## 4 Conclusions  
This research investigated many practical aspects in using the Markov chain approximation to solve photon multiple scattering through turbid slabs. Although uniform OD discretization and zenith angle discretization have already ensured an accurate prediction of the transmitted and reflected angular distribution, it has been shown in this work that non-uniform discretization can be utilized in Markov chain approximations to model multiple scattering, and the improvement in simulation performance was easily seen.  
A modified matrix inversion scheme was also proposed in this work. By converting the matrix inversion process into an iterative process, the computational cost of solving a Markov chain inversion process was reduced from 180 s to 10 s. By pre-processing the computational tasks that are dependent only on the phase function in the specific layer and incorporating the iterative process, the total computational cost was successfully reduced from 8 min to about 30 s, which will greatly contribute to the development of reconstruction/inversion algorithms.  
Overall, with both the non-uniform discretization scheme and the iterative process, the capacity of the proposed Markov chain method was improved remarkably, both in the aspects of accuracy and computational cost. The authors believed that proper OD discretization and zenith angle discretization were crucial for improving the performance of the proposed Markov chain method by adopting  
Figure 13: Absolute relative error of Markov chain angular distribution predictions compared to Monte Carlo, with OD discretization scheme 1 and modified Markov Inversion. Panel (**a**). Phase function 1, OD = 2; Panel (**b**). Phase function 1, OD = 5; Panel (**c**). Phase function 2, OD = 2; Panel (**d**). Phase function 2, OD = 5; Panel (**e**). Phase function 3, OD = 2; Panel (**f**). Phase function 3, OD = 5. Red dash lines indicate an error threshold of 1.0%, and blue dashed lines indicate a threshold of 0.1%.  
finer meshes or coarser meshes at locations with different levels of importance based on practical needs. As such, the computational cost is reduced with less total states and a similar level of accuracy. The iterative process further reduces the computational cost of Markov chain evaluations, and it enables Markov chain approximations with a much higher number of states. These methods in total open the door to accurate and computationally affordable inversion methods by reducing the computational costs of a single cost function evaluation. Such methods would lead to further improved diagnostics applications, such as spray diagnostics in the dense regime and bio-tissue characterization, which will be incorporated in future works.  
Conceptualization, X.L. and Z.M.; Methodology, X.L., Z.M., S.Y. and D.X.; Data curation, X.L. and S.Y.; Writing--original draft preparation, X.L. and S.Y.; Writing--review and editing, Z.M., D.X.; supervision, X.L.; project administration, Z.M.  
This research was funded by Shanghai Pujiang Program (grant number No. 18 PJ1406200).  
This work was carried out at the National Engineering Laboratory for Automotive Electronic Control Technology, Shanghai Jiao Tong University.  
The authors declare no conflict of interest.