# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 3 Preliminaries
### Reducing Model Complexity
#### 3.3.1 Proper Orthogonal Decomposition  
The POD method generates a set of reduced basis functions from the solutions that are computed or measured in advance. The goal is to obtain an optimal low-dimensional basis for representing high-dimensional experimental or simulation data. This low-dimensional basis can be used to formulate ROMs of models with a high complexity. The main assumption in the snapshot-based MOR techniques is that an approximation to the solution of the problem can be expressed as a linear combination of spatial modes. Let \(u(x,t)\) denote a given flow field and \(u_{r}(x,t)\) denote its approximation in the reduced space. The POD method decomposes \(u(x,t)\) into an orthonormal system of spatial modes \(\varphi_{i}(x)\) with temporal coefficients \(a_{i}(t)\) as follows:  
\[u(x,t)\approx u_{r}(x,t)=\sum_{i=1}^{N_{r}}a_{i}(t)\varphi_{i}(x). \tag{21}\]  
The spatial modes have to be carefully selected, as a correct choice of basis functions could lead to an efficient ROM with a reduced simulation time and high accuracy in the online simulation phase. The POD is capable of selecting the spatial modes with the highest energies; thus, the method is usually a good choice for fluid flow applications. The POD basis  
\[X_{N_{r}}^{POD}=span\{\varphi_{i}\},\quad i=1,\ldots,N_{r}, \tag{22}\]  
can be built from a set of snapshots of velocity solutions  
\[u_{n}(x)=u(x,t_{n}),\quad n=1,\ldots,N_{S}, \tag{23}\]  
where \(N_{r}\) denotes the number of ROM functions and \(N_{S}\) denotes the number of snapshots taken from the obtained solutions. By definition, the POD basis calculation can be thought of as the least square method, which minimizes the difference between the snapshots and the projection of the snapshots on the spatial modes in the X-norm. If the \(L^{2}\)-norm is chosen, the POD basis is optimal, considering the energy contained in the snapshots:  
\[ \begin{split} X_{N_{r}}^{POD}=\arg\min\frac{1}{N_{S}}\sum_{n=1}^ {N_{S}}||u_{n}(x)-\sum_{n=1}^{N_{r}}\left\langle u_{n}(x),\varphi(x)\right\rangle _{L^{2}}\varphi(x)||_{L^{2}}^{2},\\ \left\langle\varphi_{i}(x),\varphi_{j}(x)\right\rangle_{L^{2}}= \delta_{i,j}.\end{split}  \tag{24}\]  
In order to solve Equation (24), the following eigenvalue problem is considered:  
\[C\xi=\lambda_{i}\xi,\quad i=1,\ldots,N_{S}, \tag{25}\]  
where \(C\in\mathbb{R}^{N_{S}\times N_{S}}\) is the correlation matrix, whose components are calculated as follows:\[\left[C\right]_{kl}=\frac{1}{N_{r}}\left\langle u_{k}(x),u_{l}(x)\right\rangle_{L^{ 2}}. \tag{26}\]  
The \(\left(\lambda_{i},\xi_{i}\right)\) eigenvalue-eigenvector pair is used to construct the POD basis functions  
\[\varphi_{i}(x)=\frac{1}{\sqrt{\lambda_{i}}}\sum_{n=1}^{N_{b}}\xi_{i,n}u_{n}(x),\quad i=1,\ldots,N_{r}. \tag{27}\]  
Since the eigenvalues are sorted into descending order, the first modes have the property of retaining most of the energy present in the original solutions. The obtained basis functions are orthogonal and can be normalized in order to obtain  
\[\left\langle\varphi_{i}(x),\varphi_{j}(x)\right\rangle_{L^{2}}=\delta_{ij}, \tag{28}\]  
where \(\delta_{ij}\) is the Kronecker delta function.