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.

88 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 88-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 tracks sTiles closely, ahead on the inexpensive matrices and behind on the expensive ones; MUMPS, PaStiX, CHOLMOD, and symPACK are slower across most of the suite.

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

#	matrix	sTiles	PARDISO	MUMPS	CHOLMOD	PaStiX	symPACK
1	bcsstk10	0.0002	0.0002	0.0012	0.0007	0.0015	0.0043
2	1138_bus	0.0004	0.0001	0.0008	0.0001	0.0005	0.020
3	inla_graph_sem_n2000	0.0005	0.0015	0.0069	0.0015	0.0054	0.0095
4	bcsstk11	0.0008	0.0003	0.0020	0.0016	0.0025	0.0059
5	inla_graph_sem_n5000	0.0008	0.0035	0.015	0.0041	0.015	0.020
6	bcsstk09	0.0009	0.0003	0.0019	0.0017	0.0012	0.0056
7	inla_graph_8rtKSK	0.0010	0.0011	0.0058	0.0009	0.078	3.00
8	gyro_m	0.0013	0.0012	0.013	0.015	0.023	0.154
9	bcsstk08	0.0014	0.0004	0.0016	0.0020	0.0011	0.0076
10	bcsstk14	0.0014	0.0005	0.0034	0.0028	0.0038	0.0053
11	nasa2910	0.0014	0.0009	0.0054	0.0047	0.010	0.0076
12	nasa4704	0.0024	0.0012	0.0058	0.0069	0.0082	0.0097
13	inla_graph_sh7Pgi	0.0024	0.0020	0.0086	0.0038	0.0093	23.0
14	gyro_k	0.0027	0.0022	0.025	0.031	0.057	0.014
15	inla_graph_pedigree	0.0029	0.0070	0.082	0.0056	0.041	—
16	inla_graph_sem_n20000	0.0031	0.013	0.054	0.021	0.063	0.080
17	bcsstk15	0.0032	0.0015	0.0096	0.010	0.0078	0.013
18	inla_graph_diff	0.0032	0.0051	0.018	0.048	0.029	0.034
19	bundle1	0.0033	0.0028	0.029	0.023	0.059	0.051
20	msc10848	0.0036	0.0039	0.022	0.033	0.071	0.017
21	bcsstk18	0.0036	0.0023	0.010	0.021	0.012	0.021
22	inla_graph_net814381	0.0038	0.0063	0.032	0.013	0.031	1.21
23	inla_graph_ayaLRw	0.0038	0.0059	0.027	0.014	0.029	0.228
24	bcsstk17	0.0039	0.0024	0.015	0.020	0.026	0.013
25	bcsstk13	0.0042	0.0015	0.0068	0.0049	0.0064	0.010
26	msc23052	0.0047	0.0050	0.032	0.051	0.072	0.021
27	bcsstk16	0.0048	0.0016	0.017	0.012	0.018	0.012
28	thermal1	0.0052	0.0068	0.036	0.111	0.043	0.021
29	thermomech_TK	0.011	0.0092	0.056	0.140	0.060	0.021
30	thermomech_TC	0.011	0.0093	0.060	0.141	0.061	0.021
31	nasasrb	0.012	0.014	0.061	0.172	0.163	0.037
32	oilpan	0.013	0.012	0.065	0.160	0.219	0.050
33	inla_graph_sem_n100000	0.014	0.068	0.276	0.162	0.353	0.400
34	thermomech_dM	0.015	0.017	0.090	0.287	0.125	0.029
35	inla_graph_lgm_10010_bw2	0.015	0.012	0.036	0.044	0.048	0.023
36	ct20stif	0.018	0.020	0.073	0.152	0.191	0.069
37	inla_graph_ferris	0.021	0.012	0.036	0.143	0.116	0.087
38	s3dkt3m2	0.021	0.021	0.104	0.231	0.248	0.103
39	s3dkq4m2	0.022	0.021	0.111	0.247	0.291	0.101
40	inla_graph_animal1	0.023	0.181	0.133	0.110	0.391	0.613
41	apache1	0.024	0.023	0.081	0.273	0.079	0.127
42	smt	0.027	0.021	0.092	0.140	0.253	0.095
43	G2_circuit	0.029	0.019	0.066	0.276	0.080	0.098
44	inla_graph_bern_spd	0.029	0.044	0.126	0.511	0.306	0.775
