A fast iterative method for identifying the radiogenic source for the helium production-diffusion equation

Figures & data

Figure 1. Example 1: the relative error $e_k$ versus η with noise level $ϵ=0.01$.

Figure 2. Example 1: (a) the residual $E_k$ versus k ; (b) the relative error $e_k$ versus k.

Figure 3. Example 1: (a)–(c) The reconstructed source function obtained by the original method; (d)–(f) The reconstructed source function obtained by the accelerated method.

Table 1. Numerical comparison on the original method and the accelerated method for Example 3.1.

	Original method			Accelerated method
ϵ	$k^{†}$	$e_{k^{†}}$	CPU time(s)	$k^{\ast }$	$e_{k^{\ast }}$	CPU time(s)
0.0001	680	0.0041	229.34	124	0.0075	41.74
0.0005	371	0.0110	127.59	67	0.0140	23.54
0.001	286	0.0172	100.37	63	0.0250	21.96
0.005	158	0.0513	58.78	42	0.0638	15.11
0.01	118	0.0835	43.93	41	0.1218	15.33
0.02	82	0.1357	31.06	22	0.1333	7.94

Figure 4. Example 2: (a)–(c) The reconstructed source function obtained by the original method; (d)–(f) the reconstructed source function obtained by the accelerated method.

Table 2. Numerical comparison on the original method and the accelerated method for Example 3.2.

	Original method			Accelerated method
ϵ	$k^{†}$	$e_{k^{†}}$	CPU time(s)	$k^{\ast }$	$e_{k^{\ast }}$	CPU time(s)
0.0001	2568	0.0093	826.13	322	0.0154	102.73
0.0005	1487	0.0275	476.02	200	0.0547	63.90
0.001	1173	0.0452	377.27	130	0.0631	41.46
0.005	610	0.1334	199.12	82	0.1551	26.26
0.01	443	0.2006	145.66	61	0.1955	19.33
0.02	316	0.2927	104.66	47	0.2619	14.89

Figure 5. Example 3: (a)–(c) The reconstructed source function obtained by the original method; (d)–(f) the reconstructed source function obtained by the accelerated method.

Table 3. Numerical comparison on the original method and the accelerated method for Example 3.3.

	Original method			Accelerated method
ϵ	$k^{†}$	$e_{k^{†}}$	CPU time(s)	$k^{\ast }$	$e_{k^{\ast }}$	CPU time(s)
0.0001	2234	0.0110	591.22	212	0.0135	55.73
0.0005	966	0.0227	256.39	122	0.0280	31.76
0.001	674	0.0329	177.44	96	0.0402	25.34
0.005	290	0.0770	78.46	59	0.0955	16.07
0.01	214	0.1134	59.08	42	0.1126	11.34
0.02	158	0.1717	44.71	39	0.1911	10.91