## 3 Results  
### Determination of Optical Properties  
Here we consider reconstructions from measured data in the phantom experiment. Let us first consider initial guesses below. \[\text{Case 1:}\qquad\mu_{a}=0.01\,\text{mm}^{-1},\quad\mu_{s}^{\prime}=1.0\, \text{mm}^{-1}. \tag{41}\]  
In this Case 1, the following \(\mu_{a},\mu_{s}^{\prime}\) are obtained both by Algorithm 1 (LM) and Algorithm 3 (hybrid). \[\mu_{a}=0.016\,\text{mm}^{-1},\quad\mu_{s}^{\prime}=0.63\,\text{mm}^{-1}. \tag{42}\]  
The results for \(\mu_{a}\) and \(\mu_{s}^{\prime}\) are shown in Figures 3 and 4, respectively. In Figure 3, reconstructed values of \(\mu_{a}\) are plotted against the iteration number \(k\). In the top panel of Figure 3, we see that Algorithm 1 (LM) quickly converges. In Algorithms 2 and 3, we set \(k_{b}=99\). As is seen in the middle panel of Figure 3, the convergence of Algorithm 2 (SA) is slow. The _temperature_ is decreased from \(\sigma_{e}=10^{-6}\) to \(\sigma_{e}=10^{-7}\) at the \(k_{b}\)th Monte Carlo step. Similarly, \(\varepsilon\) is changed from 0.1 to 0.001. Algorithm 2 returns the correct values after many Monte Carlo steps; we have \(\sqrt{\hat{V}/W}=1.02,1.06\) for \(\mu_{a},\mu_{s}^{\prime}\), respectively, when \(k_{a}=9000\) and \(k_{b}=10000\). Finally, in the bottom panel of Figure 3, we see that the iteration of Algorithm 3 (hybrid) immediately converges after we switch from the Metropolis-Hastings MCMC (\(\sigma_{e}=10^{-6}\), \(\varepsilon=0.1\)) to Algorithm 1 (LM). No convergence of the Monte Carlo chain is required, and we found \(\sqrt{\hat{V}/W}=8.6\) for \(\mu_{a}\) and \(=11.0\) for \(\mu_{s}^{\prime}\) (\(k_{a}=49\), \(k_{b}=99\)). A similar behavior is observed for the reconstruction of \(\mu_{s}^{\prime}\) in Figure 4. In the top panel of Figure 4, we find that Algorithm 1 (LM) works best  
Figure 2: The function \(f_{a}(x)\) in Equation (36) is plotted for \(a=1.3\), 1.5, and 1.7.  
and converges after a few iterations, whereas Algorithm 2 (SA) in the middle panel of Figure 4 has slow convergence and reconstructed values around at \(k_{b}=99\) are still away from \(\mu_{s}^{\prime}\) in (42). By switching from MH-MCMC to Algorithm 1 (LM) using Algorithm 3 (hybrid), convergence is easily obtained as shown in the bottom panel of Figure 4.  
Now we start the simulation by setting the following initial values. \[\text{Case 2:}\qquad\mu_{a}=0.5\,\text{mm}^{-1},\quad\mu_{s}^{\prime}=1.0\, \text{mm}^{-1}. \tag{43}\]  
In this Case 2, Algorithm 1 (LM) fails and returns \(\mu_{a}=0.068\,\text{mm}^{-1}\) and \(\mu_{s}^{\prime}=1.75\,\text{mm}^{-1}\) whereas Algorithm 3 (hybrid) still gives the correct values. The reconstructed values of \(\mu_{a}\) and \(\mu_{s}^{\prime}\) at each iteration are shown in Figures 5 and 6, respectively. In Figure 5, the vertical axes of the left three panels are from 0 to 0.6 whereas the vertical axes of the right three panels are from 0 to 0.08. In Figure 6, the vertical axes of the left three panels are from 0 to 2 while the vertical axes of the right three panels are from 0 to 1.3. In the top panels of Figure 5, we see that the reconstruction of \(\mu_{a}\) is unsuccessful by Algorithm 1 (LM). Algorithm 2 (SA) approaches \(\mu_{a}\) in Equation (42) but still deviates from that value in the middle panels of Figure 5. As shown in the bottom panels of Figure 5, Algorithm 3 (hybrid) converges in a few iterations after switching to Algorithm 1 (LM). In the top panels of Figure 6, Algorithm 1 (LM) fails to reconstruct \(\mu_{s}^{\prime}\). In the middle panels of Figure 6, reconstructed values by Algorithm 2 (SA) come close to \(\mu_{s}^{\prime}\) in Equation (42) but suffer from slow convergence. In the bottom panels of Figure 6, we see that Algorithm 3 (hybrid) arrives at \(\mu_{s}^{\prime}\) in Equation (42) without any problem. Figure 4: Case 1: Reconstructed \(\mu_{s}^{\prime}\) by (top) Algorithm 1, (middle) Algorithm 2, and (bottom) Algorithm 3. The dashed lines show \(\mu_{s}^{\prime}\) in Equation (42).