## 4 Simulation Results
### Influence of Assignment Schemes on Asymptotic Capacity  
Figure 3 shows the performance of asymptotic capacity versus SNR with different assignment schemes when the number of RIS elements \(K\) are constant. We assume \(K=100\) and focus on the influence of assignment schemes in asymptotic capacity. In the figure, \"no-RIS\" means that the channel is organized by LoS links only, and hence, the asymptotic capacity results are contributed by the LoS transmission only. \"RIS-aided\" represents that the transmission consists of the LoS channel and the NLoS channel. The proposed algorithm indicates the RPSO algorithm proposed in this paper, and the distance greedy3 implies that the assignment matrix is determined based on the minimum sum of the distances (LED-RIS-PD), while the distance greedy2 depends on the minimum distance between the LED and RIS, and furthermore, the distance greedy1 is on the basis of the minimum  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline LED half-power semialeg \(\Psi_{1/2}\) & \(70^{\circ}\) \\ \hline Physical area of the PD \(A_{PD}\) & \(1\) cm\({}^{2}\) \\ \hline Optical concentrator gain \(T_{of}\) & \(1\) \\ \hline Field of view \(\psi_{\text{FoV}}\) & \(85^{\circ}\) \\ \hline Refractive index \(a\) & \(1\) \\ \hline Location of the center point of RIS & \((3,0.1,2)\) \\ \hline Location of the LED 1 & \((1.25,1.25,2.85)\) \\ \hline Location of the LED 2 & \((1.25,3.75,2.85)\) \\ \hline Area of the RIS unit & \(10\) cm \(\times\)\(10\) cm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Numerical parameters.  
Figure 2: Convergence analysis for Algorithm 1, \(K=225\), \(A=30\) dB.  
distance between the RIS and PD, the \"Relaxing results\" indicate the results modified from the results calculated by the convex optimization tools according to the relaxing problem, and \"Random selection\" means that a random matrix satisfied with the definition of the assignment matrix is applied to the RIS-aided parallel MIMO system (i.e., each RIS element is assigned to the LED randomly) to determine the scheme of service for RIS.  
For comparing the performance among the above-mentioned schemes, simulations have been carried out under different SNR. We can obtain that the application of RIS can enhance the asymptotic capacity for a parallel MIMO VLC system by comparing \"no-RIS\" with other baselines. In addition, we can know that the RPSO algorithm and the convex relaxation are effective methods to solve the assignment problems for the RIS-aided parallel MIMO VLC system. Though the complexity of the RPSO algorithm is reduced significantly compared with the exhaustive search algorithm, we think the convex relaxation is a more reasonable method to solve the problems. The capacity results calculated by the convex relaxation in Figure 3 are close to the optimal results while the computation complexity is much lower than the heuristic algorithm, and meanwhile, the conflict that the relaxed results may have interference in the primary constraints can be dealt with by choosing the solution closet to the relaxed results. However, the calculated results from the RPSO algorithm are more precise, and the complexity can be reduced significantly after determining the proper initial value or inertia weight. The evaluation of the algorithms will be our research direction in the future.