Reconstruction of multiplicative space- and time-dependent sources

Figures & data

Table 1. The RMSE (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $u(0,t),u(0.1,t),\mathit{\beta }(t)$ and $\mathit{\mu }(x)$, obtained using the BEM for the direct problem with $N=N_0\in \{10,20,40\}$, for Example 1.

$N=N_0$	u(0, t)	u(0.1, t)	$\mathit{\beta }(t)$	$\mathit{\mu }(x)$
10	5.01E$-$3	5.01E$-$3	5.64E$-$3	8.51E$-$2
20	1.03E$-$3	1.03E$-$3	1.75E$-$3	4.51E$-$2
40	8.17E$-$4	8.17E$-$4	9.69E$-$4	2.30E$-$2

Figure 1. The analytical (—–) and numerical results for (a) $\mathit{\beta }(t)$ and (b) $\mathit{\mu }(x)$ obtained using the BEM for the direct problem with $N=N_0\in \{10(-·-),20(\cdots ),40(---)\}$, for Example 1.

Figure 2. (a) The objective function ${\mathbb{F}}_0$ and the numerical results for (b) r(t), (c) s(x), (d) u(0, t), (e) u(0.1, t) obtained with no regularization $(-·-)$, for exact data for Example 1. The corresponding analytical solutions are shown by continuous line (—–) in (b)–(e) and the $p_0=100\%$ perturbed initial guesses are shown by dotted line $(\cdots )$ in (b) and (c).

Figure 3. The numerical results for (a) r(t), (b) s(x), (c) u(0, t), (d) u(0.1, t) obtained with the first-order regularization $(\cdots )$ and the second-order regularization $(---)$ with regularization parameter $\mathit{\lambda }={10}^{-5}$, for exact data for Example 1. The corresponding analytical solutions are shown by continuous line (—–).

Table 2. The RMSE (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $r(t),s(x),u(0,t),u(0.1,t)$ for exact data for Example 1.

$p_0$	$\mathit{\lambda }$	r(t)	s(x)	u(0, t)	u(0.1, t)
40%	0	6.349	18.47	4.72E$-$2	3.27E$-$2
	${10}^{-5}$	1.528	0.819	1.49E$-$2	1.53E$-$2
60%	0	9.752	26.70	1.14E$-$1	8.12E$-$2
	${10}^{-5}$	1.513	0.767	1.48E$-$2	1.50E$-$2
80%	0	25.83	44.93	1.99E$-$1	2.74E$-$1
	${10}^{-5}$	1.526	0.812	1.48E$-$2	1.47E$-$2
100%	0	53.70	54.24	2.34E$-$1	2.54E$-$1
	${10}^{-5}$	1.529	0.819	1.47E$-$2	1.48E$-$2

Figure 4. (a) The L-curve criterion, (b) the objective function ${\mathbb{F}}_{\mathit{\lambda }}$, and the numerical results $(-\circ -)$for (c) r(t), (d) s(x), (e) u(0, t), (f) u(0.1, t) obtained with the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }={10}^{-5}$ suggested by L-curve, for exact data for Example 1. The corresponding analytical solutions are shown by continuous line (—–) in (c)–(f).

Figure 5. (a) The objective function ${\mathbb{F}}_{\mathit{\lambda }}$ and the numerical results for (b) r(t), (c) s(x), (d) u(0, t), (e) u(0.1, t) obtained with the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }={10}^{-5}$ for $P\in \{1(-·-),3(\cdots ),5(---)\}\%$ noisy data for Example 1. The corresponding analytical solutions are shown by continuous line (—–) in (b)–(e).

Figure 6. The analytical (—–) and numerical results for (a) $\mathit{\beta }(t)$ and (b) $\mathit{\mu }(x)$ obtained using the BEM for the direct problem with $N=N_0\in \{5(-·-),10(\cdots ),20(---)\}$, for Example 2.

Table 3. The RMSEs (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for r(t) and s(x), for the noise levels $\mathit{\delta }\in \{0,0.01,0.1\}$, for Example 2.

Noise level	$\mathit{\lambda }$	No. of iterations	r(t)	s(x)
No noise	0	56	6.3068E–2	1.5492E–1
	0	31 (fixed)	2.6668E–2	1.4751E–1
	2E–4	28	5.6471E–2	1.1003E–1
	2E–4	23 (fixed)	1.9713E–2	6.4412E–2
$\mathit{\delta }=0.01$	4E–4	27	6.3004E–2	1.0886E–1
	4E–4	21 (fixed)	2.8829E–2	6.5281E–2
$\mathit{\delta }=0.1$	2	17	5.2212E–2	3.0714E–2

Figure 7. (a) The objective function ${\mathbb{F}}_0$, (b) the RMSEs (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $r(t)(---)$ and $s(x)(\cdots )$ obtained with no regularization for exact data, and the numerical results for (c) r(t) and (d) s(x) obtained using the minimization process after 56 unfixed iterations $(-·-)$, and 31 fixed iterations $(\circ \circ \circ )$, for Example 2. The corresponding analytical solutions (5.18(5.18) $$\begin{array}{c}\hfill r(t)=e^{3t},s(x)=4cos(x),t\in (0,0.3),x\in (0,\mathit{\pi }),\end{array}$$(5.18) ) are shown by continuous line (—–) in (c) and (d).

