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

Effect of Mixed Precision Computing on H-Matrix Vector Multiplication in BEM Analysis Rise Ooi Hokkaido University rise.ooi@frontier.hokudai.ac.jp Takeshi Iwashita Hokkaido University iwashita@iic.hokudai.ac.jp Takeshi Fukaya Hokkaido University fukaya@iic.hokudai.ac.jp Akihiro Ida The University of Tokyo ida@cc.u-tokyo.ac.jp Rio Yokota Tokyo Institute of Technology rioyokota@gsic.titech.ac.jp ABSTRACT Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time and space complexity of dense matrix vector multiplication, and hence has been applied to numerous practical problems. In this paper, we aim to accelerate the H-matrix vector multi- plication by introducing mixed precision computing, where we employ both binary64 (FP64) and binary32 (FP32) arithmetic op- erations. We propose three methods to introduce mixed precision computing to H-matrix vector multiplication, and then evaluate them in a boundary element method (BEM) analysis. The numerical tests examine the effects of mixed precision computing, particularly on the required simulation time and rate of convergence of the it- erative (BiCG-STAB) linear solver. We confirm the effectiveness of the proposed methods. CCS CONCEPTS • Computing methodologies →Massively parallel and high- performance simulations. KEYWORDS mixed precision computing, linear solver, hierarchical matrix ACM Reference Format: Rise Ooi, Takeshi Iwashita, Takeshi Fukaya, Akihiro Ida, and Rio Yokota. 2019. Effect of Mixed Precision Computing on H-Matrix Vector Multiplica- tion in BEM Analysis. In International Conference on High Performance Com- puting in Asia-Pacific Region (HPCAsia2020), January 15–17, 2020, Fukuoka, Japan. ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/3368474. 3368479 HPCAsia2020, January 15–17, 2020, Fukuoka, Japan 2019. This is the author’s version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in International Conference on High Performance Computing in Asia-Pacific Region (HPCAsia2020), January 15–17, 2020, Fukuoka, Japan, https://doi.org/10.1145/3368474.3368479. 1 INTRODUCTION Hierarchical Matrix (H-matrix) [11] is an approximation technique for dense matrices. The technique approximates a target dense ma- trix by splitting it up into multiple submatrices, where the selected submatrices are low-rank approximated. For the low-rank approxi- mation of the submatrices, methods such as random sampling, Sin- gular Value Decomposition (SVD), and Adaptive Cross Approximation (ACA) [18] are well known. In an ideal case, the technique reduces both time and space complexity of O(N 2) of (dense) matrix vector multiplication (matvecmul) into O(N loд N), where N is the dimen- sion of the matrix, assuming it is square. Therefore, H-matrices have been effectively used for practical problems, including n-body [25, 28], earthquake cycles [22], and superconductive coils [24] simulations. In this paper, we aim to accelerate the H-matrix vector multiplication by introducing mixed precision computing, where we use both binary64 (FP64) and binary32 (FP32) [1] arithmetic operations. In recent years, mixed (and lower) precision computing has been investigated in various computational kernels and scientific ap- plications. Under the demands for more computing needs, there exist modern computational devices that can process low precision computations much faster than higher ones, for example, in some types of graphics processing units (GPU). Notably, we see trends of deep learning-based applications perform sufficiently well at lower accuracy computations, especially when they are computed using binary16 (FP16) and utilizing Nvidia’s Tensor Cores [21]. Moreover, in the case of memory-bound applications such as matrix vector multiplication, the representation of data in lower precision simply reduces the amount of data transferred between CPU and memory, which naturally results in better performance. In this paper, we propose three methods to introduce mixed pre- cision computing to H-matrix vector multiplication, and evaluate them in a boundary element method (BEM) analysis. BEM [4] is one of the most popular discretization methods for partial differen- tial equation problems, and is also one of the principal application domains of H-matrix and its related techniques. In a BEM analysis, a linear system of equations with a dense coefficient matrix is to be solved. If a Krylov subspace iterative method [23] is used for the solution process, then the intensely iterated coefficient matrix vector multiplication is the most time consuming constraint part. Here, the H-matrix technique efficiently approximates the dense co- efficient matrix and thus significantly accelerates the matrix vector multiplication operations. Although when we introduce the

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