## 3 Numerical Experiments Setup
### Benchmark Problems
#### 3.2.5 Coupled Spring-Mass-System  
Three masses sliding along a frictionless surface are attached to each other by four springs, as shown in Figure 6.  
Figure 4: Shifted-rotated Rastrigin function (from left to right, high-, medium-, and low-fidelity).  
Figure 5: Plot of one- (**left**) and two-dimensional (**right**) HF and LF ALOS functions with initial HF training points.  
The following constants, variables and assumptions are going to be used in the analysis:  
\begin{tabular}{l l}
**Spring Constants** & The springs obey [PERSON]'s law with constants \(k_{i}\), \(i=1,\ldots,4\) and the mass of each spring is negligible. \\
**Fixed Ends** & The first and last spring are attached to fixed walls and have the same spring constant. \\
**Mass Constants** & \(m_{1}\), \(m_{2}\), \(m_{3}\) are point masses. \\
**Position Variables** & The variables \(x_{1}(t)\), \(x_{2}(t)\), \(x_{3}(t)\) represent the mass positions measured from their equilibrium positions (negative is left and positive is right). \\ \end{tabular}  
The equations of motion for this setup are given by  
\[\begin{array}{rcl}m_{1}\ddot{x}_{1}(t)&=&-k_{1}x_{1}(t)+k_{2}[x_{2}(t)-x_{1 }(t)]\\ m_{2}\ddot{x}_{2}(t)&=&-k_{2}[x_{2}(t)-x_{1}(t)]+k_{3}[x_{3}(t)-x_{2}(t)]\\ m_{3}\ddot{x}_{3}(t)&=&-k_{3}[x_{3}(t)-x_{2}(t)]-k_{4}x_{3}(t)\end{array} \tag{28}\]  
which can be rewritten as  
\[M\ddot{\mathbf{x}}(t)=K\mathbf{x}(t) \tag{29}\]  
where the mass matrix \(M\), stiffness matrix \(K\) and the displacement \(\mathbf{x}\) are defined as  
\[M=\left(\begin{array}{ccc}m_{1}&0&0\\ 0&m_{2}&0\\ 0&0&m_{3}\end{array}\right)\quad K=\left(\begin{array}{ccc}-k_{1}-k_{2}&k_{ 2}&0\\ k_{2}&-k_{2}-k_{3}&k_{3}\\ 0&k_{3}&-k_{3}-k_{4}\end{array}\right)\quad\mathbf{x}=\left(\begin{array}{c} x_{1}\\ x_{2}\\ x_{3}\end{array}\right)\]  
This is a constant-coefficient homogeneous system of second-order ODEs, the solution of which is given by  
\[\mathbf{x}(t)=\sum_{i=1}^{3}[a_{i}\cos(\omega_{i}t)+b_{i}\sin(\omega_{i}t)] \mathbf{v}_{i} \tag{30}\]  
where \(\omega_{i}=\sqrt{-\lambda_{i}}\) and \(\lambda_{i}\) are the eigenvalues of the matrix \(M^{-1}K\) and \(\mathbf{v}_{i}\) are the corresponding eigenvectors. The constants \(a_{i}\) and \(b_{i}\) are determined by the initial conditions \(\mathbf{x}(t=0)=\mathbf{x}_{0}\)Since the initial velocity is zero, we have \(b_{1}=b_{2}=0\) and using \(\mathbf{x}(t=0)=\mathbf{x}_{0}\) in Equation (30) yields \(a_{1}=a_{2}=0.5\) such that  
\[x_{1}(t)=0.5\cos(t)+0.5\cos(\sqrt{3}t)\quad\text{and}\quad x_{2}(t)=0.5\cos(t)-0.5 \cos(\sqrt{3}t) \tag{31}\]  
These solutions are plotted in Figure 7 for \(0\leq t\leq 30\).  
Converting Equation (29) into a system of first-order ODEs and using the fourth-order accurate Runge-Kutta time-marching method yields a multi-fidelity analysis problem by varying the time-step size \(\Delta t\) as shown in Figure 8. Note that the high-fidelity (HF) analysis with \(\Delta t=0.01\) is not distinguishable from the analytical solution given by Equation (31) and shown in Figure 7, while the low-fidelity (LF) analysis with \(\Delta t=0.6\) is exhibiting the correct trends but is somewhat inaccurate.  
Treating two spring constants as independent input variables with \(1\leq k_{1}=k_{3},\ k_{2}\leq 4\) while \(m_{1}=m_{2}=1\) and computing \(x_{1}(t=6)\) yields the two-dimensional design space shown in Figure 9. One can infer that the lower fidelity trends match the high-fidelity ones.  
In this work, a two-dimensional (by varying the two spring constants) and four-dimensional (varying two springs and two masses) version of this benchmark problem will be employed. However, other benchmark problems based on this problem for uncertainty quantification and optimization can be easily conceived [35].  
Figure 8: Numerical solution for \(x_{1}(t)\) of the example problem for \(0\leq t\leq 6\) using \(\Delta t=0.01\) labeled as high-fidelity (HF) and \(\Delta t=0.6\) labeled as low-fidelity (LF).  
Figure 7: Analytical solution for the spring-mass problem for \(0\leq t\leq 30\).