Quasi-optimal Tikhonov penalization and parameterization coarseness in space-dependent function estimation

Figures & data

Figure 1. Distance from the solution to the target $E(\mathit{\vartheta })$ as a function of the Tikhonov parameter $\mathit{\vartheta }$, for three control space dimensions $\mathrm{\Xi }$ equal to $7$, $15$ and $20$. It is seen that ${\mathit{\vartheta }}^{\ast }=argminE(\mathit{\vartheta })$ is independent of $\mathrm{\Xi }$. The quasi-optimal Tikhonov parameter found according to the discrepancy principle has also been added.

Table 1. Value of the Tikhonov parameter ${\mathit{\vartheta }}_{\text{dp}}^{\ast }$ solution of (6(6) $$\begin{array}{c}\hfill R({\mathit{\vartheta }}_{\text{dp}}^{\ast })=\sum _{i=1}^k{\left(\frac{{\mathit{\vartheta }}_{\text{dp}}^{\ast }{\breve{\mathit{\xi }}}_i}{{\mathit{\eta }}_i^2+{\mathit{\vartheta }}_{\text{dp}}^{\ast }}\right)}^2-{\mathit{ϵ}}^2=0\end{array}$$(6) ) for different discretizations and different noise magnitudes.

$\mathrm{\Xi }$$\backslash$${\mathit{ϵ}}_l^2$	${10}^{-4}$	${10}^{-3}$	${10}^{-2}$	${10}^{-1}$
$7$	$4.189\times {10}^{-11}$	$4.198\times {10}^{-10}$	$4.297\times {10}^{-9}$	$5.493\times {10}^{-8}$
$10$	$5.246\times {10}^{-11}$	$5.257\times {10}^{-10}$	$5.374\times {10}^{-9}$	$6.622\times {10}^{-8}$
$15$	$4.240\times {10}^{-11}$	$4.248\times {10}^{-10}$	$4.330\times {10}^{-9}$	$5.151\times {10}^{-8}$
$20$	$4.042\times {10}^{-11}$	$4.049\times {10}^{-10}$	$4.113\times {10}^{-9}$	$4.683\times {10}^{-8}$

Figure 2. Reconstructions ${\mathit{\phi }}_{\mathit{\vartheta }}(x_2)$ for $\mathrm{\Xi }=15$ (top), $\mathrm{\Xi }=10$ (middle) and $\mathrm{\Xi }=7$ (bottom) for under-regularization ($\mathit{\vartheta }={10}^{-15}$), over-regularization ($\mathit{\vartheta }=1$) and appropriate regularization ($\mathit{\vartheta }=5{10}^{-10}\approx {\mathit{\vartheta }}^{\ast }$). Note that points for $\mathrm{\Xi }=15$ and $\mathrm{\Xi }=10$ (over-parameterization) with $\mathit{\vartheta }={10}^{-15}$ (under-regularization) are not presented because of divergence. The noise variance ${\mathit{ϵ}}_l^2=0.001$ was used after generating the synthetic data.

Figure 3. Test medium geometry representation. Eight sources are located on the boundary. For each source (which constitutes a test), the emerging radiation is measured on all sensors.

Table 2. Dimensions of finite element spaces for $\mathit{\phi }$, $\breve{\mathit{\phi }}$ and $\mathit{\alpha }$.

	${\mathrm{\Xi }}_{\mathit{\alpha }}=dim{\mathcal{V}}_h^{\mathit{\alpha }}$	$dim{\mathcal{V}}_h^{\mathit{\phi }}$	$dim{\mathcal{V}}_h^{\breve{\mathit{\phi }}}$
Case 1	2924
Case 2	1382	2924	5336
Case 3	384

Table 3. CPU time comparisons for the generation of the error curves $E(\mathit{\vartheta })$.

	Case 1	Case 2	Case 3
CPU time	1	0.38	0.09

Figure 4. Distance from the solution to the target $E(\mathit{\vartheta })$ as a function of the Tikhonov parameter $\mathit{\vartheta }$, for $\left({\mathit{\kappa }}_{ap},{\mathit{\sigma }}_{ap}\right)=\left(0.08,20\right)c{m}^{-1}$ and three control space dimensions ${\mathrm{\Xi }}_{\mathit{\alpha }}$ equal to $384$, $1382$ and $2924$. It is seen that ${\mathit{\vartheta }}^{\ast }=argminE(\mathit{\vartheta })$ is quasi-independent of $\mathrm{\Xi }$.

Figure 5. Reconstruction of $\mathit{\kappa }$ (left) and $\mathit{\sigma }$ (right) with $\mathit{\vartheta }=0.2$, $\left({\mathit{\kappa }}_{ap},{\mathit{\sigma }}_{ap}\right)=\left(0.08,20\right){\mathrm{cm}}^{-1}$ and ${\mathrm{\Xi }}_{\mathit{\alpha }}=2924$. It should be noted the divergence for the reduced diffusion coefficient.

Figure 6. Reconstruction of $\mathit{\kappa }$ (left) and $\mathit{\sigma }$ (right) with $\mathit{\vartheta }=0.4$, $\left({\mathit{\kappa }}_{ap},{\mathit{\sigma }}_{ap}\right)=\left(0.08,20\right){\mathrm{cm}}^{-1}$ and ${\mathrm{\Xi }}_{\mathit{\alpha }}=2924$.

Figure 7. Reconstruction of $\mathit{\kappa }$ (left) and $\mathit{\sigma }$ (right) with $\mathit{\vartheta }=1.4$, $\left({\mathit{\kappa }}_{ap},{\mathit{\sigma }}_{ap}\right)=\left(0.088,24\right){\mathrm{cm}}^{-1}$ and ${\mathrm{\Xi }}_{\mathit{\alpha }}=2924$.