HLRF-BFGS-Based Algorithm for Inverse Reliability Analysis

Table 3

Comparison between inverse-FORM, the intermediate algorithm, and the BFGS-based hybrid algorithm.

Problem		True solution; (from forward analysis)	Inverse-FORM	BFGS-based algorithm
				Intermediate algorithm	BFGS-based hybrid algorithm
1a	Number of iterations	—	8	6	7
		0	0	0	0
		0.3671	0.3669	0.3535	0.3668
		2.0000	2.0002	2.0136	2.0003
1b	Number of iterations	—	53	4	42
		0	0	0.0059	0
		0.3671	0.3666	0.3079	0.3666
		2.0000	2.0005	2.0538	2.0005
1c	Number of iterations	—	62	5	30
		0	0	0.006	0
		0.3671	0.3666	0.3525	0.3666
		2.0000	2.0005	2.0082	2.0005
2a	Number of iterations	—	10	7	7
		0	0	0.0038	0.0002
		0.4291	0.4282	0.38427	0.4287
		2.0000	2.0007	2.032	2.0003
2b	Number of iterations	—	53	4	42
		0	0	0.0054	0
		0.4291	0.4284	0.3623	0.4285
		2.0000	2.0005	2.0519	2.0005
2c	Number of iterations	—	56	4	37
		0	0	0.0088	0
		0.4291	0.4284	0.3834	0.4284
		2.0000	2.0005	2.0298	2.0005
3	Number of iterations	—	4	3	3
		0	0.4211	0.0730	0.0730
		2.08	0.6986	2.0030	2.0030
		2.0000	2.0003	1.9875	1.9875
4	Number of iterations	—	3	4	3
		0	0.2535	0.0522	0.0550
		15.2593	13.3816	14.8710	14.8457
		2.0000	1.9998	1.9962	1.9861
5	Number of iterations	—	2	2	2
		0	0	0	0
		2.5	2.5	2.5	2.5
		2.5	2.5	2.5	2.5
6a	Number of iterations	—	20	6	19
		0	0	0.0998	0
		1.3838	1.3846	1.4905	1.3847
		2.0000	2.0007	2.0063	2.0008
6b	Number of iterations	—	8	4	7
		0	0	−0.0657	0
		1.3838	1.3838	1.4252	1.3839
		1.7346	1.7346	1.8274	1.7347
7	Number of iterations	—	17	6	7
		0	0	0.0937	0
		2.0000	2.0012	2.1017	2.0003
		1.5105	1.5115	1.5172	1.5107
8	Number of iterations	—	10	6	10
		0	0	0.0264	0
		1.3992	1.4000	1.4258	1.3999
		2.0000	2.0007	2.0001	2.0007
9a	Number of iterations	—	61	7	57
		0	0	−0.0680	0
		1.0	1.0007	1.0424	1.0007
		1.677	1.6776	1.7787	1.6776
9b	Number of iterations	—	9	5	9
		0	0	−0.0041	0
		1.0	1.0002	1.0333	1.0001
		1.4671	1.4672	1.4998	1.4672
10	Number of iterations	—	16	4	16
		0	0	0.0137	0
		6.0	5.9946	4.7864	5.9946
		2.0	1.9993	1.8610	1.9993
11	Number of iterations	—	78	4	69
		0	0	0.1934	0
		2.3584	2.3570	2.2964	2.3571
		1.4	1.3995	1.2382	1.3995
12	Number of iterations	—	23	6	13
		0	0	0.0752	0
		1.5551	1.5561	1.6531	1.5560
		1.0	1.0008	1.0174	1.0007
13a	Number of iterations	—	34	6	33
		0	0	0.0057	0
		0.3438	0.3443	0.3648	0.3443
		1.0	1.0004	1.0133	1.0004
13b	Number of iterations	—	12	5	11
		0	0	0.0161	0
		0.3438	0.3440	0.4305	0.3440
		0.7901	0.7903	0.8477	0.7902
14	Number of iterations	—	14	4	14
		0	0	0.0928	0
		1.922	1.9210	1.7578	1.9210
		1.0	0.9997	0.9072	0.9997
15	Number of iterations	—	119	4	108
		0	0	0.1391	0
		1.6941	1.6933	1.6515	1.6933
		1.0	0.9997	0.8844	0.9997