Figure 8. (a) The objective function ${\mathbb{F}}_{\mathit{\lambda }}$, (b) the RMSEs (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $r(t)(---)$ and $s(x)(\cdots )$ obtained using the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }=2\times {10}^{-4}$ for exact data, and the numerical results for (c) r(t) and (d) s(x) obtained using minimization process after 28 unfixed iterations $(-·-)$, and 23 fixed iterations $(\circ \circ \circ )$, for Example 2. The corresponding analytical solutions (5.18(5.18) $$\begin{array}{c}\hfill r(t)=e^{3t},s(x)=4cos(x),t\in (0,0.3),x\in (0,\mathit{\pi }),\end{array}$$(5.18) ) are shown by continuous line (—–) in (c) and (d).

Figure 9. (a) The objective function $F_{\mathit{\lambda }}$, (b) the RMSEs (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $r(t)(---)$ and $s(x)(\cdots )$ obtained using the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }=4\times {10}^{-4}$ for noise level $\mathit{\delta }=0.01$, and the numerical results for (c) r(t) and (d) s(x) obtained using the minimization process after 27 unfixed iterations $(-·-)$, and 21 fixed iterations $(\circ \circ \circ )$, for Example 2. The corresponding analytical solutions (5.18(5.18) $$\begin{array}{c}\hfill r(t)=e^{3t},s(x)=4cos(x),t\in (0,0.3),x\in (0,\mathit{\pi }),\end{array}$$(5.18) ) are shown by continuous line (—–) in (c) and (d).

Figure 10. (a) The objective function $F_{\mathit{\lambda }}$, (b) the RMSEs (5.1(5.1) $$\begin{array}{cc}\hfill {\text{RMSE}}_t& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(\mathrm{Exact}({\tilde{t}}_i)-\mathrm{Numerical}({\tilde{t}}_i)\right)}^2}},\hfill \end{array}$$(5.1) ) and (5.2(5.2) $$\begin{array}{cc}\hfill {\text{RMSE}}_x& =\sqrt{{\displaystyle {\displaystyle \frac{1}{N_0}}\sum _{k=1}^{N_0}{\left(\mathrm{Exact}({\tilde{x}}_k)-\mathrm{Numerical}({\tilde{x}}_k)\right)}^2}}.\hfill \end{array}$$(5.2) ) for $r(t)(---)$ and $s(x)(\cdots )$ obtained using the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }=2$ for noise level $\mathit{\delta }=0.1$, and the numerical results $(-·-)$ for (c) r(t) and (d) s(x) obtained using the minimization process after 17 (unfixed) iterations, for Example 2. The corresponding analytical solutions (5.18(5.18) $$\begin{array}{c}\hfill r(t)=e^{3t},s(x)=4cos(x),t\in (0,0.3),x\in (0,\mathit{\pi }),\end{array}$$(5.18) ) are shown by continuous line (—–) in (c) and (d).

Figure 11. The numerical results for (a) $\mathit{\beta }(t)$ and (b) $\mathit{\mu }(x)$ obtained using the BEM for the direct problem with $N=N_0\in \{10(-·-),20(\cdots ),40(---)\}$, for Example 3.

Figure 12. (a) The objective function $F_{\mathit{\lambda }}$ and the numerical results for (b) r(t) and (c) s(x) obtained with the hybrid-order regularization (5.12(5.12) $$\begin{array}{cc}\hfill {\mathbb{F}}_{\mathit{\lambda }}(\underline{r},\underline{s})& ={\mathbb{F}}_0(\underline{r},\underline{s})+\mathit{\lambda }({(r_1-r_2)}^2+{(-r_{N-1}+r_N)}^2+\sum _{i=2}^{N-1}{(-r_{i+1}+2r_i-r_{i-1})}^2\hfill \\ \hfill & +{(s_1-s_2)}^2+{(-s_{N_0-1}+s_{N_0})}^2+\sum _{k=2}^{N_0-1}{(-s_{k+1}+2s_k-s_{k-1})}^2).\hfill \end{array}$$(5.12) ) with regularization parameter $\mathit{\lambda }=2\times {10}^{-4}$ for $P=1\%(-·-),P=5\%(\cdots )$ and $P=10\%(---)$ noisy data for Example 3. The corresponding analytical solutions (5.21(5.21) $$\begin{array}{c}\hfill r(t)=\left\{\begin{array}{cc}t,\hfill & 0\le t\le 1/2\hfill \\ 1-t,\hfill & 1/2<t\le 1=T\hfill \end{array}\right.,s(x)=\left\{\begin{array}{cc}x,\hfill & 0\le x\le 1/20\hfill \\ 0.1-x,\hfill & 1/20<x\le 1/10=L\hfill \end{array}\right.,\end{array}$$(5.21) ) are shown by continuous line (—–) in (b) and (c).