# A Cost-Efficient Approach towards Computational Fluid Dynamics Simulations on Quantum Devices
## 3 Preliminaries
### Reducing Model Complexity
#### 3.3.2 Reduced-Order Linearized Model  
The motivation behind trajectory piecewise-linear techniques is to linearize nonlinear systems around suitably selected states and to approximate the system by combining a set of linearized reduced-order models via a weighting procedure. Let \((x_{0},u_{0}),\ldots,(x_{5-1},u_{5-1})\) be states obtained from previous simulations of the FOM on some finite time interval \([t_{start},t_{end}]\). Efficient linearized models can be extracted from the system by using a Taylor expansion of the nonlinearity around the states \((x_{i},u_{i})\)  
\[ \begin{split} f(x,u)=f(x_{i},u_{i})&+A_{i}(x-x_{i })+B_{i}(u-u_{i})\\ &+\frac{1}{2}W_{i}(x-x_{i})\otimes(x-x_{i})+\ldots,\end{split}  \tag{35}\]  
where \(\otimes\) is the Kronecker product and \(A_{i}\) and \(W_{i}\) are the Jacobian and the Hessian of \(f(\cdot)\) evaluated at the state \((x_{i},u_{i})\). The matrix \(B_{i}=\frac{\partial f}{\partial u}\) is also evaluated at \((x_{i},u_{i})\). By removing the quadratic and higher terms, the nonlinear system can be linearized around any given state to yield a state-space model of the form:  
\[ \begin{split}\dot{x}(t)&=A_{i}x(t)+B_{i}u(t)+[f(x_ {i},u_{i})-A_{i}x_{i}(t)-B_{i}u_{i}]\\ y(t)&=C_{i}x(t),\end{split}  \tag{36}\]  
where \(C_{i}=\frac{\partial t}{\partial x}\) is also evaluated at \((x_{i},u_{i})\). The vector of unknowns \(x(t)\) can be written as  
\[x(t)=x_{i}+x_{i}^{\prime}(t), \tag{37}\]  
where \(x_{i}\), fixed in time, is the value of state vector \(x\) at the linearization point \(i\), and \(x_{i}^{\prime}(t)\) contains the perturbation of the \(n\) unknown flow quantities about that linearization point \(x_{i}\). The linearized equation can then be expressed as  
\[ \begin{split}\dot{x}_{i}^{\prime}(t)&=A_{i}x_{i}^{ \prime}(t)+B_{1i}u(t)+B_{2i}\\ y(t)&=C_{i}x_{i}^{\prime}(t)+C_{0i},\end{split}  \tag{38}\]  
where \(B_{2i}=f(x_{i},u_{i})-B_{i}u_{i}\) and \(C_{0i}=C_{i}x_{i}\). Since each of the linearizations approximates the nonlinear system around the expansion point \(x_{i}\), a model including all of these linearizations could approximate the original system over a larger time interval and a larger part of the state-space. Plotted in Figure 3 are four linearization points, \(x_{0}\), \(x_{1}\), \(x_{2}\) and \(x_{3}\), along a \"training trajectory\" A, which is obtained through a simulation of the original nonlinear system. The goal is to capture the most relevant dynamics of the system through a series of simulations in the offline training phase with inputs that reflect the dynamics of interest for the given application. Each linearized model has its own range of validity (denoted by the blue circles) specifying the range where approximation by linearization can be considered as accurate. Exploiting quantum algorithms for their exponential speedup in the offline training part could prove to be the most beneficial, as it is a computationally expensive part of the TPWL. Using piecewise-linearized models for simulations might already be faster than simulating the original system. However, the linearized models can be further reduced by using previously described MOR techniques. One additional advantage of linearizing the system first is that it allows for the possibility of applying both linear and nonlinear MOR methods to the linear submodels. As a few examples for linear MOR methods, in [48], [PERSON] proposes the usage of Krylov-based reduction using the Arnoldi method. In [52] [PERSON], [PERSON] and [PERSON] introduce balanced truncation to TPWL, and [PERSON] [53] uses Poor Man's TBR as a linear MOR kernel. In this paper, we will continue the work described in Section 3.3.1 and use proper orthogonal decomposition (POD) for the reduction step. A reduced-order linearized model can be obtained by applying the projection defined before in Equation (31) to the system in Equation (38), yielding  
\[