Retrieving the variable coefficient for a nonlinear convection–diffusion problem with spectral conjugate gradient method

Figures & data

Figure 1. The flow chart of L-BFGS method.

Figure 2. The flow chart for the whole computational algorithm of SCG.

Figure 3. The recovered results based on the observation data-set H.

Figure 4. The recovered results based on the observation data-set H with noise.

Figure 5. The recovered results based on the observation data-set $\overline{H}$.

Figure 6. The recovered results based on the observation data-set $\overline{\overline{H}}$.

Figure 7. The recovered results based on the observation data-set $\overline{H}$ with noise.

Figure 8. The recovered results based on the observation data-set $\overline{\overline{H}}$ with noise.

Table 1. Effects of different levels of noise on the retrieval result using the SCG and L-BFGS method.

Data			Method
			NoR-SCG			AdR-SCG			AdR-LBFGS
			Iter.	CPU/s	Error	Iter.	CPU/s	Error	Iter.	CPU/s	Error
H	Without noise		800	25.1682	1.1056E-6	800	23.8581	1.7861E-6	800	15.0800	6.6000E-3
	Noise level	0.01	800	25.4186	2.1359E-2	800	24.4783	1.9296E-2	221	5.2741	5.8500E-2
		0.03	800	25.2125	6.2176E-2	800	24.5845	5.3894E-2	265	10.0257	6.7413E-2
		0.05	800	25.6903	7.7555E-2	800	23.9358	7.6143E-2	63	1.9440	8.5066E-2
		0.07	800	25.9120	9.4606E-2	800	23.9685	8.7593E-2	36	0.8866	1.1528E-1
		0.08	800	26.2956	1.1579E-1	800	23.9792	1.1488E-1	25	0.5013	1.3307E-1
$\overline{H}$	Without noise		800	25.1314	5.9000E-3	800	24.5823	5.5000E-3	800	23.0898	3.4000E-2
	Noise level	0.01	800	24.8626	5.8993E-2	800	23.2076	4.6337E-2	70	2.1387	9.1872E-2
		0.02	800	24.1536	1.0504E-1	800	23.8342	1.0617E-1	102	3.1797	9.3933E-2
		0.03	800	24.8353	1.9087E-1	800	22.9491	1.7027E-1	57	1.6880	9.6937E-2
$\overline{\overline{H}}$	Without noise		800	24.8658	4.1000E-2	800	24.3530	3.9500E-2	42	1.5728	1.4300E-1
	Noise level	0.007	800	24.8980	8.5541E-2	800	24.4547	8.7775E-2	30	1.1596	1.2367E-1
		0.009	800	24.3269	9.8685E-2	800	23.9578	9.8870E-2	8	0.8236	1.8503E-1
		0.01	800	24.2218	9.9909E-2	800	25.1796	9.9911E-2	0	0.0560	2.2384E-1

Note: Iter. is short for iteration and the CPU time is obtained with the Core i3-2100 CPU(3.10 GHz) and Windows XP operating system.

Figure 9. The recovered results based on the observation data-set H_T.

Figure 10. The recovered results based on the observation data-set H_T with noise level 0.01.

Figure 11. The recovered results based on the observation data-set ${\overline{H}}_T$.

Figure 12. The recovered results based on the observation data-set ${\overline{H}}_T$ with noise level 0.01.

Figure 13. Retrieval results with constant function (1).

Figure 14. Retrieval results with sinusoidal-like function (2).

Figure 15. Retrieval results with step function (3).

Figure 16. Retrieval results with triangle function (4).

Figure 17. Retrieval results with constant function (1).

Figure 18. Retrieval results with sinusoidal-like function (2).

Figure 19. Retrieval results with step function (3).

Figure 20. Retrieval results with triangle function (4).