# Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method
## 3 Weighted High-Order Gradient Reproducing Kernel Collocation Method
### Weighted Discrete Collocation Equations by HG-RK Approximation  
According to the schematic diagram in Figure 1, the following strong form by using high-order gradient reproducing kernel approximation is considered:  
\[L^{1}u_{,xx}+L^{2}u_{,yy}=m\text{ in }\Omega\cup\partial\Omega_{2}\cup \partial\Omega_{4} \tag{15}\]  
\[B_{g}u=g\text{ on }\partial\Omega_{1}\cup\partial\Omega_{3} \tag{16}\]  
\[B_{h}^{1}u_{,x}+B_{h}^{2}u_{,y}=h\text{ on }\partial\Omega_{1} \tag{17}\]  
where \(L^{1}\) and \(L^{2}\) denote the differential operators in the domain domain \(\Omega\cup\partial\Omega_{2}\cup\partial\Omega_{4}\); \(B_{h}^{1}\)Based on Equations (15)-(17), a weighted least-squares functional is established with the following expression:  
\[ \begin{array}{l}E(u)=\frac{1}{2}\int_{\Omega\cup\partial\Omega_{2}\cup\partial \Omega_{4}}\big{(}L^{1}u_{,xx}+L^{2}u_{,yy}-m\big{)}^{\mathsf{T}}\big{(}L^{1}u_{,xx}+L^{2}u_{,yy}-m\big{)}d\Omega\\ +\frac{\alpha_{h}}{2}\int_{\partial\Omega_{1}}\big{(}B_{h}^{1}u_{,x}+B_{h}^{2} u_{,y}-h\big{)}^{\mathsf{T}}\big{(}B_{h}^{1}u_{,x}+B_{h}^{2}u_{,y}-h\big{)}d \Omega_{1}\\ +\frac{\alpha_{g}}{2}\int_{\partial\Omega_{1}\cup\partial\Omega_{3}}\big{(}B_ {g}u-g\big{)}^{\mathsf{T}}\big{(}B_{g}u-g\big{)}d\Omega_{1}\end{array}  \tag{18}\]  
where these three terms correspond to least-squares residuals in the domain and on the boundary. To balance the errors of approximation in the domain and on the boundary, the weights \(\alpha_{h}\) and \(\alpha_{g}\) are further introduced. In the collocation method, the collocation points can be classified into different groups according to their locations in the problem domain. In this way, the integral form of the functional can be evaluated equivalently by treating the collocation points as the integration points in the discrete form. Then, introducing the RK approximation together with HG-RK approximation to this discrete form and using the stationary condition of the functional lead to the following weighted discrete collocation system:  
\[\left( \begin{array}{c}\mathbf{A}^{1}\\ \sqrt{\alpha_{h}}\mathbf{A}^{2}\\ \sqrt{\alpha_{g}}\mathbf{A}^{3}\end{array} \right)\mathbf{u}=\left( \begin{array} []{c}\mathbf{b}^{1}\\ \sqrt{\alpha_{h}}\mathbf{b}^{2}\\ \sqrt{\alpha_{g}}\mathbf{b}^{3}\end{array} \right) \tag{19}\]  
where the sub-matrices are given by  
\[ \begin{array}{l}\mathbf{A}^{1}=\left(\begin{array}{c}L^{1}\big{(}\mathbf{ \Psi}^{xx^{\mathsf{T}}}\big{(}\mathbf{x}_{p1}\big{)}\big{)}+L^{2}\big{(} \mathbf{\Psi}^{yy^{\mathsf{T}}}\big{(}\mathbf{x}_{p1}\big{)}\big{)}\\ L^{1}\big{(}\mathbf{\Psi}^{xx^{\mathsf{T}}}\big{(}\mathbf{x}_{p2}\big{)}\big{)} +L^{2}\big{(}\mathbf{\Psi}^{yy^{\mathsf{T}}}\big{(}\mathbf{x}_{p2}\big{)} \big{)}\\ \vdots\\ L^{1}\big{(}\mathbf{\Psi}^{xx^{\mathsf{T}}}\big{(}\mathbf{x}_{pN_{p}}\big{)} \big{)}+L^{2}\big{(}\mathbf{\Psi}^{yy^{\mathsf{T}}}\big{(}\mathbf{x}_{pN_{p}} \big{)}\big{)}\end{array}\right)\end{array}  \tag{20}\]  
\[ \begin{array}{l}\mathbf{A}^{3}=\left(\begin{array}{c}B_{g}\big{(}\mathbf{ \Psi}^{\mathsf{T}}(\mathbf{x}_{1})\big{)}\\ B_{g}\big{(}\mathbf{\Psi}^{\mathsf{T}}(\mathbf{x}_{2})\big{)}\\ \vdots\\ B_{g}\big{(}\mathbf{\Psi}^{\mathsf{T}}(\mathbf{x}_{N_{r}})\big{)}\end{array}\right) \end{array}  \tag{22}\]  
\[ \begin{array}{l}\mathbf{b}^{1}=\left(\begin{array}{c}m\Big{(}\mathbf{x}_{p1 }\Big{)}\\ m\Big{(}\mathbf{x}_{p2}\Big{)}\\ \vdots\\ m\Big{(}\mathbf{x}_{pN_{p}}\Big{)}\end{array}\right)\end{array} \mathbf{b}^{2}= \left( \begin{array}{c}h\Big{(}\mathbf{x}_{q1}\Big{)}\\ h\Big{(}\mathbf{x}_{q2}\Big{)}\\ \vdots\\ h\Big{(}\mathbf{x}_{qN_{q}}\Big{)}\end{array} \right)\ \mathbf{b}^{3}=\left( \begin{array}{c}g( \mathbf{x}_{r1})\\ g(\mathbf{x}_{r2})\\ \vdots\\ g(\mathbf{x}_{rN_{r}})\end{array} \right)\end{array} \tag{23}\]  
In Equation (19), \(\mathbf{u}\) is the vector containing generalized coefficients in both RK and HG-RK approximations, i.e., \(\mathbf{u}=[a_{1},a_{2},\ldotsldotsldots,a_{N_{s}}]^{\mathsf{T}}\).