## 2 Multi-Fidelity Monte Carlo Method
### Data-Driven Low-Fidelity Models of Nonlinear Partial Differential Equations
#### 2.3.2 Operator Reduction  
Another strategy of constructing the low-fidelity models is operator reduction [22; 31]. We first construct a reduced space from trajectories of the full-model states, and the original operator is replaced by the reduced operator to construct the low-fidelity models of nonlinear partial differential equations. Let \(\mathbf{V}_{n}\in R^{N\times n}\) be a matrix composed of the first \(n\in N\) POD basis vectors. The projection state of the full-model state \(\mathbf{y}(t)\in R^{N}\) is \(\hat{\mathbf{y}}(t)=\mathbf{V}_{n}^{T}\mathbf{y}(t)\). Let \(\mathbf{Y}(t)=\left[\mathbf{y}_{1}\left(t\right)\),..., \(\mathbf{y}_{K}(t)\right]^{T}\in R^{K\times N}\) be the trajectory and the input matrix be \(\mathbf{U}(t)=\left[u\left(t\right),\ldots,\,u_{K}(t)\right]^{T}\in R^{K\times p}\). Consider the time step \(t_{1},\ldots,t_{K}\) with time step \(\delta t\). For \(j=1,\ldots,K\), let \(\hat{\mathbf{y}}_{j}\in R^{n}\) be the approximation of the derivative \(\frac{d}{dt}\hat{\mathbf{y}}(t_{j})\) of the projection state \(\hat{\mathbf{y}}(t_{j})\) at time \(t_{j}\) that converges to \(\frac{d}{dt}\hat{\mathbf{y}}(t_{j})\) under \(L^{2}\) norm when \(\delta t\to 0\). Suppose approximates the operators \(\mathbf{A}\in R^{N\times N}\), \(\mathbf{B}\in R^{N\times p}\) and \(\mathbf{F}\in R^{N\times s}\) by the reduced operators \(\hat{\mathbf{A}}\in R^{n\times n}\), \(\hat{\mathbf{B}}\in R^{n\times p}\) and \(\hat{\mathbf{F}}\in R^{n\times s}\). The reduction operators satisfy the equation:  
\[\dot{\hat{\mathbf{y}}}_{j}=\hat{\mathbf{A}}\hat{\mathbf{y}}_{j}+\hat{\mathbf{ F}}\hat{\mathbf{y}}_{j}+\hat{\mathbf{B}}\mathbf{u}_{j},j=1,\ldots,K, \tag{17}\]  
We define the reduced operators \(\hat{\mathbf{A}}\in R^{n\times n}\), \(\hat{\mathbf{B}}\in R^{n\times p}\) and \(\hat{\mathbf{F}}\in R^{n\times s}\) as the solution of the optimization problem.  
\[\underset{\hat{\mathbf{A}}\in R^{n\times n},\hat{\mathbf{B}}\in R^{n\times p},\hat{\mathbf{F}}\in R^{n\times s}=1}{\sum\limits_{j=1}^{K}}\|\hat{\hat{ \mathbf{y}}}_{j}-\hat{\mathbf{A}}\mathbf{y}_{j}-\hat{\mathbf{F}}\hat{\mathbf{ y}}_{j}-\hat{\mathbf{B}}\mathbf{u}_{j}\|_{2}^{2}. \tag{18}\]  
Let \(\mathbf{D}=[\hat{\mathbf{Y}},\mathbf{U},\hat{\mathbf{Y}}^{(1)},\ldots,\hat{ \mathbf{Y}}^{(n)}]\in R^{K\times(n+p+s)}\) be the data matrix, where \(\hat{\mathbf{Y}}\) is the projection trajectory, \(\mathbf{U}\) is the input matrix, we define the matrix by using the finite difference scheme \(\hat{\mathbf{y}}_{j}=\frac{\hat{\mathbf{y}}_{j}-\hat{\mathbf{y}}_{j-1}}{ \delta t},j=1,\ldots,K\),  
\[\mathbf{R}=[\dot{\hat{\mathbf{y}}}_{1},\ldots,\dot{\hat{\mathbf{y}}}_{K}]^{T} \in R^{K\times n}, \tag{19}\]  
The final optimization problem (18) is transformed into the least-square problem with the data matrix \(\mathbf{D}\) and the matrix \(\mathbf{O}\):  
\[\underset{\mathbf{O}\in R^{n\times(n+p+s)}}{\text{minimize}}\|\mathbf{D} \mathbf{O}^{T}-\mathbf{R}\|_{F}^{2}, \tag{20}\]  
The solution is \(\mathbf{O}=[\hat{\mathbf{A}},\hat{\mathbf{B}},\hat{\mathbf{F}}]\in R^{n\times(n +p+s)}\).