Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

Figures & data

Figure 1. Exact and reconstruction $\mathit{\gamma }(t)$ (left) and Q(t) (right) using CGM and COMSOL for Example 6.1 for $\mathit{\varrho }=1\%$ noise in the data.

Figure 2. Exact and reconstructed $\mathit{\gamma }(t)$ (left) and Q(t) (right) using the modified CGM and COMSOL for Example 6.2 for $\mathit{\varrho }=1\%$ noise in the data.

Figure 3. Exact and reconstructed $\mathit{\gamma }(t)$ (left) and Q(t) (right) using the modified CGM and COMSOL for Example 6.3 for $\mathit{\varrho }=1\%$ noise in the data.

Figure 4. Convergence of the relative error of the reconstruction $\mathit{\gamma }$ and Q (left) and the residual accuracy error of $\mathit{\gamma }$ and Q (right) by Algorithm 5.1 for Example 6.3 with the number of iterations k.

Figure 5. Exact and numerical reconstruction of heat flux (left) and Robin coefficient (right) by Algorithm 5.1 for Example 6.1 for various levels of noise $\mathit{\varrho }$ in the data.

Table 1. The numerical results with respect to $\mathit{\gamma }$ and Q for $\mathit{\varrho }=1\%$ noise in the data using the modified CGM.

Example	$\mathrm{\Delta }t$	k	$\mathcal{R}{E}_{\mathit{\gamma }}$	$\mathcal{R}{E}_Q$	$({\mathit{\gamma }}^0,Q^0)$	Elapsed time in seconds
6.1	1/22	11	0.0264	0.0171	(3, 1.2)	9.36
6.2	1/20	11	0.0201	0.0102	(3.2, 3.2)	19.27
6.3	1/30	10	0.0112	0.0211	(3, 3)	22.30

Table 2. The numerical results with respect to $\mathit{\gamma }$ and Q for $\mathit{\varrho }=1\%$ noise in the data using COMSOL.

Example	$\mathrm{\Delta }t$	k	$\mathcal{R}{E}_{\mathit{\gamma }}$	$\mathcal{R}{E}_Q$	$({\mathit{\gamma }}^0,Q^0)$	Elapsed time in seconds
6.1	1/22	59	0.0184	0.0097	(3, 1.2)	64.33
6.2	1/20	51	0.0280	0.0063	(3.2, 3.2)	52.32
6.3	1/30	73	0.0153	0.0158	(3, 3)	75.35

Table 3. The numerical results (accuracy error) for Example 6.1 for various levels of noise $\mathit{\varrho }$.

$\mathcal{R}{E}_{\mathit{\gamma }}$	$\mathcal{R}{E}_Q$	$\mathit{\varrho }$	k	$J(\mathit{\gamma },Q)$
0.0401	0.0240	0	10	0.0088
0.0264	0.0171	$1\%$	11	0.0386
0.0508	0.0269	$2\%$	7	0.1512
0.0440	0.0318	$3\%$	6	0.3191
0.0180	0.0321	$4\%$	6	0.5389
0.0160	0.0329	$5\%$	6	0.8206