Solver Benchmarks | sTiles Test Suite

Every matrix in the suite was factored by sTiles and five reference sparse direct solvers. For each matrix and solver we report the best factorization time over a sweep of thread counts. Times below are the same measurements reported in the paper.

60 matrices 6 solvers Intel 2×Xeon Gold · 40 cores best over 1–40 core sweep Cholesky factorization

Factorization time across all matrices

Best factorization time for every matrix and solver

Best factorization time (minimum over the swept core counts) for every matrix in the 60-matrix suite and every solver, on the Intel node. Matrices are sorted along the horizontal axis by sTiles' time, so the sTiles curve is monotonic; a competitor marker below it is faster than sTiles on that matrix, above it slower. PARDISO is the closest solver: it is faster than sTiles on 15 matrices, all below 0.5 s; sTiles is faster on the other 45, including all 12 above 0.5 s. MUMPS, PaStiX, CHOLMOD, and symPACK are slower than sTiles on 97 to 100% of the matrices each of them factors.

Per-matrix best factorization time (s) · Intel node

#	matrix	sTiles	PARDISO	MUMPS	CHOLMOD	PaStiX	symPACK
1	inla_graph_sem_n2000	0.0005	0.0015	0.0069	0.0015	0.0054	0.0095
2	inla_graph_sem_n5000	0.0008	0.0035	0.015	0.0041	0.015	0.020
3	inla_graph_8rtKSK	0.0010	0.0011	0.0058	0.0009	0.078	3.00
4	nasa4704	0.0013	0.0012	0.0058	0.0069	0.0082	0.0097
5	gyro_m	0.0013	0.0013	0.013	0.015	0.023	0.154
6	bcsstk15	0.0016	0.0015	0.0096	0.010	0.0078	0.013
7	gyro_k	0.0018	0.0023	0.025	0.031	0.057	0.014
8	inla_graph_sh7Pgi	0.0026	0.0019	0.0086	0.0038	0.0093	23.0
9	inla_graph_pedigree	0.0029	0.0070	0.082	0.0056	0.041	—
10	inla_graph_sem_n20000	0.0031	0.013	0.054	0.021	0.063	0.080
11	inla_graph_diff	0.0032	0.0051	0.018	0.048	0.029	0.034
12	msc10848	0.0036	0.0039	0.022	0.033	0.071	0.017
13	inla_graph_net814381	0.0038	0.0063	0.032	0.013	0.031	1.21
14	inla_graph_ayaLRw	0.0038	0.0059	0.027	0.014	0.029	0.228
15	msc23052	0.0047	0.0050	0.032	0.051	0.072	0.021
16	thermal1	0.0052	0.0068	0.036	0.111	0.043	0.021
17	inla_graph_lgm_10010_bw2	0.010	0.013	0.036	0.044	0.048	0.023
18	nasasrb	0.012	0.014	0.061	0.172	0.163	0.037
19	oilpan	0.013	0.012	0.065	0.160	0.219	0.050
20	inla_graph_sem_n100000	0.014	0.068	0.276	0.162	0.353	0.400
21	thermomech_dM	0.015	0.017	0.090	0.287	0.125	0.029
22	ct20stif	0.018	0.020	0.073	0.152	0.191	0.069
23	s3dkt3m2	0.019	0.021	0.104	0.231	0.248	0.103
24	smt	0.021	0.022	0.092	0.140	0.253	0.095
25	inla_graph_ferris	0.021	0.013	0.036	0.143	0.116	0.087
26	s3dkq4m2	0.022	0.021	0.111	0.247	0.291	0.101
27	inla_graph_animal1	0.023	0.181	0.133	0.110	0.391	0.613
28	apache1	0.024	0.023	0.081	0.273	0.079	0.127
29	inla_graph_bern_spd	0.029	0.044	0.126	0.511	0.306	0.775
30	bmw7st_1	0.030	0.033	0.126	0.394	0.460	0.175
31	inla_graph_83o4NNNo	0.031	0.0082	0.053	0.120	0.096	0.023
32	inla_graph_stcov	0.038	0.126	0.827	1.03	1.000	1.39
33	inla_graph_spacetime	0.047	0.089	0.133	0.188	0.143	0.208
34	nd3k	0.051	0.081	0.181	0.165	0.281	0.375
35	pwtk	0.052	0.055	0.260	0.672	0.673	0.139
36	inla_graph_lidense	0.056	0.111	0.377	0.496	1.55	1.15
37	crankseg_1	0.062	0.057	0.247	0.395	0.695	0.158
38	bmw3_2	0.074	0.070	0.232	0.685	0.732	0.320
39	boneS01	0.084	0.083	0.227	0.533	0.546	0.552
40	crankseg_2	0.086	0.081	0.295	0.524	0.921	0.230
41	tmt_sym	0.087	0.089	0.254	1.20	0.453	0.176
42	ecology2	0.107	0.128	0.326	1.53	0.485	0.359
43	af_shell3	0.160	0.152	0.432	1.37	1.19	0.409
44	consph	0.188	0.206	0.512	0.760	0.719	1.81
45	nd6k	0.208	0.312	0.475	0.486	0.679	2.86
46	inline_1	0.297	0.340	0.826	2.31	2.54	0.481
47	inla_graph_yU0G1u	0.298	0.284	0.712	1.55	311	—
48	inla_graph_net1628760	0.362	0.229	0.985	3.70	3.52	0.819
49	boneS10	0.608	0.629	1.27	3.69	3.92	1.13
50	inla_graph_lgm_48600_bw2	0.723	9.43	4.34	2.36	8.10	9.45
51	nd12k	0.891	1.47	1.25	1.48	1.83	9.27
52	inla_graph_lgm_100200_bw1	1.13	2.48	1.98	2.22	4.65	1.85
53	inla_graph_lgm_50000_bw15000	1.44	5.32	2.59	5.30	2.81	28.9
54	inla_graph_lgm_100200_bw2	2.38	7.63	4.58	3.54	6.73	8.34
55	bone010	6.44	7.88	7.03	11.8	9.56	21.8
56	audikw_1	6.67	12.2	9.18	14.7	10.5	47.3
57	Fault_639	7.88	17.1	10.1	12.4	10.9	110
58	inla_graph_lgm_50400_bw2	11.9	37.1	33.5	7.16	14.9	76.7
59	Emilia_923	13.3	24.8	15.7	19.9	19.8	115
60	inla_graph_animal2	16.6	209	35.7	25.0	72.1	432

fastest solver on that matrix · — marks a matrix a solver does not factor

Best factorization time in seconds on the Intel node (minimum over the swept core counts) for every matrix and solver, sorted by sTiles' time to match the figure above. The fastest solver on each matrix is highlighted.

Speedup of sTiles over PARDISO

Per-matrix speedup of sTiles over PARDISO against matrix cost (both axes logarithmic; a value above the 1× line means sTiles is faster). sTiles is faster on 45 matrices and slower on 15, with a geometric-mean ratio of 1.50×. The two outcomes are asymmetric: every loss is on an inexpensive matrix, all 15 under 0.15 s and the largest margin 0.133 s, while the wins are on the expensive matrices and reach minutes.

Parallel efficiency to 40 cores

Parallel efficiency (speedup over one core divided by the core count; 1.0 is ideal) of a single factorization on three large finite-element matrices, sTiles (solid) against PARDISO (dashed). sTiles holds above 0.85 through 16 cores, where PARDISO has already fallen to 0.5–0.6; both drop sharply past 32 cores (shaded band), the socket's memory-bandwidth ceiling for sparse Cholesky.