based on its associated singular value.…”

Section: At the Column Level (Or Equivalently At The Row Level)mentioning

confidence: 99%

Mixed precision algorithms in numerical linear algebra

Higham¹,

Mary²

2022

Acta Numerica

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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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Effect of Mixed Precision Computing on H-Matrix Vector Multiplication in BEM Analysis

Cited by 5 publications

References 31 publications

Mixed precision low-rank approximations and their application to block low-rank LU factorization

Mixed precision low-rank approximations and their application to block low-rank LU factorization

Cost-effective massive computational speedups in simulations of high dimensional dynamical systems

Mixed precision algorithms in numerical linear algebra

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