## 6 Results
### Multilevel Case  
In this section, we investigate an MLMC-M with more than two levels. In order to demonstrate some properties and limitations of our approach, we introduce the following three pairs of Monte Carlo methods:  
* 1 LMC and 1 LMC-M: standard MC of models on 18,397 mesh elements and its extension by meta-level;
* 2 LMC and 2 LMC-M: 2-level MLMC on models with 18,397 and 2714 mesh elements, and its extension by meta-level trained on the model of 2714 mesh elements;
* 3 LMC and 3 LMC-M: 3 level MLMC with models on 18,397, 2714, and 474 mesh elements and its extension by meta-level trained on the model of 474 mesh elements  
Figure 5 shows the computational costs of these Monte Carlo methods for a different number of estimated moments \(R\). We see that 1 LMC and 1 LMC-M have a much higher cost than methods with more levels. This is the crucial observation that demonstrates the limitations of the standard MC. Let us focus on the course of the costs depending on \(R\). In all the cases, the lowest cost was obtained for \(R=2\). However, as \(R\) increases, the behaviors of the methods differ. For 1 LMC, 1 LMC-M, 2 LMC, and 2 LMC-M, we observe a similar course that is steady for \(R\gtrapprox 15\). It means we can add more moments without affecting computational cost. On the other hand, for 3 LMC and 3 LMC-M, we observe a gradual increase in cost up to \(R=35\) and \(R=75\), respectively. Regarding computational cost savings, in cases of 1 LMC and 2 LMC, there is CS \(\approx 25\%\) utilizing 1 LMC-M and 2 LMC-M, whereas 3 LMC provides us with savings, from around CS \(\approx 25\%\) for \(R=2\) to just CS \(\approx 2\%\) for \(R=100\). As mentioned in Section 5.2, the computa  
Figure 4: Cost (blue) and moments error (red) of Monte Carlo methods on six different model mesh sizes. MC (square)—standard Monte Carlo method; MC-M (circle)—standard Monte Carlo using meta-model samples; MLMC-M (triangle)—MC extended by a meta-level. \(J(\hat{\mu},\bar{\mu})\) denote error between MC moments (\(\hat{\mu}\)) and MC-M or MLMC-M moments (\(\bar{\mu}\)). tional cost distribution across levels can be described using \(\beta\) and \(\gamma\) variables. Since for 1 LMC and 2 LMC, \(\beta>\gamma\), the dominant computational cost is on the lowest-accuracy levels. Therefore, adding the meta-level results in a much higher CS (for \(R>4\)) than for \(\beta<\gamma\), which is the case with 3 LMC. Thus, the behavior of our experiments corresponds to the theoretical properties of the MLMC. A closer look at the variances of moments across levels \(\hat{V}_{l}^{\tau}\) (Figure 6) provides a rationale for the claims already presented. For clarity, we display only the first five moments (\(R=5\)), which capture the behavior observed also in cases with more moments. To recall, we employ Legendre polynomials as \(\phi(x)\); therefore, \(\hat{V}_{l}^{1}=0\). The total computational cost (see Formula (8)) depends on the number of simulation samples \(N_{l}\) that are determined by \(\hat{V}_{l}^{MAX}=\max\{\hat{V}_{l}^{\tau}\}_{r=1}^{R}\) (see Section 4.1) and \(C_{l}\), which is also displayed in the figure. To meet \(V_{t}=10^{-5}\), the pictured variances are calculated based on the following numbers of samples: \([N_{1}=54,063,N_{2}=2550,N_{3}=1426]\) for 3 LMC and \([N_{1}=53,342,N_{2}=36,638,N_{3}=3038,N_{4}=1218]\) for 3 LMC-M.  
We can see that \(\hat{V}_{l}^{MAX}\) increases with increasing \(r\) for some \(l\), leading to the observed growth of \(C\) and \(C_{M}\) with \(R\) to the point where \(\hat{V}_{l}^{MAX}\) is stable. When it comes to different \(CS\), we need to focus on the course of \(\hat{V}_{l}^{\tau}\) across levels. We observe increasing \(\hat{V}_{2}^{MAX}\) of 3 LMC-M, whereas the corresponding \(\hat{V}_{1}^{MAX}\) of 3 LMC remains constant. This trend is accentuated for \(R>5\) and causes a decrease in \(CS\). The distinguishable increase in \(\hat{V}_{1}^{MAX}\) for 3 LMC-M has just a slight impact on \(C_{M}\) due to the minor \(C_{1}=0.338\) compared to \(C_{l}\), for \(l>1\).