45	bmw7st_1	0.035	0.034	0.126	0.394	0.460	0.175
46	inla_graph_83o4NNNo	0.037	0.0081	0.053	0.120	0.096	0.023
47	inla_graph_stcov	0.038	0.126	0.827	1.03	1.000	1.39
48	hood	0.046	0.030	0.168	0.490	0.682	0.099
49	inla_graph_spacetime	0.047	0.089	0.133	0.188	0.143	0.208
50	inla_graph_lidense	0.056	0.111	0.377	0.496	1.55	1.15
51	crankseg_1	0.060	0.057	0.247	0.395	0.695	0.158
52	m_t1	0.062	0.047	0.192	0.421	0.592	0.150
53	pwtk	0.065	0.055	0.260	0.672	0.673	0.139
54	thread	0.073	0.057	0.192	0.272	0.381	0.425
55	parabolic_fem	0.075	0.058	0.209	0.908	0.392	0.107
56	bmw3_2	0.083	0.074	0.232	0.685	0.732	0.320
57	tmt_sym	0.087	0.089	0.254	1.20	0.453	0.176
58	ship_001	0.088	0.022	0.110	0.183	0.275	0.071
59	boneS01	0.092	0.082	0.227	0.533	0.546	0.552
60	crankseg_2	0.098	0.080	0.295	0.524	0.921	0.230
61	shipsec1	0.107	0.072	0.214	0.494	0.565	1.25
62	ecology2	0.107	0.128	0.326	1.53	0.485	0.359
63	shipsec8	0.114	0.078	0.202	0.437	0.496	0.551
64	nd3k	0.129	0.080	0.181	0.165	0.281	0.375
65	shipsec5	0.173	0.115	0.309	0.759	0.714	0.488
66	af_shell3	0.181	0.147	0.432	1.37	1.19	0.409
67	ship_003	0.193	0.147	0.366	0.663	0.616	0.809
68	consph	0.244	0.203	0.512	0.760	0.719	1.81
69	ldoor	0.245	0.202	0.729	2.43	3.08	1.09
70	af_0_k101	0.290	0.141	0.467	1.42	1.21	0.412
71	inla_graph_yU0G1u	0.295	0.286	0.712	1.55	311	—
72	inline_1	0.297	0.340	0.826	2.31	2.54	0.481
73	nd6k	0.312	0.301	0.475	0.486	0.679	2.86
74	inla_graph_net1628760	0.387	0.220	0.985	3.70	3.52	0.819
75	apache2	0.645	0.452	0.801	2.69	0.937	1.71
76	inla_graph_lgm_48600_bw2	0.723	9.43	4.34	2.36	8.10	9.45
77	boneS10	0.762	0.620	1.27	3.69	3.92	1.13
78	offshore	0.784	0.271	0.452	1.14	0.529	0.653
79	nd12k	0.891	1.47	1.25	1.48	1.83	9.27
80	inla_graph_lgm_100200_bw1	1.13	2.48	1.98	2.22	4.65	1.85
81	inla_graph_lgm_50000_bw15000	1.44	5.32	2.59	5.30	2.81	28.9
82	inla_graph_lgm_100200_bw2	2.38	7.63	4.58	3.54	6.73	8.34
83	bone010	6.44	7.88	7.03	11.8	9.56	21.8
84	audikw_1	6.67	12.2	9.18	14.7	10.5	47.3
85	Fault_639	7.88	17.1	10.1	12.4	10.9	110
86	inla_graph_lgm_50400_bw2	11.9	37.1	33.5	7.16	14.9	76.7
87	Emilia_923	13.3	24.8	15.7	19.9	19.8	115
88	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). By count the contest is a near-tie: sTiles is slower on 54 matrices and faster on 34, yet the geometric-mean ratio (1.03×) already favors sTiles, because its fewer wins are the larger ones. Every loss is on a cheap matrix (all but one under 0.5 s), while the wins are on the expensive matrices and reach minutes.

Strong scaling to 40 cores

Strong scaling (speedup over one core) of a single factorization on three large finite-element matrices, sTiles (solid) against PARDISO (dashed), with the ideal line. Both flatten near 32 cores (shaded band), the socket's memory-bandwidth ceiling for sparse Cholesky.