## 3 Model Problems and Numerical Results
### Numerical Results for Model Problem 2
#### 3.4.5 Level Adaptivity  
In order to better illustrate the level adaptivity of Algorithm 1, we have rerun Algorithm 2 to generate more meshes that can be used in our p-MLQMC method. The details of the hierarchy can be found in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**p-ML(Q)MC**} \\ \hline
**Level** & **Nel** & **DOF** & **Order** & **Nquad** \\ \hline
0 & 33 & 48 & 1 & 7 \\
1 & 33 & 160 & 2 & 13 \\
2 & 33 & 338 & 3 & 19 \\
3 & 33 & 582 & 4 & 25 \\
4 & 33 & 892 & 5 & 28 \\
5 & 33 & 1268 & 6 & 33 \\
6 & 33 & 1720 & 7 & 37 \\
7 & 33 & 2218 & 8 & 61 \\
8 & 33 & 2792 & 9 & 73 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Number of elements, degrees of freedom, element order and number of quadrature points for model problem 2 for additional tolerances.  
Figure 27: Uncertainty propagation towards the QoI’s for model problem 2.  
Figure 28: Runtimes for the different method in function of requested user tolerance for model problem 2.  
the location of the points where the discrete values of the random field are to be computed. The second algorithm was developed for two dimensional problems but is extensible to three dimensional problems. We observed that the p-MLQMC method drastically reduces the computational cost with respect to a classical Multilevel Monte Carlo approach. We observed computational gains ranging from a factor 20 to 100. In turn, h-MLMC is 4 to 6 times faster than standard MC. In addition to this cost reduction, we also showed that the computational cost increase in function of a user requested tolerance \(\epsilon\), is proportional to \(\epsilon^{-1}\). For model problem 2, we have shown that the decay of \(\mathbb{E}\left[\Delta P_{\ell}\right]\) for h-ML(Q)MC is much slower than for p-ML(Q)MC. The opposite holds for the decay of \(\mathbb{V}\left[\Delta P_{\ell}\right]\). This behavior could be exploited in a Multi-Index (Quasi)-Monte Carlo setting in order to further reduce the total computational cost, see [11]. Higher order Quasi-Monte Carlo methods, which use higher order digital nets instead of lattice rules, could be used in order to further decrease the error on the root mean square, and so decrease the computational cost.  
Conceptualization, P.B., P.R. and S.V.; methodology, P.B., P.R., C.V.h. and S.F.; software, P.B., P.R., C.V.h. and S.F.; validation, P.B., P.R., G.L. and S.V.; formal analysis, P.B.; investigation, P.B. and P.R.; resources, P.R., C.V.h., S.F., G.L. and S.V.; data curation, P.B., P.R. and S.F.; writing-original draft preparation, P.B., P.R. and S.V.; writing-review and editing, P.B., P.R., C.V.h., S.F., G.L. and S.V.; visualization, P.B.; supervision, G.L. and S.V.; project administration, G.L. and S.V.; funding acquisition, S.V. All authors have read and agree to the published version of the manuscript.  
The authors gratefully acknowledge the support from the Research Council of KU Leuven through project C16/17/008 \"Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints\".  
The authors would also like to thank the Structural Mechanics Section of the KU Leuven and [PERSON] for supplying their StaBL code and for providing support. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI. The stochastic part of our simulations was performed with the Julia packages **MultilevelEstimators.jl** and **GaussianRandomFields.jl**.  
The authors declare no conflict of interest.