## 4 Numerical Examples and Validation of the Software
### Comparison to Other Methods  
This section is devoted to the comparison of the augmented Lagrangian method, as described in this paper, with the classical augmented Lagrangian regime as described in [20; 22] and the Primal-Dual Active-Set Strategy, as described in [23].  
3.1 Theoretical Setup for the Classical Augmented Lagrangian Scheme and the Primal-Dual Active-Set Strategy  
This section is devoted to description of the theoretical setup for the application of the classical augmented Lagrangian scheme [20; 22] and the classical Primal-Dual Active-Set Strategy, c.f. [23].  
The classical approaches to treat (4) is to discretize the optimization problem (4) and then use an optimization algorithm for the solution, as discussed in [14; 15; 16; 19]. For an arbitrary fixed triangulation \(\mathcal{T}\) with diameter \(0<h\) and for the usage of the discretization \(X_{h}\) of \(X\), as defined in Section 3, one obtains the following discrete optimization problem:  
\[\text{Find }x_{h}=\left(c_{S,h},p_{S,h}\right)_{S\in\left\{A,B,C\right\}} \in X_{h}\text{ such that the following equality holds true} \tag{33}\] \[x_{h}=\underset{\begin{pmatrix}(p_{S,h},q_{S,h})\end{pmatrix} \text{argmin}}{\text{argmin}} LS\big{(}\big{(}v_{S,h},q_{S,h}\big{)}\big{)}\] \[\text{s.t.} 0<v_{S,h}\qquad\forall S\in A,B,C\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter Set** & \
\begin{tabular}{c} **Type of** \\ **Refinement** \\ \end{tabular} & **Geomety** &
\begin{tabular}{c} **Number of** \\ **Iterations** \\ \end{tabular} & **CPU-Time** \\ \hline Table 1 & Square & Uniform & 280 & 730.5068 s \\ Table 1 & Square & Adaptive & 166 & 615.8496 s \\ Table 1 & L-Shaped & Uniform & 53,639 & 3.1204\(\pm\) 05 s \\ Table 1 & L-Shaped & Adaptive & 188 & 662.2452 s \\ Table 2 & Square & Uniform & 287 & 881.2332 s \\ Table 2 & Square & Adaptive & 181 & 645.7126 s \\ Table 2 & L-Shaped & Uniform & 53,553 & 3.1554\(\pm\) 05 s \\ Table 2 & L-Shaped & Adaptive & 283 & 742.0187 s \\ \hline \hline \end{tabular}
\end{table}
Table 5: An overview of number of iterations of the augmented Lagrangian Algorithm 1 and CPU running times for the different parameter sets and uniform resp. adaptive mesh refinement.  
The classical approaches use the fact that \(X_{h}\) is a finite dimensional subspace of \(X\) and, thus, (33) is an finite dimensional optimization problem, for which a lot of theory is known, i.e., for the approximation of solutions (33), the augmented Lagrangian schemes can directly be applied, as described in [20; 22].  
In contrast to augmented Lagrangian methods, the Primal-Dual Active-Set Strategy, as in the form described in [23], cannot be indirectly applied to (33). For the application of Primal-Dual Active-Set Strategies, in this paper it is assumed that the operator in the ODE, resp. PDE, is linear, (2a)-(2e), is linear, i.e., the equality \(k_{1}=0\) is assumed. In this case, one obtains, by using a basis of \(X_{h}\), and using the Euler Lagrange Equation (20), one obtains a minimization problem of the following form:  
Find \(x_{h}^{v}\in\mathbb{R}^{d}\), such that the following equation holds true  
\[\begin{split} x_{h}^{v}=\operatorname*{argmin}_{y\in\mathbb{R}^{ d}}&\frac{1}{2}y^{T}Ax_{h}^{v}-y^{T}f\\ \textbf{s.t.:}& 0\leq v_{S,h}^{v}\end{split} \tag{34}\]  
In the problem above, \(d_{h}\in\mathbb{N}\) is the dimension of \(X_{h}\), i.e., \(d_{h}=\dim X_{h}\), \(x_{h}^{v}\in\mathbb{R}^{d_{h}}\) is the coefficient vector of the solution \(x_{h}\in X_{h}\) of (33) and \(v_{S,h}^{v}\) is the coefficient vector of \(v_{h}\in S^{1}(\mathcal{T})\). Furthermore, for all \(y\in\mathbb{R}^{d_{h}}\), the equality  
\[y^{T}Ay+2y^{T}f+f^{T}f=LS\big{(}y_{h}\big{)}\]  
holds true, where \(y\in\mathbb{R}^{d}\) is the coefficient vector of \(y_{h}\in X_{h}\).