1. State Key Laboratory of Software Development Environment, Beijing 100191, China 2. School of Computer Science and Engineering, Beihang University, Beijing 100191, China
Although matrix multiplication plays an essential role in a wide range of applications, previous works only focus on optimizing dense or sparse matrix multiplications. The Sparse Approximate Matrix Multiply (SpAMM) is an algorithm to accelerate the multiplication of decay matrices, the sparsity of which is between dense and sparse matrices. In addition, large-scale decay matrix multiplication is performed in scientific applications to solve cutting-edge problems. To optimize large-scale decay matrix multiplication using SpAMM on supercomputers such as Sunway Taihulight, we present swSpAMM, an optimized SpAMM algorithm by adapting the computation characteristics to the architecture features of Sunway Taihulight.
Specifically, we propose both intra-node and inter-node optimizations to accelerate swSpAMM for large-scale execution. For intra-node optimizations, we explore algorithm parallelization and block-major data layout that are tailored to better utilize the architecture advantage of Sunway processor. For inter-node optimizations, we propose a matrix organization strategy for better distributing sub-matrices across nodes and a dynamic scheduling strategy for improving load balance across nodes. We compare swSpAMM with the existing GEMM library on a single node as well as large-scale matrix multiplication methods on multiple nodes. The experiment results show that swSpAMM achieves a speedup up to 14.5× and 2.2× when compared to xMath library on a single node and 2D GEMM method on multiple nodes, respectively.
Architecture-specific (opt1)Block-major data layout (opt2)Parallelization exploration (opt3)
0.011284
0.08381
0.64348
5.069342
40.4408
Tab.4
Fig.10
Nodes
N = 8192
N = 16384
N = 32768
N = 65536
N = 131072
4
1.9526
13.452
–
–
–
8
1.4546
8.9092
–
–
–
16
0.9374
5.8864
41.677
–
–
32
0.6553
3.5677
22.399
–
–
64
0.3568
2.0592
13.032
92.038
–
128
–
1.4531
8.1913
51.82
–
256
–
0.7818
4.3437
26.848
187.04
512
–
–
3.0037
16.937
107.91
Tab.5
Nodes
N = 8192
N = 16384
N = 32768
N = 65536
N = 131072
4
1.9528
13.45
–
–
–
8
0.9466
6.6422
–
–
–
16
0.9454
5.8943
41.753
–
–
32
0.4529
2.898
20.762
–
–
64
0.2347
1.501
10.612
92.482
–
128
–
1.0257
6.4939
46.062
–
256
–
0.5126
3.2726
23.166
187.23
512
–
–
2.1272
13.311
92.964
Tab.6
Nodes
N = 8192
N = 16384
N = 32768
N = 65536
N = 131072
8
0.9451
6.6416
49.961
–
–
64
0.2194
1.4348
10.361
77.326
–
512
0.0493
0.2637
1.6654
11.584
85.988
Tab.7
Nodes
N = 8192
N = 16384
N = 32768
N = 65536
N = 131072
4
1.9879
15.016
119.33
–
–
8
1.0204
7.5894
62.057
–
–
16
0.5665
3.928
30.612
242.45
–
32
0.3705
1.8628
15.572
120.78
–
64
0.2006
1.0659
7.6402
62.472
494.92
128
–
0.7238
3.9433
29.186
249.29
256
–
0.377
1.979
14.613
113.13
512
–
–
4.6493
7.5753
57.539
Tab.8
Fig.11
Fig.12
Fig.13
Fig.14
Fig.15
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