Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions

Figures & data

Figure 1. Geometry of the inverse problem under investigation.

Figure 2. The exact (29(29) $\begin{array}{rl}{\nu }_1(t)& =\frac{5}{2}(1+t),{\nu }_2(t)=\frac{1}{4}(18+t)(1+2t).\end{array}$(29) ) and numerical solutions for (a) ${\nu }_1(t)$ and (b) ${\nu }_2(t)$, with $M_1=M_2=10$ and with various numbers of time steps $N\in \{20,40,80\}$, for direct problem.

Table 1. The rmse values ((32(32) $\begin{array}{rl}rmse({\nu }_1)& ={[\frac{1}{N}\sum _{n=1}^N({\nu }_1^{Numerical}(t_n)-{\nu }_1^{Exact}(t_n){)}^2]}^{1/2},\end{array}$(32) ) and (33(33) $\begin{array}{rl}rmse({\nu }_2)& ={[\frac{1}{N}\sum _{n=1}^N({\nu }_2^{Numerical}(t_n)-{\nu }_2^{Exact}(t_n){)}^2]}^{1/2},\end{array}$(33) )) for ${\nu }_1(t)$ and ${\nu }_2(t)$, with $M_1=M_2=10$ and with various $N\in \{20,40,80\}$, for direct problem.

$M_1=M_2=10$	$rmse({\nu }_1)$	$rmse({\nu }_2)$
N = 20	0.0353	0.0479
N = 40	0.0080	0.0073
N = 80	0.0030	0.0026

Figure 3. The objective function (19(19) $F({\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{2}})=\sum _{n=1}^N[a_1^n{u}_x(0,Y_0,t_n)-{\nu }_1(t_n){]}^2+\sum _{n=1}^N[a_2^n{u}_y(X_0,0,t_n)-{\nu }_2(t_n){]}^2,$(19) ), as a function of the number of iterations, for Example 1 with $p\in \{0,1\mathrm{\%},3\mathrm{\%},5\mathrm{\%}\}$ noise.

Figure 4. The exact (31(31) $\begin{array}{rl}{a}_1(t)& =1+t,a_2(t)=1+2t,t\in [0,1].\end{array}$(31) ) and numerical solutions for: (a) $a_1(t)$ and (b) $a_2(t)$, for Example 1 with $p\in \{0,1\mathrm{\%},3\mathrm{\%},5\mathrm{\%}\}$ noise.

Figure 5. The analytical (30(30) $\begin{array}{rl}u(x,y,t)& =1+x+3y+3xy+t{x}^{2}y+\frac{1}{10}t{y}^3,(x,y,t)\in {\overline{Q}}_T,\end{array}$(30) ) and approximate solutions for the temperature $u(x,y,1)$, for Example 1 with (a) no noise, (b) $p=1\mathrm{\%}$ noise, (c) $p=3\mathrm{\%}$ noise, and (d) $p=5\mathrm{\%}$ noise. The absolute error between them is also included.

Table 2. The number of iterations, the values of the objective function (19) at final iteration, the rmse values ((26) and (27)) and the computational time, for $p\in \{0,1,3,5\}\mathrm{\%}$, for Examples 1 and 2.

Example 1	p = 0	$p=1\mathrm{\%}$	$p=3\mathrm{\%}$	$p=5\mathrm{\%}$
Number of iterations	9	9	5	5
Minimum value of (19)	8.6E−26	1.5E−25	1.7E−25	9.5E−26
$rmse(a_1)$	0.0013	0.0279	0.0836	0.1395
$rmse(a_2)$	0.0016	0.0286	0.0860	0.1434
Computational time	8 min	8 min	6 min	6 min
Example 2	p = 0	$p=1\mathrm{\%}$	$p=3\mathrm{\%}$	$p=5\mathrm{\%}$
Number of iterations	10	10	10	7
Minimum value of (19)	6.0E−26	5.5E−26	5.8E−26	1.0E−25
$rmse(a_1)$	0.0068	0.0285	0.0838	0.1396
$rmse(a_2)$	0.0039	0.0193	0.0574	0.0958
Computational time	9 min	9 min	9 min	7 min

Figure 6. The objective function (19(19) $F({\mathbf{a}}_{\mathbf{1}},{\mathbf{a}}_{\mathbf{2}})=\sum _{n=1}^N[a_1^n{u}_x(0,Y_0,t_n)-{\nu }_1(t_n){]}^2+\sum _{n=1}^N[a_2^n{u}_y(X_0,0,t_n)-{\nu }_2(t_n){]}^2,$(19) ), as a function of the number of iterations, for Example 2 with $p\in \{0,1\mathrm{\%},3\mathrm{\%},5\mathrm{\%}\}$ noise.

Figure 7. The exact (35(35) $a_1(t)=1+{\cos }^2(2\pi t),a_2(t)=1+{\cos }^2(3\pi t),t\in [0,1].$(35) ) and numerical solutions for: (a) $a_1(t)$ and (b) $a_2(t)$, for Example 2 with $p\in \{0,1\mathrm{\%},3\mathrm{\%},5\mathrm{\%}\}$ noise.

Figure 8. The analytical (30(30) $\begin{array}{rl}u(x,y,t)& =1+x+3y+3xy+t{x}^{2}y+\frac{1}{10}t{y}^3,(x,y,t)\in {\overline{Q}}_T,\end{array}$(30) ) and approximate solutions for the temperature $u(x,y,1)$, for Example 2 with (a) no noise, (b) $p=1\mathrm{\%}$ noise, (c) $p=3\mathrm{\%}$ noise, and (d) $p=5\mathrm{\%}$ noise. The absolute error between them is also included.