# Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
## 3 Basic Discretisation
### Eigenproblem and Modal Coordinate Reduction  
Systems such as (16) or (19) come with large sparse matrices, and thus their repeated solution consequently leads to high costs in terms of computation time. In order to approach this issue, one of the natural ideas is to reduce the order of the matrices from \(N\), such as in Equation (16) or \(2N\) such as Equation (19), to a smaller order \(r\). The Modal Order Reduction(MOR) technique will offer us a suitable framework for performing this in a suitable way. With this technique, a low-dimensional system will supplant the current high-dimensional one. We recommend the reader to [19; 20] for more details on the MOR topic.  
The MOR technique itself includes a large variety of methods. Specifically, in our framework, we want to focus on the Modal Coordinate Reduction (MCR) technique. In [9], an approach was presented, which we will briefly recall now.  
The key step is the calculation of the eigenvalues \(\lambda\) and eigenvectors \(\mathbf{v}\) of the LBO \(L\mathbf{v}=\lambda\mathbf{v}\). With \(L=D^{-1}W\), we may reformulate this into the generalised eigenvalue problem  
\[W\mathbf{v}=\lambda D\mathbf{v} \tag{23}\]  
The symmetry of \(W\) and the diagonal matrix \(D\) will lead to real eigenvalues and linear independent eigenvectors. Additionally, we determine that the eigenvectors \(\mathbf{v}\) are \(D\)-orthogonal with \(\mathbf{v}_{i}^{\top}D\mathbf{v}_{j}=\delta_{ij}\). These considerations will lead to the following equations:  
\[I=V^{\top}DV,\quad L=V\Lambda V^{\top}D,\quad\Lambda=V^{\top}WV \tag{24}\]  
with \(V\) being the right eigenvector matrix of \(L\), and \(\Lambda\) being the diagonal matrix of eigenvalues. Multiplying the second equation with \(V\) from the right will give the basic property \(LV=V\Lambda\), making use of the first identify in (24). The third equation can be obtained via the multiplication of \(V^{\top}D\) from the left on \(LV=D^{-1}WV=V\Lambda\).  
To finally achieve the MCR transformation, we use these considerations and substitute Equation (16) with \(\mathbf{u}=V\mathbf{w}\), leading to  
\[V\ddot{\mathbf{w}}+kV\dot{\mathbf{w}}=LV\mathbf{w} \tag{25}\]  
With the left multiplication of \(V^{\top}D\), we obtain  
\[\ddot{\mathbf{w}}+k\dot{\mathbf{w}}=\Lambda\mathbf{w} \tag{26}\]  
Let us mention that, at this point, we still have a high-dimensional system since we consider all eigenvalues. The reduction happens when we use the first \(r\ll N\)-ordered eigenvalues \(0=\lambda_{1}<\lambda_{2}\leq\ldots\leq\lambda_{r}\). The reduced model writes as  
\[\ddot{\mathbf{w}}_{r}+k\dot{\mathbf{w}}_{r}=\Lambda_{r}\mathbf{w}_{r}\quad \text{where}\quad\mathbf{w}_{r}=V_{r}^{\top}D\mathbf{u} \tag{27}\]  
Analogously to the steps from (16) to (19), we produce from (27) the equation  
\[\dot{\mathbf{p}}_{r}(t)=H_{r}\mathbf{p}_{r}(t)\quad\text{with}\quad H_{r}= \begin{pmatrix}0&I_{r}\\ \Lambda_{r}&-kI_{r}\end{pmatrix}\in\mathbb{R}^{2r\times 2r},\quad\mathbf{p}_{r}=( \mathbf{w}_{r},\dot{\mathbf{w}}_{r})^{\top}\in\mathbb{R}^{2r} \tag{28}\]  
The last step remaining is the transformation of the initial conditions into our reduced system. Beginning with \(\mathbf{u}^{\delta}(x_{i},0)=(0,\ldots,0,|\Omega_{i}|^{-1},0,\ldots,0)^{\top}\), we obtain with \(\mathbf{w}_{r}^{\delta}=V_{r}^{\top}D\mathbf{u}^{\delta}\) the reduced condition \(\mathbf{w}_{r}^{\delta}(x_{i},0)=V_{r}^{\top}\mathbf{e}_{i}\). For the Gaussian initial condition, we receive with \(\mathbf{w}_{r}^{G}=V_{r}^{\top}D\mathbf{u}^{G}\)  
\[\mathbf{w}_{r}^{G}(x_{i},0)=V_{r}^{\top}\Big{(}u^{G}(x_{1}),\ldots,u^{G}(x_{i} ),\ldots,u^{G}(x_{N})\Big{)}^{\top} \tag{29}\]  
The initial velocity \(\dot{\mathbf{w}}_{r}(0)=\mathbf{0}\) will not change, since it was already zero.