## 2 RBF Mesh Deformation with Greedy Algorithm
### Greedy Selection  
The large-scale mesh contains huge quantity of boundary points and it results in a long time consumption. In fact, part of the boundary points have essential impact on the displacement of internal points. In other words, redundant points bring the unnecessary time cost. To solve this problem, [PERSON] proposed the greedy algorithm in RBF mesh deformation ([PERSON] et al., 2013), and it is presented as the following:  
\[\boldsymbol{\alpha}=\Phi_{c,c}^{-1}\Delta\mathbf{x}_{c}, \tag{8}\]  
\[\Delta\mathbf{x}_{in}=\Phi_{in,c}\boldsymbol{\alpha}, \tag{9}\]  
where \(\Delta\mathbf{x}_{c}\) is the actual displacement of the boundary point, \(\Phi_{c,c}\) is an \(N_{c}\times N_{c}\) matrix, \(N_{c}\) is the total quantity of control points and \(\Phi_{in,c}\) is an \(N_{in}\times N_{c}\) matrix.  
The greedy selection is the process of selecting the proper control points. Its purpose is to pick out the representative points that affects the estimated displacements most. The detailed process is as follows:  
1. Choose an arbitrary point from boundary points to initialize the set of control points;
2. Solve the predefined equation coefficient of control points by Equation (8);
3. Obtain the estimated displacements of the moving boundary points: \[\Delta\mathbf{x}_{b}^{*}=\Phi_{b,c}\boldsymbol{\alpha},\] (10) where \(\Phi_{b,c}\) is an \(N_{b}\times N_{c}\) matrix and \(\Delta\mathbf{x}_{b}^{*}\) is the estimated displacement of boundary points;
4. Obtain the boundary errors of all the unselected boundary points: \[\boldsymbol{\varepsilon}=\Delta\mathbf{x}_{b}^{*}-\Delta\mathbf{x}_{b},\] (11) where \(\boldsymbol{\varepsilon}\) is the boundary errors;  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{2}{c}{**Global**} & \multicolumn{2}{c}{**Compact**} \\ \hline
**Name** & \(\boldsymbol{f}(\boldsymbol{x})\) & **Name** & \(\boldsymbol{f}(\boldsymbol{\theta})\) \\ \hline Thin plate spline (TPS) & \(x^{2}\log(x)\) & CP C\({}^{0}\) & \((1-\theta)^{2}\) \\ Multi-quadric bi-harmonics (MQB) & \(\sqrt{a^{2}+x^{2}}\) & CP C\({}^{2}\) & \((1-\theta)^{4}(4\theta+1)\) \\ Inverse multi-quadric bi-harmonics (IMQB) & \(\sqrt{\frac{1}{a^{2}+x^{2}}}\) & CP C\({}^{6}\) & \((1-\theta)^{8}(32\theta^{3}+25\theta^{2}+8\theta+1)\) \\ Quadric bi-harmonics (QB) & \(1+x^{2}\) & CTPS C\({}^{0}\) & \((1-\theta)^{5}\) \\ Gaussian & \(e^{-x^{2}}\) & Compact TPS & \(\theta^{2}\log(\theta)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Radial basis function (\(\theta=\frac{x}{r}\), where \(x\) is the Euclidean norm and \(r\) is the support radius).  
Compare the largest boundary error with the predefined criterion threshold, and figure out whether it is larger than threshold or not: \[\frac{\|\varepsilon_{max}\|_{2}}{\|\Delta\mathbf{x}_{b}\|_{2}}>\xi_{tol},\] (12) where \(\|\mathbf{x}\|_{2}\) is the Euclidean norm of \(\mathbf{x}\) and \(\xi_{tol}\) is the artificially setting error tolerance. If yes, put the point which have the largest boundary error into the set of control points;
6. If not, end the selection;
7. Repeat from Step 2 to Step 6.  
Greedy selection effectively reduces the quantity and redundancy of operating-attended boundary points compared with the original RBF mesh deformation. However, this algorithm generates additional time cost for the RBF mesh deformation process. The extra time is mainly spent on the loop of points selection, and the cost will increase exponentially with the growth of control points quantity [17]. To accelerate the this procedure, block iteration method and parallel computing method are proposed respectively in Section 3.